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List of equivalences

2014-11-26T13:32:36Z

Emmanuel Beffara: /* Multiplicatives */ A is equivalent too

Each [[List of isomorphisms|isomorphism]] gives an equivalence of formulas.
The following equivalences are ''not'' isomorphisms.

== Multiplicatives ==

<math>
\begin{array}{rcccl}
A &\linequiv& A \tens (A\orth\parr A) &\linequiv& (A\tens A\orth)\parr A
\end{array}
</math>

== Additives ==

<math>
\begin{array}{rclcrcl}
A \with A &\linequiv& A \\
A \plus A &\linequiv& A \\
\\
A \with (A \plus B) &\linequiv& A &\quad& A \plus \top &\linequiv& \top \\
A \plus (A \with B) &\linequiv& A &\quad& A \with \zero &\linequiv& \zero
\end{array}
</math>

== Quantifiers ==

<math>
\begin{array}{rcll}
\forall X.A &\linequiv& A &\quad (X\notin A) \\
\exists X.A &\linequiv& A &\quad (X\notin A)
\end{array}
</math>

== Exponentials ==

<math>
\begin{array}{rclcrcl}
\oc A &\linequiv& \oc A\tens\oc A &\quad&
\wn A &\linequiv& \wn A\parr\wn A\\
\oc A &\linequiv& \oc\oc A &\quad& \wn A &\linequiv& \wn\wn A\\
\oc\wn A &\linequiv& \oc\wn\oc\wn A &\quad& \wn\oc A &\linequiv& \wn\oc\wn\oc A\\
\end{array}
</math>

Some of these equivalences are related with the [[lattice of exponential modalities]].

== Polarities ==

{|
| <math> \wn N \linequiv N </math>
|   (N [[Negative formula|negative]])
|-
| <math> \oc P \linequiv P </math>
|   (P [[Positive formula|positive]])
|-
| <math> \wn\oc R \linequiv R </math>
|   (R [[Regular formula|regular]])
|-
| <math> \oc\wn L \linequiv L </math>
|   (L [[Co-regular formula|co-regular]])
|}

== Second order encodings ==

<math>
\begin{array}{rclcrcl}
A &\linequiv &\forall X . (A \tens X\orth) \parr X \\
A &\linequiv &\exists X . (A \parr X\orth) \tens X \\
\\
A \with B &\linequiv& \exists X . \oc{(A \parr X\orth)} \tens \oc{(B \parr X\orth)} \tens X &\quad& \top &\linequiv& \exists X . X \\
A \plus B &\linequiv& \forall X . \wn{(A \tens X\orth)} \parr \wn{(B \tens X\orth)} \parr X &\quad& \zero &\linequiv& \forall X . X \\
\\
\bot &\linequiv& \exists X . X\tens X\orth \\
\one &\linequiv& \forall X . X\orth\parr X \\
\\
\forall \xi . A &\linequiv& \exists X . (\forall \xi . (A \parr X\orth)) \tens X \\
\exists \xi . A &\linequiv& \forall X . (\exists \xi . (A \tens X\orth)) \parr X
\end{array}
</math>

== Miscellaneous ==

<math>
\begin{array}{rcl}
\oc{\wn{(\oc{A}\with\oc{B})}} &\linequiv& \oc{(\wn{\oc{A}}\with\wn{\oc{B}})} \\
\wn{\oc{(\wn{A}\plus\wn{B})}} &\linequiv& \wn{(\oc{\wn{A}}\plus\oc{\wn{B}})}
\end{array}
</math>

Reversibility and focalization

2012-08-31T21:50:29Z

Emmanuel Beffara: /* Focalization */ improved terminology

== Reversibility ==

{{Theorem|
Negative connectives are reversible:

* A sequent <math>\vdash\Gamma,A\parr B</math> is provable if and only if <math>\vdash\Gamma,A,B</math> is provable.
* A sequent <math>\vdash\Gamma,A\with B</math> is provable if and only if <math>\vdash\Gamma,A</math> and <math>\vdash\Gamma,B</math> are provable.
* A sequent <math>\vdash\Gamma,\bot</math> is provable if and only if <math>\vdash\Gamma</math> is provable.
* A sequent <math>\vdash\Gamma,\forall\xi A</math> is provable if and only if <math>\vdash\Gamma,A</math> is provable, for some fresh variable <math>\xi</math>.
}}

{{Proof|
We start with the case of the <math>\parr</math> connective.
If <math>\vdash\Gamma,A,B</math> is provable, then by the introduction rule for <math>\parr</math>
we know that <math>\vdash\Gamma,A\parr B</math> is provable.
For the reverse implication we proceed by induction on a proof <math>\pi</math> of
<math>\vdash\Gamma,A\parr B</math>.

* If the last rule of <math>\pi</math> is the introduction of the <math>\parr</math> in <math>A\parr B</math>, then the premiss is exacty <math>\vdash\Gamma,A,B</math> so we can conclude.
* The other case where the last rule introduces <math>A\parr B</math> is when <math>\pi</math> is an axiom rule, hence <math>\Gamma=A\orth\tens B\orth</math>. Then we can conclude with the proof
* Otherwise <math>A\parr B</math> is in the context of the last rule. If the last rule is a tensor, then <math>\pi</math> has the shape
* or the same with <math>A\parr B</math> in the conclusion of <math>\pi_2</math> instead. By induction hypothesis on <math>\pi_1</math> we get a proof <math>\pi'_1</math> of <math>\vdash\Gamma_1,A,B,C</math>, then we can conclude with the proof
* The case of the cut rule has the same structure as the tensor rule.
* In the case of the <math>\with</math> rule, we have <math>A\with B</math> in both premisses and we conclude similarly, using the induction hypothesis on both <math>\pi_1</math> and <math>\pi_2</math>.
* If <math>A\parr B</math> is in the context of a rules for <math>\parr</math>, <math>\plus</math>, <math>\bot</math> or quantifiers, or in the context of a dereliction, weakening or contraction, the situation is similar as for <math>\tens</math> except that we have only one premiss.
* If <math>A\parr B</math> is in the context of <math>\top</math> rules, we can freely change the context of the rule to get the expected one.
* The two remaining cases are if the last rule is the rule for <math>1</math> or a promotion. By the constraints these rules impose on the contexts, these cases cannot happen.
The <math>\with</math> connective is treated in the same way.
In this cases where <math>A\with B</math> is in the context of a rule with two
premisses, the premiss where this formula is not present will be duplicated,
with one copy in the premiss for <math>A</math> and one in the premiss for <math>B</math>.

The <math>\forall</math> connective is also treated similarly.
Its peculiarity is that introducing <math>\forall\xi</math> requires that <math>\xi</math> does
not appear free in the context.
For all rules with one premiss except the quantifier rules, the set of fresh
variables in the same in the premiss and the conclusion, so everything works
well.
Other rules might change the set of free variables, but problems are avoided
by choosing for <math>\xi</math> a variable that is fresh for the whole proof we are
considering.
}}

Remark that this result is proved using only commutation rules, except when the
formula is introduced by an axiom rule.
Furthermore, if axioms are applied only on atoms, this particular case
disappears.

A consequence of this fact is that, when searching for a proof of some
sequent <math>\vdash\Gamma</math>, one can always start by decomposing negative
connectives in <math>\Gamma</math> without losing provability.
Applying this result to successive connectives, we can get generalized
formulations for more complex formulas. For instance:

* <math>\vdash\Gamma,(A\parr B)\parr(B\with C)</math> is provable
* iff <math>\vdash\Gamma,A\parr B,B\with C</math> is provable
* iff <math>\vdash\Gamma,A\parr B,B</math> and <math>\vdash\Gamma,A\parr B,C</math> are provable
* iff <math>\vdash\Gamma,A,B,B</math> and <math>\vdash\Gamma,A,B,C</math> are provable
So without loss of generality, we can assume that any proof of
<math>\vdash\Gamma,(A\parr B)\parr(B\with C)</math> ends like

<math>
\AxRule{ \vdash \Gamma, A, B, B }
\UnaRule{ \vdash \Gamma, A\parr B, B }
\AxRule{ \vdash \Gamma, A, B, C }
\UnaRule{ \vdash \Gamma, A\parr B, C }
\BinRule{ \vdash \Gamma, A\parr B, B\with C }
\UnaRule{ \vdash \Gamma, (A\parr B)\parr(B\with C) }
\DisplayProof
</math>

In order to define a general statement for compound formulas, as well as an
analogous result for positive connectives, we need to give a proper status to
clusters of connectives of the same polarity.

== Generalized connectives and rules ==

{{Definition|
A ''positive generalized connective'' is a parametrized formula
<math>P[X_1,\ldots,X_n]</math> made from the variables <math>X_i</math> using the connectives
<math>\tens</math>, <math>\plus</math>, <math>\one</math>, <math>\zero</math>.

A ''negative generalized connective'' is a parametrized formula
<math>N[X_1,\ldots,X_n]</math> made from the variables <math>X_i</math> using the connectives
<math>\parr</math>, <math>\with</math>, <math>\bot</math>, <math>\top</math>.

If <math>C[X_1,\ldots,X_n]</math> is a generalized connectives (of any polarity), the
''dual'' of <math>C</math> is the connective <math>C^*</math> such that
<math>C^*[X_1\orth,\ldots,X_n\orth]=C[X_1,\ldots,X_n]\orth</math>.
}}

It is clear that dualization of generalized connectives is involutive and
exchanges polarities.
We do not include quantifiers in this definition, mainly for simplicity.
Extending the notion to quantifiers would only require taking proper care of
the scope of variables.

Sequent calculus provides introduction rules for each connective. Negative
connectives have one rule, positive ones may have any number of rules, namely
2 for <math>\plus</math> and 0 for <math>\zero</math>. We can derive introduction rules for the
generalized connectives by combining the different possible introduction rules
for each of their components.

Considering the previous example
<math>N[X_1,X_2,X_3]=(X_1\parr X_2)\parr(X_2\with X_3)</math>, we can derive an
introduction rule for <math>N</math> as

<math>
\AxRule{ \vdash \Gamma, X_1, X_2, X_2 }
\UnaRule{ \vdash \Gamma, X_1\parr X_2, X_2 }
\AxRule{ \vdash \Gamma, X_1, X_2, X_3 }
\UnaRule{ \vdash \Gamma, X_1\parr X_2, X_3 }
\BinRule{ \vdash \Gamma, X_1\parr X_2, X_2\with X_3 }
\UnaRule{ \vdash \Gamma, (X_1\parr X_2)\parr(X_2\with X_3) }
\DisplayProof
\quad\text{or}\quad
\AxRule{ \vdash \Gamma, X_1, X_2, X_2 }
\AxRule{ \vdash \Gamma, X_1, X_2, X_3 }
\BinRule{ \vdash \Gamma, X_1, X_2, X_2\with X_3 }
\UnaRule{ \vdash \Gamma, X_1\parr X_2, X_2\with X_3 }
\UnaRule{ \vdash \Gamma, (X_1\parr X_2)\parr(X_2\with X_3) }
\DisplayProof
</math>

but these rules only differ by the commutation of independent rules.
In particular, their premisses are the same.
The dual of <math>N</math> is <math>P[X_1,X_2,X_3]=(X_1\tens X_2)\tens(X_2\plus X_3)</math>, for
which we have two possible derivations:

<math>
\AxRule{ \vdash \Gamma_1, X_1 }
\AxRule{ \vdash \Gamma_2, X_2 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, X_1\tens X_2 }
\AxRule{ \vdash \Gamma_3, X_2 }
\UnaRule{ \vdash \Gamma_3, X_2\plus X_3 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, \Gamma_3, (X_1\tens X_2)\tens(X_2\plus X_3) }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma_1, X_1 }
\AxRule{ \vdash \Gamma_2, X_2 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, X_1\tens X_2 }
\AxRule{ \vdash \Gamma_3, X_3 }
\UnaRule{ \vdash \Gamma_3, X_2\plus X_3 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, \Gamma_3, (X_1\tens X_2)\tens(X_2\plus X_3) }
\DisplayProof
</math>

These are actually different, in particular their premisses differ.
Each possible derivation corresponds to the choice of one side of the <math>\plus</math>
connective.

We can remark that the branches of the negative inference precisely correspond
to the possible positive inferences:

* the first branch of the negative inference has a premiss <math>X_1,X_2,X_2</math> and the first positive inference has three premisses, holding <math>X_1</math>, <math>X_2</math> and <math>X_2</math> respectively.
* the second branch of the negative inference has a premiss <math>X_1,X_2,X_3</math> and the second positive inference has three premisses, holding <math>X_1</math>, <math>X_2</math> and <math>X_3</math> respectively.
This phenomenon extends to all generalized connectives.

{{Definition|
The ''branching'' of a generalized connective <math>P[X_1,\ldots,X_n]</math> is the
multiset <math>\mathcal{I}_P</math> of multisets over <math>\{1,\ldots,n\}</math> defined
inductively as

<math> \mathcal{I}_{P\tens Q} := [ I+J \mid I\in\mathcal{I}_P, J\in\mathcal{I}_Q ] </math>,
<math> \mathcal{I}_{P\plus Q} := \mathcal{I}_P + \mathcal{I}_Q </math>,
<math> \mathcal{I}_\one := [[]] </math>,
<math> \mathcal{I}_\zero := [] </math>,
<math> \mathcal{I}_{X_i} := [[i]] </math>.

The branching of a negative generalized connective is the branching of its
dual. Elements of a branching are called branches.
}}

In the example above, the branching will be <math>[[1,2,2],[1,2,3]]</math>, which
corresponds to the granches of the negative inference and to the cases of
positive inference.

{{Definition|
Let <math>\mathcal{I}</math> be a branching.
Write <math>\mathcal{I}</math> as <math>[I_1,\ldots,I_k]</math> and write each <math>I_j</math> as
<math>[i_{j,1},\ldots,i_{j,\ell_j}]</math>.
The derived rule for a negative generalized connective <math>N</math> with
branching <math>\mathcal{I}</math> is

<math>
\AxRule{ \vdash \Gamma, A_{i_{1,1}}, \ldots, A_{i_{1,\ell_1}} }
\AxRule{ \cdots }
\AxRule{ \vdash \Gamma, A_{i_{k,1}}, \ldots, A_{i_{k,\ell_k}} }
\LabelRule{N}
\TriRule{ \vdash \Gamma, N[A_1,\ldots,A_n] }
\DisplayProof
</math>

For each branch <math>I=[i_1,\ldots,i_\ell]</math> of a positive generalized connective
<math>P</math>, the derived rule for branch <math>I</math> of <math>P</math> is

<math>
\AxRule{ \vdash \Gamma_1, A_{i_1} }
\AxRule{ \cdots }
\AxRule{ \vdash \Gamma_\ell, A_{i_\ell} }
\LabelRule{P_I}
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, P[A_1,\ldots,A_n] }
\DisplayProof
</math>
}}

The reversibility property of negative connectives can be rephrased in a
generalized way as

{{Theorem|
Let <math>N</math> be a negative generalized connective. A sequent
<math>\vdash\Gamma,N[A_1,\ldots,A_n]</math> is provable if and only if, for each
<math>[i_1,\ldots,i_k]\in\mathcal{I}_N</math>, the sequent
<math>\vdash\Gamma,A_{i_1},\ldots,A_{i_k}</math> is provable.
}}

The corresponding property for positive connectives is the focalization
property, defined in the next section.

== Focalization ==

{{Definition|
A formula is ''positive'' if it has a main connective among
<math>\tens</math>, <math>\plus</math>, <math>\one</math>, <math>\zero</math>.
It is called ''negative'' if it has a main connective among
<math>\parr</math>, <math>\with</math>, <math>\bot</math>, <math>\top</math>.
It is called ''neutral'' if it is neither positive nor negative.
}}

If we extended the theory to include quantifiers in generalized connectives,
then the definition of positive and negative formulas would be extended to
include them too.

{{Definition|
A proof <math>\pi\vdash\Gamma,A</math> is said to be ''positively focused on <math>A</math>'' if it has the shape

<math>
\AxRule{ \pi_1 \vdash \Gamma_1, A_{i_1} }
\AxRule{ \cdots }
\AxRule{ \pi_\ell \vdash \Gamma_\ell, A_{i_\ell} }
\LabelRule{P_{[i_1,\ldots,i_\ell]}}
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, P[A_1,\ldots,A_n] }
\DisplayProof
</math>

where <math>P</math> is a positive generalized connective, the <math>A_i</math> ar non-positive
and <math>A=P[A_1,\ldots,A_n]</math>. The formula <math>A</math> is called the ''focus'' of the
proof <math>\pi</math>.
}}

In other words, a proof is positively focused on a conclusion <math>A</math> if its last rules
build <math>A</math> from some of its non-positive subformulas in one cluster of
inferences. Note that this notion only makes sense for a sequent made only
of positive formulas, since by this definition a proof is obviously positively focused on
any of its non-positive conclusions, using the degenerate generalized
connective <math>P[X]=X</math>.

{{Theorem|
A sequent <math>\vdash\Gamma</math> is cut-free provable if and only if it is provable
by a cut-free proof that is positively focused.
}}

{{Proof|
We reason by induction on a proof <math>\pi</math> of <math>\Gamma</math>.
As noted above, the result trivially holds if <math>\Gamma</math> has a non-positive
formula.
We can thus assume that <math>\Gamma</math> contains only positive formulas and reason
on the nature of the last rule, which is necessarily the introduction of a
positive connective (it cannot be an axiom rule because an axiom always has
at least on non-positive conclusion).

Suppose that the last rule of <math>\pi</math> introduces a tensor, so that <math>\pi</math> is

<math>
\AxRule{ \rho \vdash \Gamma, A }
\AxRule{ \theta \vdash \Delta, B }
\BinRule{ \vdash \Gamma, \Delta, A\tens B }
\DisplayProof
</math>

By induction hypothesis, there are positively focused proofs <math>\rho'\vdash\Gamma,A</math>
and <math>\theta'\vdash\Delta,B</math>.
If <math>A</math> is the focus of <math>\rho'</math> and <math>B</math> is the focus of <math>\theta'</math>, then the
proof

<math>
\AxRule{ \rho' \vdash \Gamma, A }
\AxRule{ \theta' \vdash \Delta, B }
\BinRule{ \vdash \Gamma, \Delta, A\tens B }
\DisplayProof
</math>

is positively focused on <math>A\tens B</math>, so we can conclude.
Otherwise, one of the two proofs is positively focused on another conclusion.
Without loss of generality, suppose that <math>\rho'</math> is not positively focused on <math>A</math>.
Then it decomposes as

<math>
\AxRule{ \rho_1 \vdash \Gamma_1, C_{i_1} }
\AxRule{ \cdots }
\AxRule{ \rho_\ell \vdash \Gamma_\ell, C_{i_\ell} }
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, P[C_1,\ldots,C_n] }
\DisplayProof
</math>

where the <math>C_i</math> are not positive and <math>A</math> belongs to some context <math>\Gamma_j</math>
that we will write <math>\Gamma'_j,A</math>.
Then we can conclude with the proof

<math>
\AxRule{ \rho_1 \vdash \Gamma_1, C_{i_1} \quad\cdots }
\AxRule{ \rho_j \vdash \Gamma_j, A, C_{i_j} }
\AxRule{ \theta \vdash \Delta, B }
\BinRule{ \vdash \Gamma_j, \Delta, A\tens B, C_{i_j} }
\AxRule{ \cdots\quad \rho_\ell \vdash \Gamma_\ell, C_{i_\ell} }
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, \Delta, A\tens B, P[C_1,\ldots,C_n] }
\DisplayProof
</math>

which is positively focused on <math>P[C_1,\ldots,C_n]</math>.

If the last rule of <math>\pi</math> introduces a <math>\plus</math>, we proceed the same way
except that there is only one premiss.
If the last rule of <math>\pi</math> introduces a <math>\one</math>, then it is the only rule of
<math>\pi</math>, which is thus positively focused on this <math>\one</math>.
}}

As in the reversibility theorem, this proof only makes use of commutation of
independent rules.

These results say nothing about exponential modalities, because they respect
neither reversibility nor focalization. However, if we consider the fragment
of LL which consists only of multiplicative and additive connectives, we can
restrict the proof rules to enforce focalization without loss of
expressiveness.

Reversibility and focalization

2012-08-31T21:27:46Z

Emmanuel Beffara: /* Generalized connectives and rules */

== Reversibility ==

{{Theorem|
Negative connectives are reversible:

* A sequent <math>\vdash\Gamma,A\parr B</math> is provable if and only if <math>\vdash\Gamma,A,B</math> is provable.
* A sequent <math>\vdash\Gamma,A\with B</math> is provable if and only if <math>\vdash\Gamma,A</math> and <math>\vdash\Gamma,B</math> are provable.
* A sequent <math>\vdash\Gamma,\bot</math> is provable if and only if <math>\vdash\Gamma</math> is provable.
* A sequent <math>\vdash\Gamma,\forall\xi A</math> is provable if and only if <math>\vdash\Gamma,A</math> is provable, for some fresh variable <math>\xi</math>.
}}

{{Proof|
We start with the case of the <math>\parr</math> connective.
If <math>\vdash\Gamma,A,B</math> is provable, then by the introduction rule for <math>\parr</math>
we know that <math>\vdash\Gamma,A\parr B</math> is provable.
For the reverse implication we proceed by induction on a proof <math>\pi</math> of
<math>\vdash\Gamma,A\parr B</math>.

* If the last rule of <math>\pi</math> is the introduction of the <math>\parr</math> in <math>A\parr B</math>, then the premiss is exacty <math>\vdash\Gamma,A,B</math> so we can conclude.
* The other case where the last rule introduces <math>A\parr B</math> is when <math>\pi</math> is an axiom rule, hence <math>\Gamma=A\orth\tens B\orth</math>. Then we can conclude with the proof
* Otherwise <math>A\parr B</math> is in the context of the last rule. If the last rule is a tensor, then <math>\pi</math> has the shape
* or the same with <math>A\parr B</math> in the conclusion of <math>\pi_2</math> instead. By induction hypothesis on <math>\pi_1</math> we get a proof <math>\pi'_1</math> of <math>\vdash\Gamma_1,A,B,C</math>, then we can conclude with the proof
* The case of the cut rule has the same structure as the tensor rule.
* In the case of the <math>\with</math> rule, we have <math>A\with B</math> in both premisses and we conclude similarly, using the induction hypothesis on both <math>\pi_1</math> and <math>\pi_2</math>.
* If <math>A\parr B</math> is in the context of a rules for <math>\parr</math>, <math>\plus</math>, <math>\bot</math> or quantifiers, or in the context of a dereliction, weakening or contraction, the situation is similar as for <math>\tens</math> except that we have only one premiss.
* If <math>A\parr B</math> is in the context of <math>\top</math> rules, we can freely change the context of the rule to get the expected one.
* The two remaining cases are if the last rule is the rule for <math>1</math> or a promotion. By the constraints these rules impose on the contexts, these cases cannot happen.
The <math>\with</math> connective is treated in the same way.
In this cases where <math>A\with B</math> is in the context of a rule with two
premisses, the premiss where this formula is not present will be duplicated,
with one copy in the premiss for <math>A</math> and one in the premiss for <math>B</math>.

The <math>\forall</math> connective is also treated similarly.
Its peculiarity is that introducing <math>\forall\xi</math> requires that <math>\xi</math> does
not appear free in the context.
For all rules with one premiss except the quantifier rules, the set of fresh
variables in the same in the premiss and the conclusion, so everything works
well.
Other rules might change the set of free variables, but problems are avoided
by choosing for <math>\xi</math> a variable that is fresh for the whole proof we are
considering.
}}

Remark that this result is proved using only commutation rules, except when the
formula is introduced by an axiom rule.
Furthermore, if axioms are applied only on atoms, this particular case
disappears.

A consequence of this fact is that, when searching for a proof of some
sequent <math>\vdash\Gamma</math>, one can always start by decomposing negative
connectives in <math>\Gamma</math> without losing provability.
Applying this result to successive connectives, we can get generalized
formulations for more complex formulas. For instance:

* <math>\vdash\Gamma,(A\parr B)\parr(B\with C)</math> is provable
* iff <math>\vdash\Gamma,A\parr B,B\with C</math> is provable
* iff <math>\vdash\Gamma,A\parr B,B</math> and <math>\vdash\Gamma,A\parr B,C</math> are provable
* iff <math>\vdash\Gamma,A,B,B</math> and <math>\vdash\Gamma,A,B,C</math> are provable
So without loss of generality, we can assume that any proof of
<math>\vdash\Gamma,(A\parr B)\parr(B\with C)</math> ends like

<math>
\AxRule{ \vdash \Gamma, A, B, B }
\UnaRule{ \vdash \Gamma, A\parr B, B }
\AxRule{ \vdash \Gamma, A, B, C }
\UnaRule{ \vdash \Gamma, A\parr B, C }
\BinRule{ \vdash \Gamma, A\parr B, B\with C }
\UnaRule{ \vdash \Gamma, (A\parr B)\parr(B\with C) }
\DisplayProof
</math>

In order to define a general statement for compound formulas, as well as an
analogous result for positive connectives, we need to give a proper status to
clusters of connectives of the same polarity.

== Generalized connectives and rules ==

{{Definition|
A ''positive generalized connective'' is a parametrized formula
<math>P[X_1,\ldots,X_n]</math> made from the variables <math>X_i</math> using the connectives
<math>\tens</math>, <math>\plus</math>, <math>\one</math>, <math>\zero</math>.

A ''negative generalized connective'' is a parametrized formula
<math>N[X_1,\ldots,X_n]</math> made from the variables <math>X_i</math> using the connectives
<math>\parr</math>, <math>\with</math>, <math>\bot</math>, <math>\top</math>.

If <math>C[X_1,\ldots,X_n]</math> is a generalized connectives (of any polarity), the
''dual'' of <math>C</math> is the connective <math>C^*</math> such that
<math>C^*[X_1\orth,\ldots,X_n\orth]=C[X_1,\ldots,X_n]\orth</math>.
}}

It is clear that dualization of generalized connectives is involutive and
exchanges polarities.
We do not include quantifiers in this definition, mainly for simplicity.
Extending the notion to quantifiers would only require taking proper care of
the scope of variables.

Sequent calculus provides introduction rules for each connective. Negative
connectives have one rule, positive ones may have any number of rules, namely
2 for <math>\plus</math> and 0 for <math>\zero</math>. We can derive introduction rules for the
generalized connectives by combining the different possible introduction rules
for each of their components.

Considering the previous example
<math>N[X_1,X_2,X_3]=(X_1\parr X_2)\parr(X_2\with X_3)</math>, we can derive an
introduction rule for <math>N</math> as

<math>
\AxRule{ \vdash \Gamma, X_1, X_2, X_2 }
\UnaRule{ \vdash \Gamma, X_1\parr X_2, X_2 }
\AxRule{ \vdash \Gamma, X_1, X_2, X_3 }
\UnaRule{ \vdash \Gamma, X_1\parr X_2, X_3 }
\BinRule{ \vdash \Gamma, X_1\parr X_2, X_2\with X_3 }
\UnaRule{ \vdash \Gamma, (X_1\parr X_2)\parr(X_2\with X_3) }
\DisplayProof
\quad\text{or}\quad
\AxRule{ \vdash \Gamma, X_1, X_2, X_2 }
\AxRule{ \vdash \Gamma, X_1, X_2, X_3 }
\BinRule{ \vdash \Gamma, X_1, X_2, X_2\with X_3 }
\UnaRule{ \vdash \Gamma, X_1\parr X_2, X_2\with X_3 }
\UnaRule{ \vdash \Gamma, (X_1\parr X_2)\parr(X_2\with X_3) }
\DisplayProof
</math>

but these rules only differ by the commutation of independent rules.
In particular, their premisses are the same.
The dual of <math>N</math> is <math>P[X_1,X_2,X_3]=(X_1\tens X_2)\tens(X_2\plus X_3)</math>, for
which we have two possible derivations:

<math>
\AxRule{ \vdash \Gamma_1, X_1 }
\AxRule{ \vdash \Gamma_2, X_2 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, X_1\tens X_2 }
\AxRule{ \vdash \Gamma_3, X_2 }
\UnaRule{ \vdash \Gamma_3, X_2\plus X_3 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, \Gamma_3, (X_1\tens X_2)\tens(X_2\plus X_3) }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma_1, X_1 }
\AxRule{ \vdash \Gamma_2, X_2 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, X_1\tens X_2 }
\AxRule{ \vdash \Gamma_3, X_3 }
\UnaRule{ \vdash \Gamma_3, X_2\plus X_3 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, \Gamma_3, (X_1\tens X_2)\tens(X_2\plus X_3) }
\DisplayProof
</math>

These are actually different, in particular their premisses differ.
Each possible derivation corresponds to the choice of one side of the <math>\plus</math>
connective.

We can remark that the branches of the negative inference precisely correspond
to the possible positive inferences:

* the first branch of the negative inference has a premiss <math>X_1,X_2,X_2</math> and the first positive inference has three premisses, holding <math>X_1</math>, <math>X_2</math> and <math>X_2</math> respectively.
* the second branch of the negative inference has a premiss <math>X_1,X_2,X_3</math> and the second positive inference has three premisses, holding <math>X_1</math>, <math>X_2</math> and <math>X_3</math> respectively.
This phenomenon extends to all generalized connectives.

{{Definition|
The ''branching'' of a generalized connective <math>P[X_1,\ldots,X_n]</math> is the
multiset <math>\mathcal{I}_P</math> of multisets over <math>\{1,\ldots,n\}</math> defined
inductively as

<math> \mathcal{I}_{P\tens Q} := [ I+J \mid I\in\mathcal{I}_P, J\in\mathcal{I}_Q ] </math>,
<math> \mathcal{I}_{P\plus Q} := \mathcal{I}_P + \mathcal{I}_Q </math>,
<math> \mathcal{I}_\one := [[]] </math>,
<math> \mathcal{I}_\zero := [] </math>,
<math> \mathcal{I}_{X_i} := [[i]] </math>.

The branching of a negative generalized connective is the branching of its
dual. Elements of a branching are called branches.
}}

In the example above, the branching will be <math>[[1,2,2],[1,2,3]]</math>, which
corresponds to the granches of the negative inference and to the cases of
positive inference.

{{Definition|
Let <math>\mathcal{I}</math> be a branching.
Write <math>\mathcal{I}</math> as <math>[I_1,\ldots,I_k]</math> and write each <math>I_j</math> as
<math>[i_{j,1},\ldots,i_{j,\ell_j}]</math>.
The derived rule for a negative generalized connective <math>N</math> with
branching <math>\mathcal{I}</math> is

<math>
\AxRule{ \vdash \Gamma, A_{i_{1,1}}, \ldots, A_{i_{1,\ell_1}} }
\AxRule{ \cdots }
\AxRule{ \vdash \Gamma, A_{i_{k,1}}, \ldots, A_{i_{k,\ell_k}} }
\LabelRule{N}
\TriRule{ \vdash \Gamma, N[A_1,\ldots,A_n] }
\DisplayProof
</math>

For each branch <math>I=[i_1,\ldots,i_\ell]</math> of a positive generalized connective
<math>P</math>, the derived rule for branch <math>I</math> of <math>P</math> is

<math>
\AxRule{ \vdash \Gamma_1, A_{i_1} }
\AxRule{ \cdots }
\AxRule{ \vdash \Gamma_\ell, A_{i_\ell} }
\LabelRule{P_I}
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, P[A_1,\ldots,A_n] }
\DisplayProof
</math>
}}

The reversibility property of negative connectives can be rephrased in a
generalized way as

{{Theorem|
Let <math>N</math> be a negative generalized connective. A sequent
<math>\vdash\Gamma,N[A_1,\ldots,A_n]</math> is provable if and only if, for each
<math>[i_1,\ldots,i_k]\in\mathcal{I}_N</math>, the sequent
<math>\vdash\Gamma,A_{i_1},\ldots,A_{i_k}</math> is provable.
}}

The corresponding property for positive connectives is the focalization
property, defined in the next section.

== Focalization ==

{{Definition|
A formula is ''positive'' if it has a main connective among
<math>\tens</math>, <math>\plus</math>, <math>\one</math>, <math>\zero</math>.
It is called ''negative'' if it has a main connective among
<math>\parr</math>, <math>\with</math>, <math>\bot</math>, <math>\top</math>.
It is called ''neutral'' if it is neither positive nor negative.
}}

If we extended the theory to include quantifiers in generalized connectives,
then the definition of positive and negative formulas would be extended to
include them too.

{{Definition|
A proof <math>\pi\vdash\Gamma,A</math> is said to be ''focalized'' on <math>A</math> if it has
the shape

<math>
\AxRule{ \pi_1 \vdash \Gamma_1, A_{i_1} }
\AxRule{ \cdots }
\AxRule{ \pi_\ell \vdash \Gamma_\ell, A_{i_\ell} }
\LabelRule{P_{[i_1,\ldots,i_\ell]}}
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, P[A_1,\ldots,A_n] }
\DisplayProof
</math>

where <math>P</math> is a positive generalized connective, the <math>A_i</math> ar non-positive
and <math>A=P[A_1,\ldots,A_n]</math>. The formula <math>A</math> is called the ''focus'' of the
proof <math>\pi</math>.
}}

In other words, a proof is focalized on a conclusion <math>A</math> if its last rules
build <math>A</math> from some of its non-positive subformulas in one cluster of
inferences. Note that this notion only makes sense for a conclusion made only
of positive formulas, since a proof is obviously focalized on any of its
non-positive conclusions, using the degenerate generalized connective
<math>P[X]=X</math>.

{{Theorem|
A sequent <math>\vdash\Gamma</math> is cut-free provable if and only if it is provable
by a cut-free focalized proof.
}}

{{Proof|
We reason by induction on a proof <math>\pi</math> of <math>\Gamma</math>.
As noted above, the result trivially holds if <math>\Gamma</math> has a non-positive
formula.
We can thus assume that <math>\Gamma</math> contains only positive formulas and reason
on the nature of the last rule, which is necessarily the introduction of a
positive connective (it cannot be an axiom rule because an axiom always has
at least on non-positive conclusion).

Suppose that the last rule of <math>\pi</math> introduces a tensor, so that <math>\pi</math> is

<math>
\AxRule{ \rho \vdash \Gamma, A }
\AxRule{ \theta \vdash \Delta, B }
\BinRule{ \vdash \Gamma, \Delta, A\tens B }
\DisplayProof
</math>

By induction hypothesis, there are focalized proofs <math>\rho'\vdash\Gamma,A</math>
and <math>\theta'\vdash\Delta,B</math>.
If <math>A</math> is the focus of <math>\rho'</math> and <math>B</math> is the focus of <math>\theta'</math>, then the
proof

<math>
\AxRule{ \rho' \vdash \Gamma, A }
\AxRule{ \theta' \vdash \Delta, B }
\BinRule{ \vdash \Gamma, \Delta, A\tens B }
\DisplayProof
</math>

is focalized on <math>A\tens B</math>, so we can conclude.
Otherwise, one of the two proofs is focalized on another conclusion.
Without loss of generality, suppose that <math>\rho'</math> is not focalized on <math>A</math>.
Then it decomposes as

<math>
\AxRule{ \rho_1 \vdash \Gamma_1, C_{i_1} }
\AxRule{ \cdots }
\AxRule{ \rho_\ell \vdash \Gamma_\ell, C_{i_\ell} }
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, P[C_1,\ldots,C_n] }
\DisplayProof
</math>

where the <math>C_i</math> are not positive and <math>A</math> belongs to some context <math>\Gamma_j</math>
that we will write <math>\Gamma'_j,A</math>.
Then we can conclude with the proof

<math>
\AxRule{ \rho_1 \vdash \Gamma_1, C_{i_1} \quad\cdots }
\AxRule{ \rho_j \vdash \Gamma_j, A, C_{i_j} }
\AxRule{ \theta \vdash \Delta, B }
\BinRule{ \vdash \Gamma_j, \Delta, A\tens B, C_{i_j} }
\AxRule{ \cdots\quad \rho_\ell \vdash \Gamma_\ell, C_{i_\ell} }
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, \Delta, A\tens B, P[C_1,\ldots,C_n] }
\DisplayProof
</math>

which is focalized on <math>P[C_1,\ldots,C_n]</math>.

If the last rule of <math>\pi</math> introduces a <math>\plus</math>, we proceed the same way
except that there is only one premiss.
If the last rule of <math>\pi</math> introduces a <math>\one</math>, then it is the only rule of
<math>\pi</math>, which is thus focalized on this <math>\one</math>.
}}

As in the reversibility theorem, this proof only makes use of commutation of
independent rules.

These results say nothing about exponential modalities, because they respect
neither reversibility nor focalization. However, if we consider the fragment
of LL which consists only of multiplicative and additive connectives, we can
restrict the proof rules to enforce focalization without loss of
expressiveness.

Sequent calculus

2012-08-31T21:21:59Z

Emmanuel Beffara: /* Reversibility */ mention focalization

This article presents the language and sequent calculus of second-order
linear logic and the basic properties of this sequent calculus.
The core of the article uses the two-sided system with negation as a proper
connective; the [[#One-sided sequent calculus|one-sided system]], often used
as the definition of linear logic, is presented at the end of the page.

== Formulas ==

Atomic formulas, written <math>\alpha,\beta,\gamma</math>, are predicates of
the form <math>p(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms
from some first-order language.
The predicate symbol <math>p</math> may be either a predicate constant or a
second-order variable.
By convention we will write first-order variables as <math>x,y,z</math>,
second-order variables as <math>X,Y,Z</math>, and <math>\xi</math> for a
variable of arbitrary order (see [[Notations]]).

Formulas, represented by capital letters <math>A</math>, <math>B</math>,
<math>C</math>, are built using the following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>A\orth</math>
| negation
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists \xi.A</math>
| there exists
| <math>\forall \xi.A</math>
| for all
| quantifiers
|}

Each line (except the first one) corresponds to a particular class of
connectives, and each class consists in a pair of connectives.
Those in the left column are called [[positive formula|positive]] and those in
the right column are called [[negative formula|negative]].
The ''tensor'' and ''with'' connectives are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally read
''of course <math>A</math>'' for <math>\oc A</math> and ''why not
<math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic, as
<math>A\limp B:=A\orth\parr B</math>.

Free and bound variables and first-order substitution are defined in the
standard way.
Formulas are always considered up to renaming of bound names.
If <math>A</math> is a formula, <math>X</math> is a second-order variable and
<math>B[x_1,\ldots,x_n]</math> is a formula with variables <math>x_i</math>,
then the formula <math>A[B/X]</math> is <math>A</math> where every atom
<math>X(t_1,\ldots,t_n)</math> is replaced by <math>B[t_1,\ldots,t_n]</math>.

== Sequents and proofs ==

A sequent is an expression <math>\Gamma\vdash\Delta</math> where
<math>\Gamma</math> and <math>\Delta</math> are finite multisets of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation
<math>\wn\Gamma</math> represents the multiset
<math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:
* Identity group: <math>
\LabelRule{\rulename{axiom}}
\NulRule{ A \vdash A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{\rulename{cut}}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>
* Negation: <math>
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\LabelRule{n_L}
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\LabelRule{n_R}
\DisplayProof
</math>
* Multiplicative group:
** tensor: <math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
</math>
** par: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma', B \vdash \Delta' }
\LabelRule{ \parr_L }
\BinRule{ \Gamma, \Gamma', A \parr B \vdash \Delta, \Delta' }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, B, \Delta }
\LabelRule{ \parr_R }
\UnaRule{ \Gamma \vdash A \parr B, \Delta }
\DisplayProof
</math>
** one: <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>
** bottom: <math>
\LabelRule{ \bot_L }
\NulRule{ \bot \vdash }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \bot_R }
\UnaRule{ \Gamma \vdash \bot, \Delta }
\DisplayProof
</math>
* Additive group:
** plus: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
</math>
** with: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ \with_{L1} }
\UnaRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \with_{L2} }
\UnaRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \with_R }
\BinRule{ \Gamma \vdash A \with B, \Delta }
\DisplayProof
</math>
** zero: <math>
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>
** top: <math>
\LabelRule{ \top_R }
\NulRule{ \Gamma \vdash \top, \Delta }
\DisplayProof
</math>
* Exponential group:
** of course: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B ,\wn B_1, \ldots, \wn B_m }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B ,\wn B_1, \ldots, \wn B_m }
\DisplayProof
</math>
** why not: <math>
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ d_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \wn A, \wn A, \Delta }
\LabelRule{ c_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \oc A_1, \ldots, \oc A_n, A \vdash \wn B_1, \ldots, \wn B_m }
\LabelRule{ \wn_L }
\UnaRule{ \oc A_1, \ldots, \oc A_n, \wn A \vdash \wn B_1, \ldots, \wn B_m }
\DisplayProof
</math>
* Quantifier group (in the <math>\exists_L</math> and <math>\forall_R</math> rules, <math>\xi</math> must not occur free in <math>\Gamma</math> or <math>\Delta</math>):
** there exists: <math>
\AxRule{ \Gamma , A \vdash \Delta }
\LabelRule{ \exists_L }
\UnaRule{ \Gamma, \exists\xi.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A[t/x] }
\LabelRule{ \exists^1_R }
\UnaRule{ \Gamma \vdash \Delta, \exists x.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A[B/X] }
\LabelRule{ \exists^2_R }
\UnaRule{ \Gamma \vdash \Delta, \exists X.A }
\DisplayProof
</math>
** for all: <math>
\AxRule{ \Gamma, A[t/x] \vdash \Delta }
\LabelRule{ \forall^1_L }
\UnaRule{ \Gamma, \forall x.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, A[B/X] \vdash \Delta }
\LabelRule{ \forall^2_L }
\UnaRule{ \Gamma, \forall X.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A }
\LabelRule{ \forall_R }
\UnaRule{ \Gamma \vdash \Delta, \forall\xi.A }
\DisplayProof
</math>

The left rules for ''of course'' and right rules for ''why not'' are called
''dereliction'', ''weakening'' and ''contraction'' rules.
The right rule for ''of course'' and the left rule for ''why not'' are called
''promotion'' rules.
Note the fundamental fact that there are no contraction and weakening rules
for arbitrary formulas, but only for the formulas starting with the
<math>\wn</math> modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An alternative presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rules:
: <math>
\AxRule{ \Gamma_1, A, B, \Gamma_2 \vdash \Delta }
\LabelRule{\rulename{exchange}_L}
\UnaRule{ \Gamma_1, B, A, \Gamma_2 \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta_1, A, B, \Delta_2 }
\LabelRule{\rulename{exchange}_R}
\UnaRule{ \Gamma \vdash \Delta_1, B, A, \Delta_2 }
\DisplayProof
</math>

== Equivalences ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent,
written <math>A\linequiv B</math>, if both implications <math>A\limp B</math>
and <math>B\limp A</math> are provable.
Equivalently, <math>A\linequiv B</math> if both <math>A\vdash B</math> and
<math>B\vdash A</math> are provable.
Another formulation of <math>A\linequiv B</math> is that, for all
<math>\Gamma</math> and <math>\Delta</math>, <math>\Gamma\vdash\Delta,A</math>
is provable if and only if <math>\Gamma\vdash\Delta,B</math> is provable.

Two related notions are [[isomorphism]] (stronger than equivalence) and
[[equiprovability]] (weaker than equivalence).

=== De Morgan laws ===

Negation is involutive:
: <math>A\linequiv A\biorth</math>
Duality between connectives:
{|
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>\linequiv A\orth \parr B\orth </math>
|width=30|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>\linequiv A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>\linequiv \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>\linequiv \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>\linequiv A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>\linequiv A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>\linequiv \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>\linequiv \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>\linequiv \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>\linequiv \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists \xi.A )\orth </math>
| <math>\linequiv \forall \xi.( A\orth ) </math>
|
|align="right"| <math> ( \forall \xi.A )\orth </math>
| <math>\linequiv \exists \xi.( A\orth ) </math>
|}

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality:
*: <math>
A \tens (B \tens C) \linequiv (A \tens B) \tens C </math> &emsp; <math>
A \tens B \linequiv B \tens A </math> &emsp; <math>
A \tens \one \linequiv A </math>
*: <math>
A \parr (B \parr C) \linequiv (A \parr B) \parr C </math> &emsp; <math>
A \parr B \linequiv B \parr A </math> &emsp; <math>
A \parr \bot \linequiv A </math>
*: <math>
A \plus (B \plus C) \linequiv (A \plus B) \plus C </math> &emsp; <math>
A \plus B \linequiv B \plus A </math> &emsp; <math>
A \plus \zero \linequiv A </math>
*: <math>
A \with (B \with C) \linequiv (A \with B) \with C </math> &emsp; <math>
A \with B \linequiv B \with A </math> &emsp; <math>
A \with \top \linequiv A </math>
* Idempotence of additives:
*: <math>
A \plus A \linequiv A </math> &emsp; <math>
A \with A \linequiv A </math>
* Distributivity of multiplicatives over additives:
*: <math>
A \tens (B \plus C) \linequiv (A \tens B) \plus (A \tens C) </math> &emsp; <math>
A \tens \zero \linequiv \zero </math>
*: <math>
A \parr (B \with C) \linequiv (A \parr B) \with (A \parr C) </math> &emsp; <math>
A \parr \top \linequiv \top </math>
* Defining property of exponentials:
*: <math>
\oc(A \with B) \linequiv \oc A \tens \oc B </math> &emsp; <math>
\oc\top \linequiv \one </math>
*: <math>
\wn(A \plus B) \linequiv \wn A \parr \wn B </math> &emsp; <math>
\wn\zero \linequiv \bot </math>
* Monoidal structure of exponentials:
*: <math>
\oc A \tens \oc A \linequiv \oc A </math> &emsp; <math>
\oc \one \linequiv \one </math>
*: <math>
\wn A \parr \wn A \linequiv \wn A </math> &emsp; <math>
\wn \bot \linequiv \bot </math>
* Digging:
*: <math>
\oc\oc A \linequiv \oc A </math> &emsp; <math>
\wn\wn A \linequiv \wn A </math>
* Other properties of exponentials:
*: <math>
\oc\wn\oc\wn A \linequiv \oc\wn A </math> &emsp; <math>
\oc\wn \one \linequiv \one </math>
*: <math>
\wn\oc\wn\oc A \linequiv \wn\oc A </math> &emsp; <math>
\wn\oc \bot \linequiv \bot </math>
These properties of exponentials lead to the [[lattice of exponential modalities]].
* Commutation of quantifiers (<math>\zeta</math> does not occur in <math>A</math>):
*: <math>
\exists \xi. \exists \psi. A \linequiv \exists \psi. \exists \xi. A </math> &emsp; <math>
\exists \xi.(A \plus B) \linequiv \exists \xi.A \plus \exists \xi.B </math> &emsp; <math>
\exists \zeta.(A\tens B) \linequiv A\tens\exists \zeta.B </math> &emsp; <math>
\exists \zeta.A \linequiv A </math>
*: <math>
\forall \xi. \forall \psi. A \linequiv \forall \psi. \forall \xi. A </math> &emsp; <math>
\forall \xi.(A \with B) \linequiv \forall \xi.A \with \forall \xi.B </math> &emsp; <math>
\forall \zeta.(A\parr B) \linequiv A\parr\forall \zeta.B </math> &emsp; <math>
\forall \zeta.A \linequiv A </math>

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:
* <math> \zero \linequiv \forall X.X </math>
* <math> \one \linequiv \forall X.(X \limp X) </math>
* <math> A \plus B \linequiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X) </math>
The constants <math>\top</math> and <math>\bot</math> and the connective
<math>\with</math> can be defined by duality.

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

== Properties of proofs ==

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\Gamma\vdash\Delta</math>, there is a proof of
<math>\Gamma\vdash\Delta</math> if and only if there is a proof of
<math>\Gamma\vdash\Delta</math> that does not use the cut rule.}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas of its immediate subformulas:
* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,
* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,
* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas have no immediate subformula,
* the immediate subformulas of <math>\exists x.A</math> and <math>\forall x.A</math> are all the <math>A[t/x]</math> for all first-order terms <math>t</math>,
* the immediate subformulas of <math>\exists X.A</math> and <math>\forall X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math> (with the appropriate number of parameters).}}

{{Theorem|title=subformula property|
A sequent <math>\Gamma\vdash\Delta</math> is provable if and only if it is the conclusion of a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math> and <math>\Delta</math>.}}
{{Proof|By the cut elimination theorem, if a sequent is provable, then it is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its
negation <math>A\orth</math>; it is impossible to prove <math>\zero</math> or
<math>\bot</math>.}}
{{Proof|If a sequent is provable, then it is the conclusion of a cut-free proof.
In each rule except the cut rule, there is at least one formula in conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.
The other properties are immediate consequences: if <math>\vdash A\orth</math>
and <math>\vdash A</math> are provable, then by the left negation rule
<math>A\orth\vdash</math> is provable, and by the cut rule one gets empty
conclusion, which is not possible.
As particular cases, since <math>\one</math> and <math>\top</math> are
provable, <math>\bot</math> and <math>\zero</math> are not, since they are
equivalent to <math>\one\orth</math> and <math>\top\orth</math>
respectively.}}

=== Expansion of identities ===

Let us write <math>\pi:\Gamma\vdash\Delta</math> to signify that
<math>\pi</math> is a proof with conclusion <math>\Gamma\vdash\Delta</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the
same conclusion as <math>\pi</math> in which the axiom rule is only used with
atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.}}
{{Proof|It suffices to prove that for every formula <math>A</math>, the sequent
<math>A\vdash A</math> has a cut-free proof in which the axiom rule is used
only for atomic formulas.
We prove this by induction on <math>A</math>.
* If <math>A</math> is atomic, then <math>A\vdash A</math> is an instance of the atomic axiom rule.
* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 : A_1 \vdash A_1 }
\AxRule{ \pi_2 : A_2 \vdash A_2 }
\LabelRule{ \tens_R }
\BinRule{ A_1, A_2 \vdash A_1 \tens A_2 }
\LabelRule{ \tens_L }
\UnaRule{ A_1 \tens A_2 \vdash A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* If <math>A=A_1\parr A_2</math> then we have <math>
\AxRule{ \pi_1 : A_1 \vdash A_1 }
\AxRule{ \pi_2 : A_2 \vdash A_2 }
\LabelRule{ \parr_L }
\BinRule{ A_1 \parr A_2 \vdash A_1, A_2 }
\LabelRule{ \parr_R }
\UnaRule{ A_1 \parr A_2 \vdash A_1 \parr A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* All other connectives follow the same pattern.}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if
* for every proof <math>\pi:\Gamma\vdash\Delta,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,
* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are reversible.}}
{{Proof|Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only applied to atomic formulas.
Then each top-level connective is introduced either by its associated (left or
right) rule or in an instance of the <math>\zero_L</math> or
<math>\top_R</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\Gamma\vdash\Delta,A\parr
B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr_R</math> rule (not
necessarily the last rule in <math>\pi</math>), then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction on <math>\pi</math>).
If it is introduced in the context of a <math>\zero_L</math> or
<math>\top_R</math> rule, then this rule can be changed so that
<math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the
expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a
<math>\bot_R</math> rule, then remove this rule to get a proof of
<math>\vdash\Gamma</math>, if it is introduced by a <math>\zero_L</math> or
<math>\top_R</math> rule, remove the <math>\bot</math> from this rule, then
apply the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof
<math>\pi:\Gamma\vdash\Delta,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is
applied in a context like

<math>
\AxRule{ \pi_1 \Gamma' \vdash \Delta', A }
\AxRule{ \pi_2 \Gamma' \vdash \Delta', B }
\LabelRule{ \with }
\BinRule{ \Gamma' \vdash \Delta', A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except
as context), if we replace this step by <math>\pi_1</math> in <math>\pi</math>
we finally get a proof <math>\pi'_1:\Gamma\vdash\Delta,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof
<math>\pi'_2:\Gamma\vdash\Delta,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final
<math>\with</math> rule we finally get the expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math>
rule is solved as before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as
an instance of the <math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math>, consider a proof
<math>\pi:\Gamma\vdash\Delta,\forall\xi.A</math>.
Up to renaming, we can assume that <math>\xi</math> occurs free only above the
rule that introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we
remove this rule, we can check that we get a proof of
<math>\Gamma\vdash\Delta,A</math> on which we can finally apply the
<math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math>
rule is solved as before.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the
original proof.}}

A corresponding property for positive connectives is [[Reversibility and focalization|focalization]], which states that clusters of positive formulas can be treated in one step, under certain circumstances.

== One-sided sequent calculus ==

The sequent calculus presented above is very symmetric: for every left
introduction rule, there is a right introduction rule for the dual connective
that has the exact same structure.
Moreover, because of the involutivity of negation, a sequent
<math>\Gamma,A\vdash\Delta</math> is provable if and only if the sequent
<math>\Gamma\vdash A\orth,\Delta</math> is provable.
From these remarks, we can define an equivalent one-sided sequent calculus:
* Formulas are considered up to De Morgan duality. Equivalently, one can consider that negation is not a connective but a syntactically defined operation on formulas. In this case, negated atoms <math>\alpha\orth</math> must be considered as another kind of atomic formulas.
* Sequents have the form <math>\vdash\Gamma</math>.
The inference rules are essentially the same except that the left hand side of
sequents is kept empty:
* Identity group:
*: <math>
\LabelRule{\rulename{axiom}}
\NulRule{ \vdash A\orth, A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{\rulename{cut}}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>
* Multiplicative group:
*: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>
* Additive group:
*: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>
* Exponential group:
*: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>
* Quantifier group (in the <math>\forall</math> rule, <math>\xi</math> must not occur free in <math>\Gamma</math>):
*: <math>
\AxRule{ \vdash \Gamma, A[t/x] }
\LabelRule{ \exists^1 }
\UnaRule{ \vdash \Gamma, \exists x.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists^2 }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall \xi.A }
\DisplayProof
</math>

{{Theorem|A two-sided sequent <math>\Gamma\vdash\Delta</math> is provable if
and only if the sequent <math>\vdash\Gamma\orth,\Delta</math> is provable in
the one-sided system.}}

The one-sided system enjoys the same properties as the two-sided one,
including cut elimination, the subformula property, etc.
This formulation is often used when studying proofs because it is much lighter
than the two-sided form while keeping the same expressiveness.
In particular, [[proof-nets]] can be seen as a quotient of one-sided sequent
calculus proofs under commutation of rules.

== Variations ==

=== Exponential rules ===

* The promotion rule, on the right-hand side for example,
<math>
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B, \wn B_1, \ldots, \wn B_m }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B, \wn B_1, \ldots, \wn B_m }
\DisplayProof
</math>
can be replaced by a ''multi-functorial'' promotion rule
<math>
\AxRule{ A_1, \ldots, A_n \vdash B, B_1, \ldots, B_m }
\LabelRule{ \oc_R \rulename{mf}}
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B, \wn B_1, \ldots, \wn B_m }
\DisplayProof
</math>
and a ''digging'' rule
<math>
\AxRule{ \Gamma \vdash \wn\wn A, \Delta }
\LabelRule{ \wn\wn}
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math>,
without modifying the provability.

Note that digging violates the subformula property.

* In presence of the digging rule <math>
\AxRule{ \Gamma \vdash \wn\wn A, \Delta }
\LabelRule{ \wn\wn}
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math>, the multiplexing rule <math>
\AxRule{\Gamma\vdash A^{(n)},\Delta}
\LabelRule{\rulename{mplex}}
\UnaRule{\Gamma\vdash \wn A,\Delta}
\DisplayProof
</math> (where <math>A^{(n)}</math> stands for n occurrences of formula <math>A</math>) is equivalent (for provability) to the triple of rules: contraction, weakening, dereliction.

=== Non-symmetric sequents ===

The same remarks that lead to the definition of the one-sided calculus can
lead the definition of other simplified systems:
* A one-sided variant with sequents of the form <math>\Gamma\vdash</math> could be defined.
* When considering formulas up to De Morgan duality, an equivalent system is obtained by considering only the left and right rules for positive connectives (or the ones for negative connectives only, obviously).
* [[Intuitionistic linear logic]] is the two-sided system where the right-hand side is constrained to always contain exactly one formula (with a few associated restrictions).
* Similar restrictions are used in various [[semantics]] and [[proof search]] formalisms.

=== Mix rules ===

It is quite common to consider [[Mix|mix rules]]:
<math>
\LabelRule{\rulename{Mix}_0}
\NulRule{\vdash}
\DisplayProof
\qquad
\AxRule{\Gamma \vdash \Delta}
\AxRule{\Gamma' \vdash \Delta'}
\LabelRule{\rulename{Mix}_2}
\BinRule{\Gamma,\Gamma' \vdash \Delta,\Delta'}
\DisplayProof
</math>

Reversibility and focalization

2012-08-31T21:10:58Z

Emmanuel Beffara: Creation

== Reversibility ==

{{Theorem|
Negative connectives are reversible:

* A sequent <math>\vdash\Gamma,A\parr B</math> is provable if and only if <math>\vdash\Gamma,A,B</math> is provable.
* A sequent <math>\vdash\Gamma,A\with B</math> is provable if and only if <math>\vdash\Gamma,A</math> and <math>\vdash\Gamma,B</math> are provable.
* A sequent <math>\vdash\Gamma,\bot</math> is provable if and only if <math>\vdash\Gamma</math> is provable.
* A sequent <math>\vdash\Gamma,\forall\xi A</math> is provable if and only if <math>\vdash\Gamma,A</math> is provable, for some fresh variable <math>\xi</math>.
}}

{{Proof|
We start with the case of the <math>\parr</math> connective.
If <math>\vdash\Gamma,A,B</math> is provable, then by the introduction rule for <math>\parr</math>
we know that <math>\vdash\Gamma,A\parr B</math> is provable.
For the reverse implication we proceed by induction on a proof <math>\pi</math> of
<math>\vdash\Gamma,A\parr B</math>.

* If the last rule of <math>\pi</math> is the introduction of the <math>\parr</math> in <math>A\parr B</math>, then the premiss is exacty <math>\vdash\Gamma,A,B</math> so we can conclude.
* The other case where the last rule introduces <math>A\parr B</math> is when <math>\pi</math> is an axiom rule, hence <math>\Gamma=A\orth\tens B\orth</math>. Then we can conclude with the proof
* Otherwise <math>A\parr B</math> is in the context of the last rule. If the last rule is a tensor, then <math>\pi</math> has the shape
* or the same with <math>A\parr B</math> in the conclusion of <math>\pi_2</math> instead. By induction hypothesis on <math>\pi_1</math> we get a proof <math>\pi'_1</math> of <math>\vdash\Gamma_1,A,B,C</math>, then we can conclude with the proof
* The case of the cut rule has the same structure as the tensor rule.
* In the case of the <math>\with</math> rule, we have <math>A\with B</math> in both premisses and we conclude similarly, using the induction hypothesis on both <math>\pi_1</math> and <math>\pi_2</math>.
* If <math>A\parr B</math> is in the context of a rules for <math>\parr</math>, <math>\plus</math>, <math>\bot</math> or quantifiers, or in the context of a dereliction, weakening or contraction, the situation is similar as for <math>\tens</math> except that we have only one premiss.
* If <math>A\parr B</math> is in the context of <math>\top</math> rules, we can freely change the context of the rule to get the expected one.
* The two remaining cases are if the last rule is the rule for <math>1</math> or a promotion. By the constraints these rules impose on the contexts, these cases cannot happen.
The <math>\with</math> connective is treated in the same way.
In this cases where <math>A\with B</math> is in the context of a rule with two
premisses, the premiss where this formula is not present will be duplicated,
with one copy in the premiss for <math>A</math> and one in the premiss for <math>B</math>.

The <math>\forall</math> connective is also treated similarly.
Its peculiarity is that introducing <math>\forall\xi</math> requires that <math>\xi</math> does
not appear free in the context.
For all rules with one premiss except the quantifier rules, the set of fresh
variables in the same in the premiss and the conclusion, so everything works
well.
Other rules might change the set of free variables, but problems are avoided
by choosing for <math>\xi</math> a variable that is fresh for the whole proof we are
considering.
}}

Remark that this result is proved using only commutation rules, except when the
formula is introduced by an axiom rule.
Furthermore, if axioms are applied only on atoms, this particular case
disappears.

A consequence of this fact is that, when searching for a proof of some
sequent <math>\vdash\Gamma</math>, one can always start by decomposing negative
connectives in <math>\Gamma</math> without losing provability.
Applying this result to successive connectives, we can get generalized
formulations for more complex formulas. For instance:

* <math>\vdash\Gamma,(A\parr B)\parr(B\with C)</math> is provable
* iff <math>\vdash\Gamma,A\parr B,B\with C</math> is provable
* iff <math>\vdash\Gamma,A\parr B,B</math> and <math>\vdash\Gamma,A\parr B,C</math> are provable
* iff <math>\vdash\Gamma,A,B,B</math> and <math>\vdash\Gamma,A,B,C</math> are provable
So without loss of generality, we can assume that any proof of
<math>\vdash\Gamma,(A\parr B)\parr(B\with C)</math> ends like

<math>
\AxRule{ \vdash \Gamma, A, B, B }
\UnaRule{ \vdash \Gamma, A\parr B, B }
\AxRule{ \vdash \Gamma, A, B, C }
\UnaRule{ \vdash \Gamma, A\parr B, C }
\BinRule{ \vdash \Gamma, A\parr B, B\with C }
\UnaRule{ \vdash \Gamma, (A\parr B)\parr(B\with C) }
\DisplayProof
</math>

In order to define a general statement for compound formulas, as well as an
analogous result for positive connectives, we need to give a proper status to
clusters of connectives of the same polarity.

== Generalized connectives and rules ==

{{Definition|
A ''positive generalized connective'' is a parametrized formula
<math>P[X_1,\ldots,X_n]</math> made from the variables <math>X_i</math> using the connectives
<math>\tens</math>, <math>\plus</math>, <math>\one</math>, <math>\zero</math>.

A ''negative generalized connective'' is a parametrized formula
<math>N[X_1,\ldots,X_n]</math> made from the variables <math>X_i</math> using the connectives
<math>\parr</math>, <math>\with</math>, <math>\bot</math>, <math>\top</math>.

If <math>C[X_1,\ldots,X_n]</math> is a generalized connectives (of any polarity), the
''dual'' of <math>C</math> is the connective <math>C^*</math> such that
<math>C^*[X_1\orth,\ldots,X_n\orth]=C[X_1,\ldots,X_n]\orth</math>.
}}

It is clear that dualization of generalized connectives is involutive and
exchanges polarities.
We do not include quantifiers in this definition, mainly for simplicity.
Extending the notion to quantifiers would only require taking proper care of
the scope of variables.

Sequent calculus provides introduction rules for each connective. Negative
connectives have one rule, positive ones may have any number of rules, namely
2 for <math>\plus</math> and 0 for <math>\zero</math>. We can derive introduction rules for the
generalized connectives by combining the different possible introduction rules
for each of their components.

Considering the previous example
<math>N[X_1,X_2,X_3]=(X_1\parr X_2)\parr(X_2\with X_3)</math>, we can derive an
introduction rule for <math>N</math> as

<math>
\AxRule{ \vdash \Gamma, X_1, X_2, X_2 }
\UnaRule{ \vdash \Gamma, X_1\parr X_2, X_2 }
\AxRule{ \vdash \Gamma, X_1, X_2, X_3 }
\UnaRule{ \vdash \Gamma, X_1\parr X_2, X_3 }
\BinRule{ \vdash \Gamma, X_1\parr X_2, X_2\with X_3 }
\UnaRule{ \vdash \Gamma, (X_1\parr X_2)\parr(X_2\with X_3) }
\DisplayProof
\quad\text{or}\quad
\AxRule{ \vdash \Gamma, X_1, X_2, X_2 }
\AxRule{ \vdash \Gamma, X_1, X_2, X_3 }
\BinRule{ \vdash \Gamma, X_1, X_2, X_2\with X_3 }
\UnaRule{ \vdash \Gamma, X_1\parr X_2, X_2\with X_3 }
\UnaRule{ \vdash \Gamma, (X_1\parr X_2)\parr(X_2\with X_3) }
\DisplayProof
</math>

but these rules only differ by the commutation of independent rules.
In particular, their premisses are the same.
The dual of <math>N</math> is <math>P[X_1,X_2,X_3]=(X_1\tens X_2)\tens(X_2\plus X_3)</math>, for
which we have two possible derivations:

<math>
\AxRule{ \vdash \Gamma_1, X_1 }
\AxRule{ \vdash \Gamma_2, X_2 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, X_1\tens X_2 }
\AxRule{ \vdash \Gamma_3, X_2 }
\UnaRule{ \vdash \Gamma_3, X_2\plus X_3 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, \Gamma_3, (X_1\tens X_2)\tens(X_2\plus X_3) }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma_1, X_1 }
\AxRule{ \vdash \Gamma_2, X_2 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, X_1\tens X_2 }
\AxRule{ \vdash \Gamma_3, X_3 }
\UnaRule{ \vdash \Gamma_3, X_2\plus X_3 }
\BinRule{ \vdash \Gamma_1, \Gamma_2, \Gamma_3, (X_1\tens X_2)\tens(X_2\plus X_3) }
\DisplayProof
</math>

These are actually different, in particular their premisses differ.
Each possible derivation corresponds to the choice of one side of the <math>\plus</math>
connective.

We can remark that the branches of the negative inference precisely correspond
to the possible positive inferences:

* the first branch of the negative inference has a premiss <math>X_1,X_2,X_2</math> and the first positive inference has three premisses, holding <math>X_1</math>, <math>X_2</math> and <math>X_2</math> respectively.
* the second branch of the negative inference has a premiss <math>X_1,X_2,X_3</math> and the second positive inference has three premisses, holding <math>X_1</math>, <math>X_2</math> and <math>X_3</math> respectively.
This phenomenon extends to all generalized connectives.

{{Definition|
The ''branching'' of a generalized connective <math>P[X_1,\ldots,X_n]</math> is the
multiset <math>\mathcal{I}_P</math> of multisets over <math>\{1,\ldots,n\}</math> defined
inductively as

{|
|-
| <math>
\mathcal{I}_{P\tens Q} </math>
| <math>:= [ I+J \mid I\in\mathcal{I}_P, J\in\mathcal{I}_Q ]
</math>
| <math> \mathcal{I}_{P\plus Q} </math>
| <math>:= \mathcal{I}_P + \mathcal{I}_Q
</math>
| <math> \mathcal{I}_\one </math>
| <math>:= [[]]
</math>
| <math> \mathcal{I}_\zero </math>
| <math>:= []
</math>
| <math> \mathcal{I}_{X_i} </math>
| <math>:= [[i]]
</math>
|}

The branching of a negative generalized connective is the branching of its
dual. Elements of a branching are called branches.
}}

In the example above, the branching will be <math>[[1,2,2],[1,2,3]]</math>, which
corresponds to the granches of the negative inference and to the cases of
positive inference.

{{Definition|
Let <math>\mathcal{I}</math> be a branching.
Write <math>\mathcal{I}</math> as <math>[I_1,\ldots,I_k]</math> and write each <math>I_j</math> as
<math>[i_{j,1},\ldots,i_{j,\ell_j}]</math>.
The derived rule for a negative generalized connective <math>N</math> with
branching <math>\mathcal{I}</math> is

<math>
\AxRule{ \vdash \Gamma, A_{i_{1,1}}, \ldots, A_{i_{1,\ell_1}} }
\AxRule{ \cdots }
\AxRule{ \vdash \Gamma, A_{i_{k,1}}, \ldots, A_{i_{k,\ell_k}} }
\LabelRule{N}
\TriRule{ \vdash \Gamma, N[A_1,\ldots,A_n] }
\DisplayProof
</math>

For each branch <math>I=[i_1,\ldots,i_\ell]</math> of a positive generalized connective
<math>P</math>, the derived rule for branch <math>I</math> of <math>P</math> is

<math>
\AxRule{ \vdash \Gamma_1, A_{i_1} }
\AxRule{ \cdots }
\AxRule{ \vdash \Gamma_\ell, A_{i_\ell} }
\LabelRule{P_I}
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, P[A_1,\ldots,A_n] }
\DisplayProof
</math>
}}

The reversibility property of negative connectives can be rephrased in a
generalized way as

{{Theorem|
Let <math>N</math> be a negative generalized connective. A sequent
<math>\vdash\Gamma,N[A_1,\ldots,A_n]</math> is provable if and only if, for each
<math>[i_1,\ldots,i_k]\in\mathcal{I}_N</math>, the sequent
<math>\vdash\Gamma,A_{i_1},\ldots,A_{i_k}</math> is provable.
}}

The corresponding property for positive connectives is the focalization
property, defined in the next section.

== Focalization ==

{{Definition|
A formula is ''positive'' if it has a main connective among
<math>\tens</math>, <math>\plus</math>, <math>\one</math>, <math>\zero</math>.
It is called ''negative'' if it has a main connective among
<math>\parr</math>, <math>\with</math>, <math>\bot</math>, <math>\top</math>.
It is called ''neutral'' if it is neither positive nor negative.
}}

If we extended the theory to include quantifiers in generalized connectives,
then the definition of positive and negative formulas would be extended to
include them too.

{{Definition|
A proof <math>\pi\vdash\Gamma,A</math> is said to be ''focalized'' on <math>A</math> if it has
the shape

<math>
\AxRule{ \pi_1 \vdash \Gamma_1, A_{i_1} }
\AxRule{ \cdots }
\AxRule{ \pi_\ell \vdash \Gamma_\ell, A_{i_\ell} }
\LabelRule{P_{[i_1,\ldots,i_\ell]}}
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, P[A_1,\ldots,A_n] }
\DisplayProof
</math>

where <math>P</math> is a positive generalized connective, the <math>A_i</math> ar non-positive
and <math>A=P[A_1,\ldots,A_n]</math>. The formula <math>A</math> is called the ''focus'' of the
proof <math>\pi</math>.
}}

In other words, a proof is focalized on a conclusion <math>A</math> if its last rules
build <math>A</math> from some of its non-positive subformulas in one cluster of
inferences. Note that this notion only makes sense for a conclusion made only
of positive formulas, since a proof is obviously focalized on any of its
non-positive conclusions, using the degenerate generalized connective
<math>P[X]=X</math>.

{{Theorem|
A sequent <math>\vdash\Gamma</math> is cut-free provable if and only if it is provable
by a cut-free focalized proof.
}}

{{Proof|
We reason by induction on a proof <math>\pi</math> of <math>\Gamma</math>.
As noted above, the result trivially holds if <math>\Gamma</math> has a non-positive
formula.
We can thus assume that <math>\Gamma</math> contains only positive formulas and reason
on the nature of the last rule, which is necessarily the introduction of a
positive connective (it cannot be an axiom rule because an axiom always has
at least on non-positive conclusion).

Suppose that the last rule of <math>\pi</math> introduces a tensor, so that <math>\pi</math> is

<math>
\AxRule{ \rho \vdash \Gamma, A }
\AxRule{ \theta \vdash \Delta, B }
\BinRule{ \vdash \Gamma, \Delta, A\tens B }
\DisplayProof
</math>

By induction hypothesis, there are focalized proofs <math>\rho'\vdash\Gamma,A</math>
and <math>\theta'\vdash\Delta,B</math>.
If <math>A</math> is the focus of <math>\rho'</math> and <math>B</math> is the focus of <math>\theta'</math>, then the
proof

<math>
\AxRule{ \rho' \vdash \Gamma, A }
\AxRule{ \theta' \vdash \Delta, B }
\BinRule{ \vdash \Gamma, \Delta, A\tens B }
\DisplayProof
</math>

is focalized on <math>A\tens B</math>, so we can conclude.
Otherwise, one of the two proofs is focalized on another conclusion.
Without loss of generality, suppose that <math>\rho'</math> is not focalized on <math>A</math>.
Then it decomposes as

<math>
\AxRule{ \rho_1 \vdash \Gamma_1, C_{i_1} }
\AxRule{ \cdots }
\AxRule{ \rho_\ell \vdash \Gamma_\ell, C_{i_\ell} }
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, P[C_1,\ldots,C_n] }
\DisplayProof
</math>

where the <math>C_i</math> are not positive and <math>A</math> belongs to some context <math>\Gamma_j</math>
that we will write <math>\Gamma'_j,A</math>.
Then we can conclude with the proof

<math>
\AxRule{ \rho_1 \vdash \Gamma_1, C_{i_1} \quad\cdots }
\AxRule{ \rho_j \vdash \Gamma_j, A, C_{i_j} }
\AxRule{ \theta \vdash \Delta, B }
\BinRule{ \vdash \Gamma_j, \Delta, A\tens B, C_{i_j} }
\AxRule{ \cdots\quad \rho_\ell \vdash \Gamma_\ell, C_{i_\ell} }
\TriRule{ \vdash \Gamma_1, \ldots, \Gamma_\ell, \Delta, A\tens B, P[C_1,\ldots,C_n] }
\DisplayProof
</math>

which is focalized on <math>P[C_1,\ldots,C_n]</math>.

If the last rule of <math>\pi</math> introduces a <math>\plus</math>, we proceed the same way
except that there is only one premiss.
If the last rule of <math>\pi</math> introduces a <math>\one</math>, then it is the only rule of
<math>\pi</math>, which is thus focalized on this <math>\one</math>.
}}

As in the reversibility theorem, this proof only makes use of commutation of
independent rules.

These results say nothing about exponential modalities, because they respect
neither reversibility nor focalization. However, if we consider the fragment
of LL which consists only of multiplicative and additive connectives, we can
restrict the proof rules to enforce focalization without loss of
expressiveness.

Sequent calculus

2010-12-09T14:30:46Z

Emmanuel Beffara: /* Sequents and proofs */

This article presents the language and sequent calculus of second-order
linear logic and the basic properties of this sequent calculus.
The core of the article uses the two-sided system with negation as a proper
connective; the [[#One-sided sequent calculus|one-sided system]], often used
as the definition of linear logic, is presented at the end of the page.

== Formulas ==

Atomic formulas, written <math>\alpha,\beta,\gamma</math>, are predicates of
the form <math>p(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms
from some first-order language.
The predicate symbol <math>p</math> may be either a predicate constant or a
second-order variable.
By convention we will write first-order variables as <math>x,y,z</math>,
second-order variables as <math>X,Y,Z</math>, and <math>\xi</math> for a
variable of arbitrary order (see [[Notations]]).

Formulas, represented by capital letters <math>A</math>, <math>B</math>,
<math>C</math>, are built using the following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>A\orth</math>
| negation
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists \xi.A</math>
| there exists
| <math>\forall \xi.A</math>
| for all
| quantifiers
|}

Each line (except the first one) corresponds to a particular class of
connectives, and each class consists in a pair of connectives.
Those in the left column are called [[positive formula|positive]] and those in
the right column are called [[negative formula|negative]].
The ''tensor'' and ''with'' connectives are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally read
''of course <math>A</math>'' for <math>\oc A</math> and ''why not
<math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic, as
<math>A\limp B:=A\orth\parr B</math>.

Free and bound variables and first-order substitution are defined in the
standard way.
Formulas are always considered up to renaming of bound names.
If <math>A</math> is a formula, <math>X</math> is a second-order variable and
<math>B[x_1,\ldots,x_n]</math> is a formula with variables <math>x_i</math>,
then the formula <math>A[B/X]</math> is <math>A</math> where every atom
<math>X(t_1,\ldots,t_n)</math> is replaced by <math>B[t_1,\ldots,t_n]</math>.

== Sequents and proofs ==

A sequent is an expression <math>\Gamma\vdash\Delta</math> where
<math>\Gamma</math> and <math>\Delta</math> are finite multisets of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation
<math>\wn\Gamma</math> represents the multiset
<math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:
* Identity group: <math>
\LabelRule{\rulename{axiom}}
\NulRule{ A \vdash A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{\rulename{cut}}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>
* Negation: <math>
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\LabelRule{n_L}
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\LabelRule{n_R}
\DisplayProof
</math>
* Multiplicative group:
** tensor: <math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
</math>
** par: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma', B \vdash \Delta' }
\LabelRule{ \parr_L }
\BinRule{ \Gamma, \Gamma', A \parr B \vdash \Delta, \Delta' }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, B, \Delta }
\LabelRule{ \parr_R }
\UnaRule{ \Gamma \vdash A \parr B, \Delta }
\DisplayProof
</math>
** one: <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>
** bottom: <math>
\LabelRule{ \bot_L }
\NulRule{ \bot \vdash }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \bot_R }
\UnaRule{ \Gamma \vdash \bot, \Delta }
\DisplayProof
</math>
* Additive group:
** plus: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
</math>
** with: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ \with_{L1} }
\UnaRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \with_{L2} }
\UnaRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \with_R }
\BinRule{ \Gamma \vdash A \with B, \Delta }
\DisplayProof
</math>
** zero: <math>
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>
** top: <math>
\LabelRule{ \top_R }
\NulRule{ \Gamma \vdash \top, \Delta }
\DisplayProof
</math>
* Exponential group:
** of course: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B ,\wn B_1, \ldots, \wn B_m }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B ,\wn B_1, \ldots, \wn B_m }
\DisplayProof
</math>
** why not: <math>
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ d_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \wn A, \wn A, \Delta }
\LabelRule{ c_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \oc A_1, \ldots, \oc A_n, A \vdash \wn B_1, \ldots, \wn B_m }
\LabelRule{ \wn_L }
\UnaRule{ \oc A_1, \ldots, \oc A_n, \wn A \vdash \wn B_1, \ldots, \wn B_m }
\DisplayProof
</math>
* Quantifier group (in the <math>\exists_L</math> and <math>\forall_R</math> rules, <math>\xi</math> must not occur free in <math>\Gamma</math> or <math>\Delta</math>):
** there exists: <math>
\AxRule{ \Gamma , A \vdash \Delta }
\LabelRule{ \exists_L }
\UnaRule{ \Gamma, \exists\xi.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A[t/x] }
\LabelRule{ \exists^1_R }
\UnaRule{ \Gamma \vdash \Delta, \exists x.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A[B/X] }
\LabelRule{ \exists^2_R }
\UnaRule{ \Gamma \vdash \Delta, \exists X.A }
\DisplayProof
</math>
** for all: <math>
\AxRule{ \Gamma, A[t/x] \vdash \Delta }
\LabelRule{ \forall^1_L }
\UnaRule{ \Gamma, \forall x.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, A[B/X] \vdash \Delta }
\LabelRule{ \forall^2_L }
\UnaRule{ \Gamma, \forall X.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A }
\LabelRule{ \forall_R }
\UnaRule{ \Gamma \vdash \Delta, \forall\xi.A }
\DisplayProof
</math>

The left rules for ''of course'' and right rules for ''why not'' are called
''dereliction'', ''weakening'' and ''contraction'' rules.
The right rule for ''of course'' and the left rule for ''why not'' are called
''promotion'' rules.
Note the fundamental fact that there are no contraction and weakening rules
for arbitrary formulas, but only for the formulas starting with the
<math>\wn</math> modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An alternative presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rules:
: <math>
\AxRule{ \Gamma_1, A, B, \Gamma_2 \vdash \Delta }
\LabelRule{\rulename{exchange}_L}
\UnaRule{ \Gamma_1, B, A, \Gamma_2 \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta_1, A, B, \Delta_2 }
\LabelRule{\rulename{exchange}_R}
\UnaRule{ \Gamma \vdash \Delta_1, B, A, \Delta_2 }
\DisplayProof
</math>

== Equivalences ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent,
written <math>A\linequiv B</math>, if both implications <math>A\limp B</math>
and <math>B\limp A</math> are provable.
Equivalently, <math>A\linequiv B</math> if both <math>A\vdash B</math> and
<math>B\vdash A</math> are provable.
Another formulation of <math>A\linequiv B</math> is that, for all
<math>\Gamma</math> and <math>\Delta</math>, <math>\Gamma\vdash\Delta,A</math>
is provable if and only if <math>\Gamma\vdash\Delta,B</math> is provable.

Two related notions are [[isomorphism]] (stronger than equivalence) and
[[equiprovability]] (weaker than equivalence).

=== De Morgan laws ===

Negation is involutive:
: <math>A\linequiv A\biorth</math>
Duality between connectives:
{|
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>\linequiv A\orth \parr B\orth </math>
|width=30|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>\linequiv A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>\linequiv \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>\linequiv \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>\linequiv A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>\linequiv A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>\linequiv \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>\linequiv \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>\linequiv \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>\linequiv \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists \xi.A )\orth </math>
| <math>\linequiv \forall \xi.( A\orth ) </math>
|
|align="right"| <math> ( \forall \xi.A )\orth </math>
| <math>\linequiv \exists \xi.( A\orth ) </math>
|}

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality:
*: <math>
A \tens (B \tens C) \linequiv (A \tens B) \tens C </math> &emsp; <math>
A \tens B \linequiv B \tens A </math> &emsp; <math>
A \tens \one \linequiv A </math>
*: <math>
A \parr (B \parr C) \linequiv (A \parr B) \parr C </math> &emsp; <math>
A \parr B \linequiv B \parr A </math> &emsp; <math>
A \parr \bot \linequiv A </math>
*: <math>
A \plus (B \plus C) \linequiv (A \plus B) \plus C </math> &emsp; <math>
A \plus B \linequiv B \plus A </math> &emsp; <math>
A \plus \zero \linequiv A </math>
*: <math>
A \with (B \with C) \linequiv (A \with B) \with C </math> &emsp; <math>
A \with B \linequiv B \with A </math> &emsp; <math>
A \with \top \linequiv A </math>
* Idempotence of additives:
*: <math>
A \plus A \linequiv A </math> &emsp; <math>
A \with A \linequiv A </math>
* Distributivity of multiplicatives over additives:
*: <math>
A \tens (B \plus C) \linequiv (A \tens B) \plus (A \tens C) </math> &emsp; <math>
A \tens \zero \linequiv \zero </math>
*: <math>
A \parr (B \with C) \linequiv (A \parr B) \with (A \parr C) </math> &emsp; <math>
A \parr \top \linequiv \top </math>
* Defining property of exponentials:
*: <math>
\oc(A \with B) \linequiv \oc A \tens \oc B </math> &emsp; <math>
\oc\top \linequiv \one </math>
*: <math>
\wn(A \plus B) \linequiv \wn A \parr \wn B </math> &emsp; <math>
\wn\zero \linequiv \bot </math>
* Monoidal structure of exponentials:
*: <math>
\oc A \tens \oc A \linequiv \oc A </math> &emsp; <math>
\oc \one \linequiv \one </math>
*: <math>
\wn A \parr \wn A \linequiv \wn A </math> &emsp; <math>
\wn \bot \linequiv \bot </math>
* Digging:
*: <math>
\oc\oc A \linequiv \oc A </math> &emsp; <math>
\wn\wn A \linequiv \wn A </math>
* Other properties of exponentials:
*: <math>
\oc\wn\oc\wn A \linequiv \oc\wn A </math> &emsp; <math>
\oc\wn \one \linequiv \one </math>
*: <math>
\wn\oc\wn\oc A \linequiv \wn\oc A </math> &emsp; <math>
\wn\oc \bot \linequiv \bot </math>
These properties of exponentials lead to the [[lattice of exponential modalities]].
* Commutation of quantifiers (<math>\zeta</math> does not occur in <math>A</math>):
*: <math>
\exists \xi. \exists \psi. A \linequiv \exists \psi. \exists \xi. A </math> &emsp; <math>
\exists \xi.(A \plus B) \linequiv \exists \xi.A \plus \exists \xi.B </math> &emsp; <math>
\exists \zeta.(A\tens B) \linequiv A\tens\exists \zeta.B </math> &emsp; <math>
\exists \zeta.A \linequiv A </math>
*: <math>
\forall \xi. \forall \psi. A \linequiv \forall \psi. \forall \xi. A </math> &emsp; <math>
\forall \xi.(A \with B) \linequiv \forall \xi.A \with \forall \xi.B </math> &emsp; <math>
\forall \zeta.(A\parr B) \linequiv A\parr\forall \zeta.B </math> &emsp; <math>
\forall \zeta.A \linequiv A </math>

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:
* <math> \zero \linequiv \forall X.X </math>
* <math> \one \linequiv \forall X.(X \limp X) </math>
* <math> A \plus B \linequiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X) </math>
The constants <math>\top</math> and <math>\bot</math> and the connective
<math>\with</math> can be defined by duality.

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

== Properties of proofs ==

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\Gamma\vdash\Delta</math>, there is a proof of
<math>\Gamma\vdash\Delta</math> if and only if there is a proof of
<math>\Gamma\vdash\Delta</math> that does not use the cut rule.}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas of its immediate subformulas:
* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,
* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,
* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas have no immediate subformula,
* the immediate subformulas of <math>\exists x.A</math> and <math>\forall x.A</math> are all the <math>A[t/x]</math> for all first-order terms <math>t</math>,
* the immediate subformulas of <math>\exists X.A</math> and <math>\forall X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math> (with the appropriate number of parameters).}}

{{Theorem|title=subformula property|
A sequent <math>\Gamma\vdash\Delta</math> is provable if and only if it is the conclusion of a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math> and <math>\Delta</math>.}}
{{Proof|By the cut elimination theorem, if a sequent is provable, then it is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its
negation <math>A\orth</math>; it is impossible to prove <math>\zero</math> or
<math>\bot</math>.}}
{{Proof|If a sequent is provable, then it is the conclusion of a cut-free proof.
In each rule except the cut rule, there is at least one formula in conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.
The other properties are immediate consequences: if <math>\vdash A\orth</math>
and <math>\vdash A</math> are provable, then by the left negation rule
<math>A\orth\vdash</math> is provable, and by the cut rule one gets empty
conclusion, which is not possible.
As particular cases, since <math>\one</math> and <math>\top</math> are
provable, <math>\bot</math> and <math>\zero</math> are not, since they are
equivalent to <math>\one\orth</math> and <math>\top\orth</math>
respectively.}}

=== Expansion of identities ===

Let us write <math>\pi:\Gamma\vdash\Delta</math> to signify that
<math>\pi</math> is a proof with conclusion <math>\Gamma\vdash\Delta</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the
same conclusion as <math>\pi</math> in which the axiom rule is only used with
atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.}}
{{Proof|It suffices to prove that for every formula <math>A</math>, the sequent
<math>A\vdash A</math> has a cut-free proof in which the axiom rule is used
only for atomic formulas.
We prove this by induction on <math>A</math>.
* If <math>A</math> is atomic, then <math>A\vdash A</math> is an instance of the atomic axiom rule.
* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 : A_1 \vdash A_1 }
\AxRule{ \pi_2 : A_2 \vdash A_2 }
\LabelRule{ \tens_R }
\BinRule{ A_1, A_2 \vdash A_1 \tens A_2 }
\LabelRule{ \tens_L }
\UnaRule{ A_1 \tens A_2 \vdash A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* If <math>A=A_1\parr A_2</math> then we have <math>
\AxRule{ \pi_1 : A_1 \vdash A_1 }
\AxRule{ \pi_2 : A_2 \vdash A_2 }
\LabelRule{ \parr_L }
\BinRule{ A_1 \parr A_2 \vdash A_1, A_2 }
\LabelRule{ \parr_R }
\UnaRule{ A_1 \parr A_2 \vdash A_1 \parr A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* All other connectives follow the same pattern.}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if
* for every proof <math>\pi:\Gamma\vdash\Delta,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,
* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are reversible.}}
{{Proof|Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only applied to atomic formulas.
Then each top-level connective is introduced either by its associated (left or
right) rule or in an instance of the <math>\zero_L</math> or
<math>\top_R</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\Gamma\vdash\Delta,A\parr
B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr_R</math> rule (not
necessarily the last rule in <math>\pi</math>), then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction on <math>\pi</math>).
If it is introduced in the context of a <math>\zero_L</math> or
<math>\top_R</math> rule, then this rule can be changed so that
<math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the
expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a
<math>\bot_R</math> rule, then remove this rule to get a proof of
<math>\vdash\Gamma</math>, if it is introduced by a <math>\zero_L</math> or
<math>\top_R</math> rule, remove the <math>\bot</math> from this rule, then
apply the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof
<math>\pi:\Gamma\vdash\Delta,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is
applied in a context like

<math>
\AxRule{ \pi_1 \Gamma' \vdash \Delta', A }
\AxRule{ \pi_2 \Gamma' \vdash \Delta', B }
\LabelRule{ \with }
\BinRule{ \Gamma' \vdash \Delta', A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except
as context), if we replace this step by <math>\pi_1</math> in <math>\pi</math>
we finally get a proof <math>\pi'_1:\Gamma\vdash\Delta,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof
<math>\pi'_2:\Gamma\vdash\Delta,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final
<math>\with</math> rule we finally get the expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math>
rule is solved as before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as
an instance of the <math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math>, consider a proof
<math>\pi:\Gamma\vdash\Delta,\forall\xi.A</math>.
Up to renaming, we can assume that <math>\xi</math> occurs free only above the
rule that introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we
remove this rule, we can check that we get a proof of
<math>\Gamma\vdash\Delta,A</math> on which we can finally apply the
<math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math>
rule is solved as before.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the
original proof.}}

== One-sided sequent calculus ==

The sequent calculus presented above is very symmetric: for every left
introduction rule, there is a right introduction rule for the dual connective
that has the exact same structure.
Moreover, because of the involutivity of negation, a sequent
<math>\Gamma,A\vdash\Delta</math> is provable if and only if the sequent
<math>\Gamma\vdash A\orth,\Delta</math> is provable.
From these remarks, we can define an equivalent one-sided sequent calculus:
* Formulas are considered up to De Morgan duality. Equivalently, one can consider that negation is not a connective but a syntactically defined operation on formulas. In this case, negated atoms <math>\alpha\orth</math> must be considered as another kind of atomic formulas.
* Sequents have the form <math>\vdash\Gamma</math>.
The inference rules are essentially the same except that the left hand side of
sequents is kept empty:
* Identity group:
*: <math>
\LabelRule{\rulename{axiom}}
\NulRule{ \vdash A\orth, A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{\rulename{cut}}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>
* Multiplicative group:
*: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>
* Additive group:
*: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>
* Exponential group:
*: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>
* Quantifier group (in the <math>\forall</math> rule, <math>\xi</math> must not occur free in <math>\Gamma</math>):
*: <math>
\AxRule{ \vdash \Gamma, A[t/x] }
\LabelRule{ \exists^1 }
\UnaRule{ \vdash \Gamma, \exists x.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists^2 }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall \xi.A }
\DisplayProof
</math>

{{Theorem|A two-sided sequent <math>\Gamma\vdash\Delta</math> is provable if
and only if the sequent <math>\vdash\Gamma\orth,\Delta</math> is provable in
the one-sided system.}}

The one-sided system enjoys the same properties as the two-sided one,
including cut elimination, the subformula property, etc.
This formulation is often used when studying proofs because it is much lighter
than the two-sided form while keeping the same expressiveness.
In particular, [[proof-nets]] can be seen as a quotient of one-sided sequent
calculus proofs under commutation of rules.

== Variations ==

=== Exponential rules ===

* The promotion rule, on the right-hand side for example,
<math>
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B, \wn B_1, \ldots, \wn B_m }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B, \wn B_1, \ldots, \wn B_m }
\DisplayProof
</math>
can be replaced by a ''multi-functorial'' promotion rule
<math>
\AxRule{ A_1, \ldots, A_n \vdash B, B_1, \ldots, B_m }
\LabelRule{ \oc_R \rulename{mf}}
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B, \wn B_1, \ldots, \wn B_m }
\DisplayProof
</math>
and a ''digging'' rule
<math>
\AxRule{ \Gamma \vdash \wn\wn A, \Delta }
\LabelRule{ \wn\wn}
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math>,
without modifying the provability.

Note that digging violates the subformula property.

* In presence of the digging rule <math>
\AxRule{ \Gamma \vdash \wn\wn A, \Delta }
\LabelRule{ \wn\wn}
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math>, the multiplexing rule <math>
\AxRule{\Gamma\vdash A^{(n)},\Delta}
\LabelRule{\rulename{mplex}}
\UnaRule{\Gamma\vdash \wn A,\Delta}
\DisplayProof
</math> (where <math>A^{(n)}</math> stands for n occurrences of formula <math>A</math>) is equivalent (for provability) to the triple of rules: contraction, weakening, dereliction.

=== Non-symmetric sequents ===

The same remarks that lead to the definition of the one-sided calculus can
lead the definition of other simplified systems:
* A one-sided variant with sequents of the form <math>\Gamma\vdash</math> could be defined.
* When considering formulas up to De Morgan duality, an equivalent system is obtained by considering only the left and right rules for positive connectives (or the ones for negative connectives only, obviously).
* [[Intuitionistic linear logic]] is the two-sided system where the right-hand side is constrained to always contain exactly one formula (with a few associated restrictions).
* Similar restrictions are used in various [[semantics]] and [[proof search]] formalisms.

Geometry of interaction

2010-04-29T10:19:56Z

Emmanuel Beffara: references fixed

The ''geometry of interaction'', GoI in short, was defined in the early nineties by Girard as an interpretation of linear logic into operators algebra: formulae were interpreted by Hilbert spaces and proofs by partial isometries.

This was a striking novelty as it was the first time that a mathematical model of logic (lambda-calculus) didn't interpret a proof of <math>A\limp B</math> as a morphism ''from'' <math>A</math> ''to'' <math>B</math><ref>to be precise one should say from ''the space interpreting'' <math>A</math> to the space interpreting'' <math>B</math></ref>and proof composition (cut rule) as the composition of morphisms. Rather the proof was interpreted as an operator acting ''on'' <math>A\limp B</math>, that is a morphism from <math>A\limp B</math> to <math>A\limp B</math>. For proof composition the problem was then, given an operator on <math>A\limp B</math> and another one on <math>B\limp C</math> to construct a new operator on <math>A\limp C</math>. This problem was solved by the ''execution formula'' that bares some formal analogies with Kleene's formula for recursive functions. For this reason GoI was claimed to be an ''operational semantics'', as opposed to traditionnal [[Semantics|denotational semantics]].

The first instance of the GoI was restricted to the <math>MELL</math> fragment of linear logic (Multiplicative and Exponential fragment) which is enough to encode lambda-calculus. Since then Girard proposed several improvements: firstly the extension to the additive connectives known as ''Geometry of Interaction 3'' and more recently a complete reformulation using Von Neumann algebras that allows to deal with some aspects of [[Light linear logics|implicit complexity]]

The GoI has been a source of inspiration for various authors. Danos and Regnier have reformulated the original model exhibiting its combinatorial nature using a theory of reduction of paths in proof-nets and showing the link with abstract machines; in particular the execution formula appears as the composition of two automata that interact one with the other through their common interface. Also the execution formula has rapidly been understood as expressing the composition of strategies in game semantics. It has been used in the theory of sharing reduction for lambda-calculus in the Abadi-Gonthier-Lévy reformulation and simplification of Lamping's representation of sharing. Finally the original GoI for the <math>MELL</math> fragment has been reformulated in the framework of traced monoidal categories following an idea originally proposed by Joyal.

= The Geometry of Interaction as operators =

The original construction of GoI by Girard follows a general pattern already mentionned in [[coherent semantics]] under the name ''symmetric reducibility'' and that was first put to use in [[phase semantics]]. First set a general space <math>P</math> called the ''proof space'' because this is where the interpretations of proofs will live. Make sure that <math>P</math> is a (not necessarily commutative) monoid. In the case of GoI, the proof space is a subset of the space of bounded operators on <math>\ell^2</math>.

Second define a particular subset of <math>P</math> that will be denoted by <math>\bot</math>; then derive a duality on <math>P</math>: for <math>u,v\in P</math>, <math>u</math> and <math>v</math> are dual<ref>In modern terms one says that <math>u</math> and <math>v</math> are ''polar''.</ref>iff <math>uv\in\bot</math>.

For the GoI, two dualities have proved to work; we will consider the first one: nilpotency, ''ie'', <math>\bot</math> is the set of nilpotent operators in <math>P</math>. Let us explicit this: two operators <math>u</math> and <math>v</math> are dual if there is a nonegative integer <math>n</math> such that <math>(uv)^n = 0</math>. Note in particular that <math>uv\in\bot</math> iff <math>vu\in\bot</math>.

When <math>X</math> is a subset of <math>P</math> define <math>X\orth</math> as the set of elements of <math>P</math> that are dual to all elements of <math>X</math>:
: <math>X\orth = \{u\in P, \forall v\in X, uv\in\bot\}</math>.

This construction has a few properties that we will use without mention in the sequel. Given two subsets <math>X</math> and <math>Y</math> of <math>P</math> we have:
* if <math>X\subset Y</math> then <math>Y\orth\subset X</math>;
* <math>X\subset X\biorth</math>;
* <math>X\triorth = X\orth</math>.

Last define a ''type'' as a subset <math>T</math> of the proof space that is equal to its bidual: <math>T = T\biorth</math>. This means that <math>u\in T</math> iff for all operator <math>v\in T\orth</math>, that is such that <math>u'v\in\bot</math> for all <math>u'\in T</math>, we have <math>uv\in\bot</math>.

The real work<ref>The difficulty is to find the right duality that will make logical operations interpretable. General conditions that allows to achieve this have been formulated by Hyland and Schalk thanks to their theory of ''double gluing''.</ref>is now to interpret logical operations, that is to associate a type to each formula, an object to each proof and show the ''adequacy lemma'': if <math>u</math> is the interpretation of a proof of the formula <math>A</math> then <math>u</math> belongs to the type associated to <math>A</math>.

== Preliminaries ==

We will denote by <math>H</math> the Hilbert space <math>\ell^2(\mathbb{N})</math> of sequences <math>(x_n)_{n\in\mathbb{N}}</math> of complex numbers such that the series <math>\sum_{n\in\mathbb{N}}|x_n|^2</math> converges. If <math>x = (x_n)_{n\in\mathbb{N}}</math> and <math>y = (y_n)_{n\in\mathbb{N}}</math> are two vectors of <math>H</math> their ''scalar product'' is:
: <math>\langle x, y\rangle = \sum_{n\in\mathbb{N}} x_n\bar y_n</math>.

Two vectors of <math>H</math> are ''othogonal'' if their scalar product is nul. We will say that two subspaces are ''disjoint'' when any two vectors taken in each subspace are orthorgonal. Note that this notion is different from the set theoretic one, in particular two disjoint subspaces always have exactly one vector in common: <math>0</math>.

The ''norm'' of a vector is the square root of the scalar product with itself:
: <math>\|x\| = \sqrt{\langle x, x\rangle}</math>.

Let us denote by <math>(e_k)_{k\in\mathbb{N}}</math> the canonical ''hilbertian basis'' of <math>H</math>: <math>e_k = (\delta_{kn})_{n\in\mathbb{N}}</math> where <math>\delta_{kn}</math> is the Kroenecker symbol: <math>\delta_{kn}=1</math> if <math>k=n</math>, <math>0</math> otherwise. Thus if <math>x=(x_n)_{n\in\mathbb{N}}</math> is a sequence in <math>H</math> we have:
: <math> x = \sum_{n\in\mathbb{N}} x_ne_n</math>.

An ''operator'' on <math>H</math> is a ''continuous'' linear map from <math>H</math> to <math>H</math>.<ref>Continuity is equivalent to the fact that operators are ''bounded'', which means that one may define the ''norm'' of an operator <math>u</math> as the sup on the unit ball of the norms of its values:
: <math>\|u\| = \sup_{\{x\in H,\, \|x\| = 1\}}\|u(x)\|</math>.</ref>The set of (bounded) operators is denoted by <math>\mathcal{B}(H)</math>.

The ''range'' or ''codomain'' of the operator <math>u</math> is the set of images of vectors; the ''kernel'' of <math>u</math> is the set of vectors that are anihilated by <math>u</math>; the ''domain'' of <math>u</math> is the set of vectors orthogonal to the kernel, ''ie'', the maximal subspace disjoint with the kernel:

* <math>\mathrm{Codom}(u) = \{u(x),\, x\in H\}</math>;
* <math>\mathrm{Ker}(u) = \{x\in H,\, u(x) = 0\}</math>;
* <math>\mathrm{Dom}(u) = \{x\in H,\, \forall y\in\mathrm{Ker}(u), \langle x, y\rangle = 0\}</math>.

These three sets are closed subspaces of <math>H</math>.

The ''adjoint'' of an operator <math>u</math> is the operator <math>u^*</math> defined by <math>\langle u(x), y\rangle = \langle x, u^*(y)\rangle</math> for any <math>x,y\in H</math>. Adjointness is well behaved w.r.t. composition of operators:
: <math>(uv)^* = v^*u^*</math>.

A ''projector'' is an idempotent operator of norm <math>0</math> (the projector
on the null subspace) or <math>1</math>, that is an operator <math>p</math>
such that <math>p^2 = p</math> and <math>\|p\| = 0</math> or <math>1</math>. A projector is auto-adjoint and its domain is equal to its codomain.

A ''partial isometry'' is an operator <math>u</math> satisfying <math>uu^* u =
u</math>; this condition entails that we also have <math>u^*uu^* =
u^*</math>. As a consequence <math>uu^*</math> and <math>uu^*</math> are both projectors, called respectively the ''initial'' and the ''final'' projector of <math>u</math> because their (co)domains are respectively the domain and the codomain of <math>u</math>:
* <math>\mathrm{Dom}(u^*u) = \mathrm{Codom}(u^*u) = \mathrm{Dom}(u)</math>;
* <math>\mathrm{Dom}(uu^*) = \mathrm{Codom}(uu^*) = \mathrm{Codom}(u)</math>.

The restriction of <math>u</math> to its domain is an isometry. Projectors are particular examples of partial isometries.

If <math>u</math> is a partial isometry then <math>u^*</math> is also a partial isometry the domain of which is the codomain of <math>u</math> and the codomain of which is the domain of <math>u</math>.

If the domain of <math>u</math> is <math>H</math> that is if <math>u^* u = 1</math> we say that <math>u</math> has ''full domain'', and similarly for codomain. If <math>u</math> and <math>v</math> are two partial isometries, the equation <math>uu^* + vv^* = 1</math> means that the codomains of <math>u</math> and <math>v</math> are disjoint but their direct sum is <math>H</math>.

=== Partial permutations and partial isometries ===

We will now define our proof space which turns out to be the set of partial isometries acting as permutations on the canonical basis <math>(e_n)_{n\in\mathbb{N}}</math>.

More precisely a ''partial permutation'' <math>\varphi</math> on <math>\mathbb{N}</math> is a one-to-one map defined on a subset <math>D_\varphi</math> of <math>\mathbb{N}</math> onto a subset <math>C_\varphi</math> of <math>\mathbb{N}</math>. <math>D_\varphi</math> is called the ''domain'' of <math>\varphi</math> and <math>C_\varphi</math> its ''codomain''. Partial permutations may be composed: if <math>\psi</math> is another partial permutation on <math>\mathbb{N}</math> then <math>\varphi\circ\psi</math> is defined by:

* <math>n\in D_{\varphi\circ\psi}</math> iff <math>n\in D_\psi</math> and <math>\psi(n)\in D_\varphi</math>;
* if <math>n\in D_{\varphi\circ\psi}</math> then <math>\varphi\circ\psi(n) = \varphi(\psi(n))</math>;
* the codomain of <math>\varphi\circ\psi</math> is the image of the domain: <math>C_{\varphi\circ\psi} = \{\varphi(\psi(n)), n\in D_{\varphi\circ\psi}\}</math>.

Partial permutations are well known to form a structure of ''inverse monoid'' that we detail now.

A ''partial identitie'' is a partial permutation <math>1_D</math> whose domain and codomain are both equal to a subset <math>D</math> on which <math>1_D</math> is the identity function. Partial identities are idempotent for composition.

Among partial identities one finds the identity on the empty subset, that is the empty map, that we will denote by <math>0</math> and the identity on <math>\mathbb{N}</math> that we will denote by <math>1</math>. This latter permutation is the neutral for composition.

If <math>\varphi</math> is a partial permutation there is an inverse partial permutation <math>\varphi^{-1}</math> whose domain is <math>D_{\varphi^{-1}} = C_{\varphi}</math> and who satisfies:

: <math>\varphi^{-1}\circ\varphi = 1_{D_\varphi}</math>
: <math>\varphi\circ\varphi^{-1} = 1_{C_\varphi}</math>

Given a partial permutation <math>\varphi</math> one defines a partial isometry <math>u_\varphi</math> by:
: <math>u_\varphi(e_n) =
\begin{cases}
e_{\varphi(n)} & \text{ if }n\in D_\varphi,\\
0 & \text{ otherwise.}
\end{cases}
</math>
In other terms if <math>x=(x_n)_{n\in\mathbb{N}}</math> is a sequence in <math>\ell^2</math> then <math>u_\varphi(x)</math> is the sequence <math>(y_n)_{n\in\mathbb{N}}</math> defined by:
: <math>y_n = x_{\varphi^{-1}(n)}</math> if <math>n\in C_\varphi</math>, <math>0</math> otherwise.

We will (not so abusively) write <math>e_{\varphi(n)} = 0</math> when <math>\varphi(n)</math> is undefined so that the definition of <math>u_\varphi</math> reads:
: <math>u_\varphi(e_n) = e_{\varphi(n)}</math>.

The domain of <math>u_\varphi</math> is the subspace spanned by the family <math>(e_n)_{n\in D_\varphi}</math> and the codomain of <math>u_\varphi</math> is the subspace spanned by <math>(e_n)_{n\in C_\varphi}</math>. In particular if <math>\varphi</math> is <math>1_D</math> then <math>u_\varphi</math> is the projector on the subspace spanned by <math>(e_n)_{n\in D}</math>.

{{Proposition|
Let <math>\varphi</math> and <math>\psi</math> be two partial permutations. We have:
: <math>u_\varphi u_\psi = u_{\varphi\circ\psi}</math>.

The adjoint of <math>u_\varphi</math> is:
: <math>u_\varphi^* = u_{\varphi^{-1}}</math>.

In particular the initial projector of <math>u_{\varphi}</math> is given by:
: <math>u_\varphi u^*_\varphi = u_{1_{D_\varphi}}</math>.

and the final projector of <math>u_\varphi</math> is:
: <math>u^*_\varphi u_\varphi = u_{1_{C_\varphi}}</math>.

Projectors generated by partial identities commute:
: <math>u_\varphi u_\varphi^*u_\psi u_\psi^* = u_\psi u_\psi^*u_\varphi u_\varphi^*</math>.
}}
Note that this entails all the other commutations of projectors: <math>u^*_\varphi u_\varphi u_\psi u^*\psi = u_\psi u^*_\psi u^*_\varphi u_\varphi</math> and <math>u^*_\varphi u_\varphi u^*_\psi u\psi = u^*_\psi u_\psi u^*_\varphi u_\varphi</math>.

{{Definition|
We call ''<math>p</math>-isometry'' a partial isometry of the form <math>u_\varphi</math> where <math>\varphi</math> is a partial permutation on <math>\mathbb{N}</math>. The ''proof space'' <math>\mathcal{P}</math> is the set of all <math>p</math>-isometries.
}}

In particular note that <math>0</math> is a <math>p</math>-isometry. The set <math>\mathcal{P}</math> is a submonoid of <math>\mathcal{B}(H)</math> but it is not a subalgebra.<ref><math>\mathcal{P}</math> is the normalizing groupoid of the maximal commutative subalgebra of <math>\mathcal{B}(H)</math> consisiting of all operators ''diagonalizable'' in the canonical basis.</ref>In general given <math>u,v\in\mathcal{P}</math> we don't necessarily have <math>u+v\in\mathcal{P}</math>. However we have:

{{Proposition|
Let <math>u, v\in\mathcal{P}</math>. Then <math>u+v\in\mathcal{P}</math> iff <math>u</math> and <math>v</math> have disjoint domains and disjoint codomains, that is:
: <math>u+v\in\mathcal{P}</math> iff <math>uu^*vv^* = u^*uv^*v = 0</math>.
}}

{{Proof|
Suppose for contradiction that <math>e_n</math> is in the domains of <math>u</math> and <math>v</math>. There are integers <math>p</math> and <math>q</math> such that <math>u(e_n) = e_p</math> and <math>v(e_n) = e_q</math> thus <math>(u+v)(e_n) = e_p + e_q</math> which is not a basis vector; therefore <math>u+v</math> is not a <math>p</math>-permutation.
}}

As a corollary note that if <math>u+v=0</math> then <math>u=v=0</math>.

=== From operators to matrices: internalization/externalization ===

It will be convenient to view operators on <math>H</math> as acting on <math>H\oplus H</math>, and conversely. For this purpose we define an isomorphism <math>H\oplus H \cong H</math> by <math>x\oplus y\rightsquigarrow p(x)+q(y)</math> where <math>p:H\mapsto H</math> and <math>q:H\mapsto H</math> are partial isometries given by:

: <math>p(e_n) = e_{2n}</math>,
: <math>q(e_n) = e_{2n+1}</math>.

From the definition <math>p</math> and <math>q</math> have full domain, that is
satisfy <math>p^* p = q^* q = 1</math>. On the other hand their codomains are
disjoint, thus we have <math>p^*q = q^*p = 0</math>. As the sum of their
codomains is the full space <math>H</math> we also have <math>pp^* + qq^* = 1</math>.

Note that we have choosen <math>p</math> and <math>q</math> in <math>\mathcal{P}</math>. However the choice is arbitrary: any two <math>p</math>-isometries with full domain and disjoint codomains would do the job.

Given an operator <math>u</math> on <math>H</math> we may ''externalize'' it obtaining an operator <math>U</math> on <math>H\oplus H</math> defined by the matrix:
: <math>U = \begin{pmatrix}
u_{11} & u_{12}\\
u_{21} & u_{22}
\end{pmatrix}</math>
where the <math>u_{ij}</math>'s are given by:
: <math>u_{11} = p^*up</math>;
: <math>u_{12} = p^*uq</math>;
: <math>u_{21} = q^*up</math>;
: <math>u_{22} = q^*uq</math>.

The <math>u_{ij}</math>'s are called the ''external components'' of <math>u</math>. The externalization is functorial in the sense that if <math>v</math> is another operator externalized as:
: <math>V = \begin{pmatrix}
v_{11} & v_{12}\\
v_{21} & v_{22}
\end{pmatrix}
= \begin{pmatrix}
p^*vp & p^*vq\\
q^*vp & q^*vq
\end{pmatrix}
</math>
then the externalization of <math>uv</math> is <math>UV</math>.

As <math>pp^* + qq^* = 1</math> we have:
: <math>u = (pp^*+qq^*)u(pp^*+qq^*) = pu_{11}p^* + pu_{12}q^* + qu_{21}p^* + qu_{22}q^*</math>
which entails that externalization is reversible, its converse being called ''internalization''.

If we suppose that <math>u</math> is a <math>p</math>-isometry then so are the components <math>u_{ij}</math>'s. Thus the formula above entails that the four terms of the sum have pairwise disjoint domains and pairwise disjoint codomains from which we deduce:

{{Proposition|
If <math>u</math> is a <math>p</math>-isometry and <math>u_{ij}</math> are its external components then:
* <math>u_{1j}</math> and <math>u_{2j}</math> have disjoint domains, that is <math>u_{1j}^*u_{1j}u_{2j}^*u_{2j} = 0</math> for <math>j=1,2</math>;
* <math>u_{i1}</math> and <math>u_{i2}</math> have disjoint codomains, that is <math>u_{i1}u_{i1}^*u_{i2}u_{i2}^* = 0</math> for <math>i=1,2</math>.
}}

As an example of computation in <math>\mathcal{P}</math> let us check that the product of the final projectors of <math>pu_{11}p^*</math> and <math>pu_{12}q^*</math> is null:
: <math>\begin{align}
(pu_{11}p^*)(pu^*_{11}p^*)(pu_{12}q^*)(qu_{12}^*p^*)
&= pu_{11}u_{11}^*u_{12}u_{12}^*p^*\\
&= pp^*upp^*u^*pp^*uqq^*u^*pp^*\\
&= pp^*u(pp^*)(u^*pp^*u)qq^*u^*pp^*\\
&= pp^*u(u^*pp^*u)(pp^*)qq^*u^*pp^*\\
&= pp^*uu^*pp^*u(pp^*)(qq^*)u^*pp^*\\
&= 0
\end{align}</math>
where we used the fact that all projectors in <math>\mathcal{P}</math> commute, which is in particular the case of <math>pp^*</math> and <math>u^*pp^*u</math>.

== Interpreting the multiplicative connectives ==

Recall that when <math>u</math> and <math>v</math> are <math>p</math>-isometries we say they are dual when <math>uv</math> is nilpotent, and that <math>\bot</math> denotes the set of nilpotent operators. A ''type'' is a subset of <math>\mathcal{P}</math> that is equal to its bidual. In particular <math>X\orth</math> is a type for any <math>X\subset\mathcal{P}</math>. We say that <math>X</math> ''generates'' the type <math>X\biorth</math>.

=== The tensor and the linear application ===

If <math>u</math> and <math>v</math> are two <math>p</math>-isometries summing them doesn't in general produces a <math>p</math>-isometry. However as <math>pup^*</math> and <math>qvq^*</math> have disjoint domains and disjoint codomains it is true that <math>pup^* + qvq^*</math> is a <math>p</math>-isometry. Given two types <math>A</math> and <math>B</math>, we thus define their ''tensor'' by:

: <math>A\tens B = \{pup^* + qvq^*, u\in A, v\in B\}\biorth</math>

Note the closure by bidual to make sure that we obtain a type.

From what precedes we see that <math>A\tens B</math> is generated by the internalizations of operators on <math>H\oplus H</math> of the form:
: <math>\begin{pmatrix}
u & 0\\
0 & v
\end{pmatrix}</math>

{{Remark|
This so-called tensor resembles a sum rather than a product. We will stick to this terminology though because it defines the interpretation of the tensor connective of linear logic.
}}

The linear implication is derived from the tensor by duality: given two types <math>A</math> and <math>B</math> the type <math>A\limp B</math> is defined by:
: <math>A\limp B = (A\tens B\orth)\orth</math>.

Unfolding this definition we get:
: <math>A\limp B = \{u\in\mathcal{P}\text{ s.t. } \forall v\in A, \forall w\in B\orth,\, u.(pvp^* + qwq^*) \in\bot\}</math>.

=== The identity ===

Given a type <math>A</math> we are to find an operator <math>\iota</math> in type <math>A\limp A</math>, thus satisfying:
: <math>\forall u\in A, v\in A\orth,\, \iota(pup^* + qvq^*)\in\bot</math>.

An easy solution is to take <math>\iota = pq^* + qp^*</math>. In this way we get <math>\iota(pup^* + qvq^*) = qup^* + pvq^*</math>. Therefore <math>(\iota(pup^* + qvq^*))^2 = quvq^* + pvup^*</math>, from which one deduces that this operator is nilpotent iff <math>uv</math> is nilpotent. It is the case since <math>u</math> is in <math>A</math> and <math>v</math> in <math>A\orth</math>.

It is interesting to note that the <math>\iota</math> thus defined is actually the internalization of the operator on <math>H\oplus H</math> given by the matrix:
: <math>\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}</math>.

We will see once the composition is defined that the <math>\iota</math> operator is the interpretation of the identity proof, as expected.

=== The execution formula, version 1: application ===

{{Definition|
Let <math>u</math> and <math>v</math> be two operators; as above denote by <math>u_{ij}</math> the external components of <math>u</math>. If <math>u_{11}v</math> is nilpotent we define the ''application of <math>u</math> to <math>v</math>'' by:
: <math>\mathrm{App}(u,v) = u_{22} + u_{21}v\sum_k(u_{11}v)^ku_{12}</math>.
}}

Note that the hypothesis that <math>u_{11}v</math> is nilpotent entails that the sum <math>\sum_k(u_{11}v)^k</math> is actually finite. It would be enough to assume that this sum converges. For simplicity we stick to the nilpotency condition, but we should mention that weak nilpotency would do as well.

{{Theorem|
If <math>u</math> and <math>v</math> are <math>p</math>-isometries such that <math>u_{11}v</math> is nilpotent, then <math>\mathrm{App}(u,v)</math> is also a <math>p</math>-isometry.
}}

{{Proof|
Let us note <math>E_k = u_{21}v(u_{11}v)^ku_{12}</math>. Recall that <math>u_{22}</math> and <math>u_{12}</math> being external components of the <math>p</math>-isometry <math>u</math>, they have disjoint domains. Thus it is also the case of <math>u_{22}</math> and <math>E_k</math>. Similarly <math>u_{22}</math> and <math>E_k</math> have disjoint codomains because <math>u_{22}</math> and <math>u_{21}</math> have disjoint codomains.

Let now <math>k</math> and <math>l</math> be two integers such that <math>k>l</math> and let us compute for example the intersection of the codomains of <math>E_k</math> and <math>E_l</math>:
: <math>
E_kE^*_kE_lE^*_l = (u_{21}v(u_{11}v)^ku_{12})(u^*_{12}(v^*u^*_{11})^kv^*u^*_{21})(u_{21}v(u_{11}v)^lu_{12})(u^*_{12}(v^*u^*_{11})^lv^*u_{21}^*)
</math>
As <math>k>l</math> we may write <math>(v^*u_{11}^*)^l = (v^*u^*_{11})^{k-l-1}v^*u^*_{11}(v^*u^*_{11})^l</math>. Let us note <math>E = u^*_{11}(v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^lu_{12}</math> so that <math>E_kE^*_kE_lE^*_l = u_{21}v(u_{11}v)^ku_{12}u^*_{12}(v^*u^*_{11})^{k-l-1}v^*Eu^*_{12}(v^*u^*_{11})^lv^*u_{21}^*</math>. We have:
: <math>\begin{align}
E &= u^*_{11}(v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^lu_{12}\\
&= (u^*_{11}u_{11}u^*_{11})(v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^lu_{12}\\
&= u^*_{11}(u_{11}u^*_{11})\bigl((v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^l\bigr)u_{12}\\
&= u^*_{11}\bigl((v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^l\bigr)(u_{11}u^*_{11})u_{12}\\
&= u^*_{11}(v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^lu_{11}u^*_{11}u_{12}\\
&= 0
\end{align}</math>
because <math>u_{11}</math> and <math>u_{12}</math> have disjoint codomains, thus <math>u^*_{11}u_{12} = 0</math>.

Similarly we can show that <math>E_k</math> and <math>E_l</math> have disjoint domains. Therefore we have proved that all terms of the sum <math>\mathrm{App}(u,v)</math> have disjoint domains and disjoint codomains. Consequently <math>\mathrm{App}(u,v)</math> is a <math>p</math>-isometry.
}}

{{Theorem|
Let <math>A</math> and <math>B</math> be two types and <math>u</math> a <math>p</math>-isometry. Then the two following conditions are equivalent:
# <math>u\in A\limp B</math>;
# for any <math>v\in A</math> we have:
#* <math>u_{11}v</math> is nilpotent and
#* <math>\mathrm{App}(u, v)\in B</math>.
}}

{{Proof|
Let <math>v</math> and <math>w</math> be two <math>p</math>-isometries. If we compute
: <math>(u.(pvp^* + qwq^*))^n = \bigl((pu_{11}p^* + pu_{12}q^* + qu_{21}p^* + qu_{22}q^*)(pvp^* + qwq^*)\bigr)^n</math>
we get a finite sum of monomial operators of the form:
# <math>p(u_{11}v)^{i_0}u_{12}w(u_{22}w)^{i_1}\dots u_{21}v(u_{11}v)^{i_m}p^*</math>
# <math>p(u_{11}v)^{i_0}u_{12}w(u_{22}w)^{i_1}\dots u_{12}w(u_{22}w)^{i_m}q^*</math>,
# <math>q(u_{22}w)^{i_0}u_{21}v(u_{11}v)^{i_1}\dots u_{21}v(u_{11}v)^{i_m}p^*</math> or
# <math>q(u_{22}w)^{i_0}u_{21}v(u_{11}v)^{i_1}\dots u_{12}w(u_{22}w)^{i_m}q^*</math>,
for all tuples of (nonnegative) integers <math>(i_1,\dots, i_m)</math> such that <math>i_0+\cdots+i_m+m = n</math>.

Each of these monomial is a <math>p</math>-isometry. Furthermore they have disjoint domains and disjoint codomains because their sum is the <math>p</math>-isometry <math>(u.(pvp^* + qwq^*))^n</math>. This entails that <math>(u.(pvp^* + qwq^*))^n = 0</math> iff all these monomials are null.

Suppose <math>u_{11}v</math> is nilpotent and consider:
: <math>\bigl(\mathrm{App}(u,v)w\bigr)^n = \biggl(\bigl(u_{22} + u_{21}v\sum_k(u_{11}v)^k u_{12}\bigr)w\biggr)^n</math>.
Developping we get a finite sum of monomials of the form:
: 5. <math>(u_{22}w)^{l_0}u_{21}v(u_{11}v)^{k_1}u_{12}w(u_{22}w)^{l_1}\dots u_{21}v(u_{11}v)^{k_m}u_{12}w(u_{22}w)^{l_m}</math>
for all tuples <math>(l_0, k_1, l_1,\dots, k_m, l_m)</math> such that <math>l_0\cdots l_m + m = n</math> and <math>k_i</math> is less than the degree of nilpotency of <math>u_{11}v</math> for all <math>i</math>.

Again as these monomials are <math>p</math>-isometries and their sum is the <math>p</math>-isometry <math>(\mathrm{App}(u,v)w)^n</math>, they have pairwise disjoint domains and pairwise disjoint codomains. Note that each of these monomial is equal to <math>q^*Mq</math> where <math>M</math> is a monomial of type 4 above.

As before we thus have that <math>\bigl(\mathrm{App}(u,v)w\bigr)^n = 0</math> iff all monomials of type 5 are null.

Suppose now that <math>u\in A\limp B</math> and <math>v\in A</math>. Then, since <math>0\in B\orth</math> (<math>0</math> belongs to any type) <math>u.(pvp^*) = pu_{11}vp^*</math> is nilpotent, thus <math>u_{11}v</math> is nilpotent.

Suppose further that <math>w\in B\orth</math>. Then <math>u.(pvp^*+qwq^*)</math> is nilpotent, thus there is a <math>N</math> such that <math>(u.(pvp^* + qwq^*))^n=0</math> for any <math>n\geq N</math>. This entails that all monomials of type 1 to 4 are null. Therefore all monomials appearing in the developpment of <math>(\mathrm{App}(u,v)w)^N</math> are null which proves that <math>\mathrm{App}(u,v)w</math> is nilpotent. Thus <math>\mathrm{App}(u,v)\in B</math>.

Conversely suppose for any <math>v\in A</math> and <math>w\in B\orth</math>, <math>u_{11}v</math> and <math>\mathrm{App}(u,v)w</math> are nilpotent. Let <math>P</math> and <math>N</math> be their respective degrees of nilpotency and put <math>n=N(P+1)+N</math>. Then we claim that all monomials of type 1 to 4 appearing in the development of <math>(u.(pvp^*+qwq^*))^n</math> are null.

Consider for example a monomial of type 1:
: <math>p(u_{11}v)^{i_0}u_{12}w(u_{22}w)^{i_1}\dots u_{21}v(u_{11}v)^{i_m}p^*</math>
with <math>i_0+\cdots+i_m + m = n</math>. Note that <math>m</math> must be even.

If <math>i_{2k}\geq P</math> for some <math>0\leq k\leq m/2</math> then <math>(u_{11}v)^{i_{2k}}=0</math> thus our monomial is null. Otherwise if <math>i_{2k}<P</math> for all <math>k</math> we have:
: <math>i_1+i_3+\cdots +i_{m-1} + m/2 = n - m/2 - (i_0+i_2+\cdots +i_m)</math>
thus:
: <math>i_1+i_3+\cdots +i_{m-1} + m/2\geq n - m/2 - (1+m/2)P</math>.
Now if <math>m/2\geq N</math> then <math>i_1+\cdots+i_{m-1}+m/2 \geq N</math>. Otherwise <math>1+m/2\leq N</math> thus
: <math>i_1+i_3+\cdots +i_{m-1} + m/2\geq n - N - NP = N</math>.
Since <math>N</math> is the degree of nilpotency of <math>\mathrm{App}(u,v)w</math> we have that the monomial:
: <math>(u_{22}w)^{i_1}u_{21}v(u_{11}v)^{i_2}u_{12}w\dots(u_{11}v)^{i_{m-2}}u_{12}w(u_{22}w)^{i_{m-1}}</math>
is null, thus also the monomial of type 1 we started with.
}}

{{Corollary|
If <math>A</math> and <math>B</math> are types then we have:
: <math>A\limp B = \{u\in\mathcal{P} \text{ such that }\forall v\in A: u_{11}v\in\bot\text{ and } \mathrm{App}(u, v)\in B\}</math>.
}}

As an example if we compute the application of the interpretation of the identity <math>\iota</math> in type <math>A\limp A</math> to the operator <math>v\in A</math> then we have:
: <math>\mathrm{App}(\iota, v) = \iota_{22} + \iota_{21}v\sum(\iota_{11}v)^k\iota_{12}</math>.
Now recall that <math>\iota = pq^* + qp^*</math> so that <math>\iota_{11} = \iota_{22} = 0</math> and <math>\iota_{12} = \iota_{21} = 1</math> and we thus get:
: <math>\mathrm{App}(\iota, v) = v</math>
as expected.

=== The tensor rule ===

Let now <math>A, A', B</math> and <math>B'</math> be types and consider two operators <math>u</math> and <math>u'</math> respectively in <math>A\limp B</math> and <math>A\limp B'</math>. We define an operator denoted by <math>u\tens u'</math> by:
: <math>\begin{align}
u\tens u' &= ppp^*upp^*p^* + qpq^*upp^*p^* + ppp^*uqp^*q^* + qpq^*uqp^*q^*\\
&+ pqp^*u'pq^*p^* + qqq^*u'pq^*p^* + pqp^*u'qq^*q^* + qqq^*u'qq^*q^*
\end{align}</math>

Once again the notation is motivated by linear logic syntax and is contradictory with linear algebra practice since what we denote by <math>u\tens u'</math> actually is the internalization of the direct sum <math>u\oplus u'</math>.

Indeed if we think of <math>u</math> and <math>u'</math> as the internalizations of the matrices:
: <math>
\begin{pmatrix}u_{11} & u_{12}\\
u_{21} & u_{22}
\end{pmatrix}
</math> and <math>
\begin{pmatrix}u'_{11} & u'_{12}\\
u'_{21} & u'_{22}
\end{pmatrix}</math>
then we may write:
: <math>\begin{align}
u\tens u' &= ppu_{11}p^*p^* + qpu_{21}p^*p^* + ppu_{12}p^*q^* + qpu_{22}p^*q^*\\
&+ pqu'_{11}q^*p^* + qqu'_{21}q^*p^* + pqu'_{12}q^*q^* + qqu'_{22}q^*q^*
\end{align}</math>

Thus the components of <math>u\tens u'</math> are given by:
: <math>(u\tens u')_{ij} = pu_{ij}p^* + qu'_{ij}q^*</math>.
and we see that <math>u\tens u'</math> is actually the internalization of the matrix:
: <math>
\begin{pmatrix}
u_{11} & 0 & u_{12} & 0 \\
0 & u'_{11} & 0 & u'_{12} \\
u_{21} & 0 & u_{22} & 0 \\
0 & u'_{21} & 0 & u'_{22} \\
\end{pmatrix}
</math>

We are now to show that if we suppose <math>u</math>and <math>u'</math> are in types <math>A\limp B</math> and <math>A'\limp B'</math>, then <math>u\tens u'</math> is in <math>A\tens A'\limp B\tens B'</math>. For this we consider <math>v</math> and <math>v'</math> in respectively in <math>A</math> and <math>A'</math>, so that <math>pvp^* + qv'q^*</math> is in <math>A\tens A'</math>, and we show that <math>\mathrm{App}(u\tens u', pvp^* + qv'q^*)\in B\tens B'</math>.

Since <math>u</math> and <math>u'</math> are in <math>A\limp B</math> and <math>A'\limp B'</math> we have that <math>\mathrm{App}(u, v)</math> and <math>\mathrm{App}(u', v')</math> are respectively in <math>B</math> and <math>B'</math>, thus:
: <math>p\mathrm{App}(u, v)p^* + q\mathrm{App}(u', v')q^* \in B\tens B'</math>.

We know that both <math>u_{11}v</math> and <math>u'_{11}v'</math> are nilpotent. But we have:
: <math>\begin{align}
\bigl((u\tens u')_{11}(pvp^* + qv'q^*)\bigr)^n
&= \bigl((pu_{11} + qu'_{11}q^*)(pvp^* + qv'q^*)\bigr)^n\\
&= (pu_{11}vp^* + qu'_{11}v'q^*)^n\\
&= p(u_{11}v)^np^* + q(u'_{11}v')^nq^*
\end{align}</math>

Therefore <math>(u\tens u')_{11}(pvp^* + qv'q^*)</math> is nilpotent. So we can compute <math>\mathrm{App}(u\tens u', pvp^* + qv'q^*)</math>:
: <math>\begin{align}
&\mathrm{App}(u\tens u', pvp^* + qv'q^*)\\
&= (u\tens u')_{22} + (u\tens u')_{21}(pvp^* + qv'q^*)\sum\bigl((u\tens u')_{11}(pvp^* + qv'q^*)\bigr)^k(u\tens u')_{12}\\
&= pu_{22}p^* + qu'_{22}q^* + (pu_{21}p^* + qu'_{21}q^*)(pvp^* + qv'q^*)\sum\bigl((pu_{11}p^* + qu'_{11}q^*)(pvp^* + qv'q^*)\bigr)^k(pu_{12}p^* + qu'_{12}q^*)\\
&= p\bigl(u_{22} + u_{21}v\sum(u_{11}v)^ku_{12}\bigr)p^* + q\bigl(u'_{22} + u'_{21}v'\sum(u'_{11}v')^ku'_{12}\bigr)q^*\\
&= p\mathrm{App}(u, v)p^* + q\mathrm{App}(u', v')q^*
\end{align}</math>
thus lives in <math>B\tens B'</math>.

=== Other monoidal constructions ===

==== Contraposition ====

Let <math>A</math> and <math>B</math> be some types; we have:
: <math>A\limp B = A\orth\limpinv B\orth</math>

Indeed, <math>u\in A\limp B</math> means that for any <math>v</math> and <math>w</math> in respectively <math>A</math> and <math>B\orth</math> we have <math>u.(pvp^* + qwq^*)\in\bot</math> which is exactly the definition of <math>A\orth\limpinv B\orth</math>.

We will denote <math>u\orth</math> the operator:
: <math>u\orth = pu_{22}p^* + pu_{12}q^* + qu_{12}p^* + qu_{11}q^*</math>
where <math>u_{ij}</math> is given by externalization. Therefore the externalization of <math>u\orth</math> is:
: <math>(u\orth)_{ij} = u_{\bar i\,\bar j}</math> where <math>\bar .</math> is defined by <math>\bar1 = 2, \bar2 = 1</math>.
From this we deduce that <math>u\orth\in B\orth\limp A\orth</math> and that <math>(u\orth)\orth = u</math>.

==== Commutativity ====
Let <math>\sigma</math> be the operator:
: <math>\sigma = ppq^*q^* +pqp^*q^* + qpq^*p^* + qqp^*p^*</math>.
One can check that <math>\sigma</math> is the internalization of the operator <math>S</math> on <math>H\oplus H\oplus H\oplus H</math> defined by: <math>S(x_1\oplus x_2\oplus x_3\oplus x_4) = x_4\oplus x_3\oplus x_2\oplus x_1</math>. In particular the components of <math>\sigma</math> are:
: <math>\sigma_{11} = \sigma_{22} = 0</math>;
: <math>\sigma_{12} = \sigma_{21} = pq^* + qp^*</math>.

Let <math>A</math> and <math>B</math> be types and <math>u</math> and <math>v</math> be operators in <math>A</math> and <math>B</math>. Then <math>pup^* + qvq^*</math> is in <math>A\tens B</math> and as <math>\sigma_{11}.(pup^* + qvq^*) = 0</math> we may compute:
: <math>\begin{align}
\mathrm{App}(\sigma, pup^* + qvq^*)
&= \sigma_{22} + \sigma_{21}(pup^* + qvq^*)\sum(\sigma_{11}(pup^* + qvq^*))^k\sigma_{12}\\
&= (pq^* + qp^*)(pup^* + qvq^*)(pq^* + qp^*)\\
&= pvp^* + quq^*
\end{align}</math>
But <math>pvp^* + quq^*\in B\tens A</math>, thus we have shown that:
: <math>\sigma\in (A\tens B) \limp (B\tens A)</math>.

==== Distributivity ====
We get distributivity by considering the operator:
: <math>\delta = ppp^*p^*q^* + pqpq^*p^*q^* + pqqq^*q^* + qppp^*p^* + qpqp^*q^*p^* + qqq^*q^*p^*</math>
that is similarly shown to be in type <math>A\tens(B\tens C)\limp(A\tens B)\tens C</math> for any types <math>A</math>, <math>B</math> and <math>C</math>.

==== Weak distributivity ====
We can finally get weak distributivity thanks to the operators:
: <math>\delta_1 = pppp^*q^* + ppqp^*q^*q^* + pqq^*q^*q^* + qpp^*p^*p^* + qqp q^*p^*p^* + qqq q^*p^*</math> and
: <math>\delta_2 = ppp^*p^*q^* + pqpq^*p^*q^* + pqqq^*q^* + qppp^*p^* + qpqp^*q^*p^* + qqq^*q^*p^*</math>.

Given three types <math>A</math>, <math>B</math> and <math>C</math> then one can show that:
: <math>\delta_1</math> has type <math>((A\limp B)\tens C)\limp A\limp (B\tens C)</math> and
: <math>\delta_2</math> has type <math>(A\tens(B\limp C))\limp (A\limp B)\limp C</math>.

=== Execution formula, version 2: composition ===

Let <math>A</math>, <math>B</math> and <math>C</math> be types and <math>u</math> and <math>v</math> be operators respectively in types <math>A\limp B</math> and <math>B\limp C</math>.

As usual we will denote <math>u_{ij}</math> and <math>v_{ij}</math> the operators obtained by externalization of <math>u</math> and <math>v</math>, eg, <math>u_{11} = p^*up</math>, ...

As <math>u</math> is in <math>A\limp B</math> we have that <math>\mathrm{App}(u, 0)=u_{22}\in B</math>; similarly as <math>v\in B\limp C</math>, thus <math>v\orth\in C\orth\limp B\orth</math>, we have <math>\mathrm{App}(v\orth, 0) = v_{11}\in B\orth</math>. Thus <math>u_{22}v_{11}</math> is nilpotent.

We define the operator <math>\mathrm{Comp}(u, v)</math> by:
: <math>\begin{align}
\mathrm{Comp}(u, v) &= p(u_{11} + u_{12}\sum(v_{11}u_{22})^k\,v_{11}u_{21})p^*\\
&+ p(u_{12}\sum(v_{11}u_{22})^k\,v_{12})q^*\\
&+ q(v_{21}\sum(u_{22}v_{11})^k\,u_{21})p^*\\
&+ q(v_{22} + v_{21}\sum(u_{22}v_{11})^k\,u_{22}v_{12})q^*
\end{align}</math>

This is well defined since <math>u_{11}v_{22}</math> is nilpotent. As an example let us compute the composition of <math>u</math> and <math>\iota</math> in type <math>B\limp B</math>; recall that <math>\iota_{ij} = \delta_{ij}</math>, so we get:
: <math>
\mathrm{Comp}(u, \iota) = pu_{11}p^* + pu_{12}q^* + qu_{21}p^* + qu_{22}q^* = u
</math>
Similar computation would show that <math>\mathrm{Comp}(\iota, v) = v</math> (we use <math>pp^* + qq^* = 1</math> here).

Coming back to the general case we claim that <math>\mathrm{Comp}(u, v)</math> is in <math>A\limp C</math>: let <math>a</math> be an operator in <math>A</math>. By computation we can check that:
: <math>\mathrm{App}(\mathrm{Comp}(u, v), a) = \mathrm{App}(v, \mathrm{App}(u, a))</math>.
Now since <math>u</math> is in <math>A\limp B</math>, <math>\mathrm{App}(u, a)</math> is in <math>B</math> and since <math>v</math> is in <math>B\limp C</math>, <math>\mathrm{App}(v, \mathrm{App}(u, a))</math> is in <math>C</math>.

If we now consider a type <math>D</math> and an operator <math>w</math> in <math>C\limp D</math> then we have:
: <math>\mathrm{Comp}(\mathrm{Comp}(u, v), w) = \mathrm{Comp}(u,
\mathrm{Comp}(v, w))</math>.

Putting together the results of this section we finally have:

{{Theorem|
Let GoI(H) be defined by:
* objects are types, ''ie'' sets <math>A</math> of operators satisfying: <math>A\biorth = A</math>;
* morphisms from <math>A</math> to <math>B</math> are operators in type <math>A\limp B</math>;
* composition is given by the formula above.

Then GoI(H) is a star-autonomous category.
}}

= The Geometry of Interaction as an abstract machine =

= Notes and references =

<references/>

Geometry of interaction

2010-04-29T10:16:24Z

Emmanuel Beffara: Reverted edits by Emmanuel Beffara (Talk); changed back to last version by Laurent Regnier

The ''geometry of interaction'', GoI in short, was defined in the early nineties by Girard as an interpretation of linear logic into operators algebra: formulae were interpreted by Hilbert spaces and proofs by partial isometries.

This was a striking novelty as it was the first time that a mathematical model of logic (lambda-calculus) didn't interpret a proof of <math>A\limp B</math> as a morphism ''from'' <math>A</math> ''to'' <math>B</math><ref>to be precise one should say from ''the space interpreting'' <math>A</math> to the space interpreting'' <math>B</math></ref>, and proof composition (cut rule) as the composition of morphisms. Rather the proof was interpreted as an operator acting ''on'' <math>A\limp B</math>, that is a morphism from <math>A\limp B</math> to <math>A\limp B</math>. For proof composition the problem was then, given an operator on <math>A\limp B</math> and another one on <math>B\limp C</math> to construct a new operator on <math>A\limp C</math>. This problem was solved by the ''execution formula'' that bares some formal analogies with Kleene's formula for recursive functions. For this reason GoI was claimed to be an ''operational semantics'', as opposed to traditionnal [[Semantics|denotational semantics]].

The first instance of the GoI was restricted to the <math>MELL</math> fragment of linear logic (Multiplicative and Exponential fragment) which is enough to encode lambda-calculus. Since then Girard proposed several improvements: firstly the extension to the additive connectives known as ''Geometry of Interaction 3'' and more recently a complete reformulation using Von Neumann algebras that allows to deal with some aspects of [[Light linear logics|implicit complexity]]

The GoI has been a source of inspiration for various authors. Danos and Regnier have reformulated the original model exhibiting its combinatorial nature using a theory of reduction of paths in proof-nets and showing the link with abstract machines; in particular the execution formula appears as the composition of two automata that interact one with the other through their common interface. Also the execution formula has rapidly been understood as expressing the composition of strategies in game semantics. It has been used in the theory of sharing reduction for lambda-calculus in the Abadi-Gonthier-Lévy reformulation and simplification of Lamping's representation of sharing. Finally the original GoI for the <math>MELL</math> fragment has been reformulated in the framework of traced monoidal categories following an idea originally proposed by Joyal.

= The Geometry of Interaction as operators =

The original construction of GoI by Girard follows a general pattern already mentionned in [[coherent semantics]] under the name ''symmetric reducibility'' and that was first put to use in [[phase semantics]]. First set a general space <math>P</math> called the ''proof space'' because this is where the interpretations of proofs will live. Make sure that <math>P</math> is a (not necessarily commutative) monoid. In the case of GoI, the proof space is a subset of the space of bounded operators on <math>\ell^2</math>.

Second define a particular subset of <math>P</math> that will be denoted by <math>\bot</math>; then derive a duality on <math>P</math>: for <math>u,v\in P</math>, <math>u</math> and <math>v</math> are dual<ref>In modern terms one says that <math>u</math> and <math>v</math> are ''polar''.</ref>, iff <math>uv\in\bot</math>.

For the GoI, two dualities have proved to work; we will consider the first one: nilpotency, ''ie'', <math>\bot</math> is the set of nilpotent operators in <math>P</math>. Let us explicit this: two operators <math>u</math> and <math>v</math> are dual if there is a nonegative integer <math>n</math> such that <math>(uv)^n = 0</math>. Note in particular that <math>uv\in\bot</math> iff <math>vu\in\bot</math>.

When <math>X</math> is a subset of <math>P</math> define <math>X\orth</math> as the set of elements of <math>P</math> that are dual to all elements of <math>X</math>:
: <math>X\orth = \{u\in P, \forall v\in X, uv\in\bot\}</math>.

This construction has a few properties that we will use without mention in the sequel. Given two subsets <math>X</math> and <math>Y</math> of <math>P</math> we have:
* if <math>X\subset Y</math> then <math>Y\orth\subset X</math>;
* <math>X\subset X\biorth</math>;
* <math>X\triorth = X\orth</math>.

Last define a ''type'' as a subset <math>T</math> of the proof space that is equal to its bidual: <math>T = T\biorth</math>. This means that <math>u\in T</math> iff for all operator <math>v\in T\orth</math>, that is such that <math>u'v\in\bot</math> for all <math>u'\in T</math>, we have <math>uv\in\bot</math>.

The real work<ref>The difficulty is to find the right duality that will make logical operations interpretable. General conditions that allows to achieve this have been formulated by Hyland and Schalk thanks to their theory of ''double gluing''.</ref>, is now to interpret logical operations, that is to associate a type to each formula, an object to each proof and show the ''adequacy lemma'': if <math>u</math> is the interpretation of a proof of the formula <math>A</math> then <math>u</math> belongs to the type associated to <math>A</math>.

== Preliminaries ==

We will denote by <math>H</math> the Hilbert space <math>\ell^2(\mathbb{N})</math> of sequences <math>(x_n)_{n\in\mathbb{N}}</math> of complex numbers such that the series <math>\sum_{n\in\mathbb{N}}|x_n|^2</math> converges. If <math>x = (x_n)_{n\in\mathbb{N}}</math> and <math>y = (y_n)_{n\in\mathbb{N}}</math> are two vectors of <math>H</math> their ''scalar product'' is:
: <math>\langle x, y\rangle = \sum_{n\in\mathbb{N}} x_n\bar y_n</math>.

Two vectors of <math>H</math> are ''othogonal'' if their scalar product is nul. We will say that two subspaces are ''disjoint'' when any two vectors taken in each subspace are orthorgonal. Note that this notion is different from the set theoretic one, in particular two disjoint subspaces always have exactly one vector in common: <math>0</math>.

The ''norm'' of a vector is the square root of the scalar product with itself:
: <math>\|x\| = \sqrt{\langle x, x\rangle}</math>.

Let us denote by <math>(e_k)_{k\in\mathbb{N}}</math> the canonical ''hilbertian basis'' of <math>H</math>: <math>e_k = (\delta_{kn})_{n\in\mathbb{N}}</math> where <math>\delta_{kn}</math> is the Kroenecker symbol: <math>\delta_{kn}=1</math> if <math>k=n</math>, <math>0</math> otherwise. Thus if <math>x=(x_n)_{n\in\mathbb{N}}</math> is a sequence in <math>H</math> we have:
: <math> x = \sum_{n\in\mathbb{N}} x_ne_n</math>.

An ''operator'' on <math>H</math> is a ''continuous'' linear map from <math>H</math> to <math>H</math><ref>Continuity is equivalent to the fact that operators are ''bounded'', which means that one may define the ''norm'' of an operator <math>u</math> as the sup on the unit ball of the norms of its values:
: <math>\|u\| = \sup_{\{x\in H,\, \|x\| = 1\}}\|u(x)\|</math>.</ref>. The set of (bounded) operators is denoted by <math>\mathcal{B}(H)</math>.

The ''range'' or ''codomain'' of the operator <math>u</math> is the set of images of vectors; the ''kernel'' of <math>u</math> is the set of vectors that are anihilated by <math>u</math>; the ''domain'' of <math>u</math> is the set of vectors orthogonal to the kernel, ''ie'', the maximal subspace disjoint with the kernel:

* <math>\mathrm{Codom}(u) = \{u(x),\, x\in H\}</math>;
* <math>\mathrm{Ker}(u) = \{x\in H,\, u(x) = 0\}</math>;
* <math>\mathrm{Dom}(u) = \{x\in H,\, \forall y\in\mathrm{Ker}(u), \langle x, y\rangle = 0\}</math>.

These three sets are closed subspaces of <math>H</math>.

The ''adjoint'' of an operator <math>u</math> is the operator <math>u^*</math> defined by <math>\langle u(x), y\rangle = \langle x, u^*(y)\rangle</math> for any <math>x,y\in H</math>. Adjointness is well behaved w.r.t. composition of operators:
: <math>(uv)^* = v^*u^*</math>.

A ''projector'' is an idempotent operator of norm <math>0</math> (the projector
on the null subspace) or <math>1</math>, that is an operator <math>p</math>
such that <math>p^2 = p</math> and <math>\|p\| = 0</math> or <math>1</math>. A projector is auto-adjoint and its domain is equal to its codomain.

A ''partial isometry'' is an operator <math>u</math> satisfying <math>uu^* u =
u</math>; this condition entails that we also have <math>u^*uu^* =
u^*</math>. As a consequence <math>uu^*</math> and <math>uu^*</math> are both projectors, called respectively the ''initial'' and the ''final'' projector of <math>u</math> because their (co)domains are respectively the domain and the codomain of <math>u</math>:
* <math>\mathrm{Dom}(u^*u) = \mathrm{Codom}(u^*u) = \mathrm{Dom}(u)</math>;
* <math>\mathrm{Dom}(uu^*) = \mathrm{Codom}(uu^*) = \mathrm{Codom}(u)</math>.

The restriction of <math>u</math> to its domain is an isometry. Projectors are particular examples of partial isometries.

If <math>u</math> is a partial isometry then <math>u^*</math> is also a partial isometry the domain of which is the codomain of <math>u</math> and the codomain of which is the domain of <math>u</math>.

If the domain of <math>u</math> is <math>H</math> that is if <math>u^* u = 1</math> we say that <math>u</math> has ''full domain'', and similarly for codomain. If <math>u</math> and <math>v</math> are two partial isometries, the equation <math>uu^* + vv^* = 1</math> means that the codomains of <math>u</math> and <math>v</math> are disjoint but their direct sum is <math>H</math>.

=== Partial permutations and partial isometries ===

We will now define our proof space which turns out to be the set of partial isometries acting as permutations on the canonical basis <math>(e_n)_{n\in\mathbb{N}}</math>.

More precisely a ''partial permutation'' <math>\varphi</math> on <math>\mathbb{N}</math> is a one-to-one map defined on a subset <math>D_\varphi</math> of <math>\mathbb{N}</math> onto a subset <math>C_\varphi</math> of <math>\mathbb{N}</math>. <math>D_\varphi</math> is called the ''domain'' of <math>\varphi</math> and <math>C_\varphi</math> its ''codomain''. Partial permutations may be composed: if <math>\psi</math> is another partial permutation on <math>\mathbb{N}</math> then <math>\varphi\circ\psi</math> is defined by:

* <math>n\in D_{\varphi\circ\psi}</math> iff <math>n\in D_\psi</math> and <math>\psi(n)\in D_\varphi</math>;
* if <math>n\in D_{\varphi\circ\psi}</math> then <math>\varphi\circ\psi(n) = \varphi(\psi(n))</math>;
* the codomain of <math>\varphi\circ\psi</math> is the image of the domain: <math>C_{\varphi\circ\psi} = \{\varphi(\psi(n)), n\in D_{\varphi\circ\psi}\}</math>.

Partial permutations are well known to form a structure of ''inverse monoid'' that we detail now.

A ''partial identitie'' is a partial permutation <math>1_D</math> whose domain and codomain are both equal to a subset <math>D</math> on which <math>1_D</math> is the identity function. Partial identities are idempotent for composition.

Among partial identities one finds the identity on the empty subset, that is the empty map, that we will denote by <math>0</math> and the identity on <math>\mathbb{N}</math> that we will denote by <math>1</math>. This latter permutation is the neutral for composition.

If <math>\varphi</math> is a partial permutation there is an inverse partial permutation <math>\varphi^{-1}</math> whose domain is <math>D_{\varphi^{-1}} = C_{\varphi}</math> and who satisfies:

: <math>\varphi^{-1}\circ\varphi = 1_{D_\varphi}</math>
: <math>\varphi\circ\varphi^{-1} = 1_{C_\varphi}</math>

Given a partial permutation <math>\varphi</math> one defines a partial isometry <math>u_\varphi</math> by:
: <math>u_\varphi(e_n) =
\begin{cases}
e_{\varphi(n)} & \text{ if }n\in D_\varphi,\\
0 & \text{ otherwise.}
\end{cases}
</math>
In other terms if <math>x=(x_n)_{n\in\mathbb{N}}</math> is a sequence in <math>\ell^2</math> then <math>u_\varphi(x)</math> is the sequence <math>(y_n)_{n\in\mathbb{N}}</math> defined by:
: <math>y_n = x_{\varphi^{-1}(n)}</math> if <math>n\in C_\varphi</math>, <math>0</math> otherwise.

We will (not so abusively) write <math>e_{\varphi(n)} = 0</math> when <math>\varphi(n)</math> is undefined so that the definition of <math>u_\varphi</math> reads:
: <math>u_\varphi(e_n) = e_{\varphi(n)}</math>.

The domain of <math>u_\varphi</math> is the subspace spanned by the family <math>(e_n)_{n\in D_\varphi}</math> and the codomain of <math>u_\varphi</math> is the subspace spanned by <math>(e_n)_{n\in C_\varphi}</math>. In particular if <math>\varphi</math> is <math>1_D</math> then <math>u_\varphi</math> is the projector on the subspace spanned by <math>(e_n)_{n\in D}</math>.

{{Proposition|
Let <math>\varphi</math> and <math>\psi</math> be two partial permutations. We have:
: <math>u_\varphi u_\psi = u_{\varphi\circ\psi}</math>.

The adjoint of <math>u_\varphi</math> is:
: <math>u_\varphi^* = u_{\varphi^{-1}}</math>.

In particular the initial projector of <math>u_{\varphi}</math> is given by:
: <math>u_\varphi u^*_\varphi = u_{1_{D_\varphi}}</math>.

and the final projector of <math>u_\varphi</math> is:
: <math>u^*_\varphi u_\varphi = u_{1_{C_\varphi}}</math>.

Projectors generated by partial identities commute:
: <math>u_\varphi u_\varphi^*u_\psi u_\psi^* = u_\psi u_\psi^*u_\varphi u_\varphi^*</math>.
}}
Note that this entails all the other commutations of projectors: <math>u^*_\varphi u_\varphi u_\psi u^*\psi = u_\psi u^*_\psi u^*_\varphi u_\varphi</math> and <math>u^*_\varphi u_\varphi u^*_\psi u\psi = u^*_\psi u_\psi u^*_\varphi u_\varphi</math>.

{{Definition|
We call ''<math>p</math>-isometry'' a partial isometry of the form <math>u_\varphi</math> where <math>\varphi</math> is a partial permutation on <math>\mathbb{N}</math>. The ''proof space'' <math>\mathcal{P}</math> is the set of all <math>p</math>-isometries.
}}

In particular note that <math>0</math> is a <math>p</math>-isometry. The set <math>\mathcal{P}</math> is a submonoid of <math>\mathcal{B}(H)</math> but it is not a subalgebra<ref><math>\mathcal{P}</math> is the normalizing groupoid of the maximal commutative subalgebra of <math>\mathcal{B}(H)</math> consisiting of all operators ''diagonalizable'' in the canonical basis.</ref>. In general given <math>u,v\in\mathcal{P}</math> we don't necessarily have <math>u+v\in\mathcal{P}</math>. However we have:

{{Proposition|
Let <math>u, v\in\mathcal{P}</math>. Then <math>u+v\in\mathcal{P}</math> iff <math>u</math> and <math>v</math> have disjoint domains and disjoint codomains, that is:
: <math>u+v\in\mathcal{P}</math> iff <math>uu^*vv^* = u^*uv^*v = 0</math>.
}}

{{Proof|
Suppose for contradiction that <math>e_n</math> is in the domains of <math>u</math> and <math>v</math>. There are integers <math>p</math> and <math>q</math> such that <math>u(e_n) = e_p</math> and <math>v(e_n) = e_q</math> thus <math>(u+v)(e_n) = e_p + e_q</math> which is not a basis vector; therefore <math>u+v</math> is not a <math>p</math>-permutation.
}}

As a corollary note that if <math>u+v=0</math> then <math>u=v=0</math>.

=== From operators to matrices: internalization/externalization ===

It will be convenient to view operators on <math>H</math> as acting on <math>H\oplus H</math>, and conversely. For this purpose we define an isomorphism <math>H\oplus H \cong H</math> by <math>x\oplus y\rightsquigarrow p(x)+q(y)</math> where <math>p:H\mapsto H</math> and <math>q:H\mapsto H</math> are partial isometries given by:

: <math>p(e_n) = e_{2n}</math>,
: <math>q(e_n) = e_{2n+1}</math>.

From the definition <math>p</math> and <math>q</math> have full domain, that is
satisfy <math>p^* p = q^* q = 1</math>. On the other hand their codomains are
disjoint, thus we have <math>p^*q = q^*p = 0</math>. As the sum of their
codomains is the full space <math>H</math> we also have <math>pp^* + qq^* = 1</math>.

Note that we have choosen <math>p</math> and <math>q</math> in <math>\mathcal{P}</math>. However the choice is arbitrary: any two <math>p</math>-isometries with full domain and disjoint codomains would do the job.

Given an operator <math>u</math> on <math>H</math> we may ''externalize'' it obtaining an operator <math>U</math> on <math>H\oplus H</math> defined by the matrix:
: <math>U = \begin{pmatrix}
u_{11} & u_{12}\\
u_{21} & u_{22}
\end{pmatrix}</math>
where the <math>u_{ij}</math>'s are given by:
: <math>u_{11} = p^*up</math>;
: <math>u_{12} = p^*uq</math>;
: <math>u_{21} = q^*up</math>;
: <math>u_{22} = q^*uq</math>.

The <math>u_{ij}</math>'s are called the ''external components'' of <math>u</math>. The externalization is functorial in the sense that if <math>v</math> is another operator externalized as:
: <math>V = \begin{pmatrix}
v_{11} & v_{12}\\
v_{21} & v_{22}
\end{pmatrix}
= \begin{pmatrix}
p^*vp & p^*vq\\
q^*vp & q^*vq
\end{pmatrix}
</math>
then the externalization of <math>uv</math> is <math>UV</math>.

As <math>pp^* + qq^* = 1</math> we have:
: <math>u = (pp^*+qq^*)u(pp^*+qq^*) = pu_{11}p^* + pu_{12}q^* + qu_{21}p^* + qu_{22}q^*</math>
which entails that externalization is reversible, its converse being called ''internalization''.

If we suppose that <math>u</math> is a <math>p</math>-isometry then so are the components <math>u_{ij}</math>'s. Thus the formula above entails that the four terms of the sum have pairwise disjoint domains and pairwise disjoint codomains from which we deduce:

{{Proposition|
If <math>u</math> is a <math>p</math>-isometry and <math>u_{ij}</math> are its external components then:
* <math>u_{1j}</math> and <math>u_{2j}</math> have disjoint domains, that is <math>u_{1j}^*u_{1j}u_{2j}^*u_{2j} = 0</math> for <math>j=1,2</math>;
* <math>u_{i1}</math> and <math>u_{i2}</math> have disjoint codomains, that is <math>u_{i1}u_{i1}^*u_{i2}u_{i2}^* = 0</math> for <math>i=1,2</math>.
}}

As an example of computation in <math>\mathcal{P}</math> let us check that the product of the final projectors of <math>pu_{11}p^*</math> and <math>pu_{12}q^*</math> is null:
: <math>\begin{align}
(pu_{11}p^*)(pu^*_{11}p^*)(pu_{12}q^*)(qu_{12}^*p^*)
&= pu_{11}u_{11}^*u_{12}u_{12}^*p^*\\
&= pp^*upp^*u^*pp^*uqq^*u^*pp^*\\
&= pp^*u(pp^*)(u^*pp^*u)qq^*u^*pp^*\\
&= pp^*u(u^*pp^*u)(pp^*)qq^*u^*pp^*\\
&= pp^*uu^*pp^*u(pp^*)(qq^*)u^*pp^*\\
&= 0
\end{align}</math>
where we used the fact that all projectors in <math>\mathcal{P}</math> commute, which is in particular the case of <math>pp^*</math> and <math>u^*pp^*u</math>.

== Interpreting the multiplicative connectives ==

Recall that when <math>u</math> and <math>v</math> are <math>p</math>-isometries we say they are dual when <math>uv</math> is nilpotent, and that <math>\bot</math> denotes the set of nilpotent operators. A ''type'' is a subset of <math>\mathcal{P}</math> that is equal to its bidual. In particular <math>X\orth</math> is a type for any <math>X\subset\mathcal{P}</math>. We say that <math>X</math> ''generates'' the type <math>X\biorth</math>.

=== The tensor and the linear application ===

If <math>u</math> and <math>v</math> are two <math>p</math>-isometries summing them doesn't in general produces a <math>p</math>-isometry. However as <math>pup^*</math> and <math>qvq^*</math> have disjoint domains and disjoint codomains it is true that <math>pup^* + qvq^*</math> is a <math>p</math>-isometry. Given two types <math>A</math> and <math>B</math>, we thus define their ''tensor'' by:

: <math>A\tens B = \{pup^* + qvq^*, u\in A, v\in B\}\biorth</math>

Note the closure by bidual to make sure that we obtain a type.

From what precedes we see that <math>A\tens B</math> is generated by the internalizations of operators on <math>H\oplus H</math> of the form:
: <math>\begin{pmatrix}
u & 0\\
0 & v
\end{pmatrix}</math>

{{Remark|
This so-called tensor resembles a sum rather than a product. We will stick to this terminology though because it defines the interpretation of the tensor connective of linear logic.
}}

The linear implication is derived from the tensor by duality: given two types <math>A</math> and <math>B</math> the type <math>A\limp B</math> is defined by:
: <math>A\limp B = (A\tens B\orth)\orth</math>.

Unfolding this definition we get:
: <math>A\limp B = \{u\in\mathcal{P}\text{ s.t. } \forall v\in A, \forall w\in B\orth,\, u.(pvp^* + qwq^*) \in\bot\}</math>.

=== The identity ===

Given a type <math>A</math> we are to find an operator <math>\iota</math> in type <math>A\limp A</math>, thus satisfying:
: <math>\forall u\in A, v\in A\orth,\, \iota(pup^* + qvq^*)\in\bot</math>.

An easy solution is to take <math>\iota = pq^* + qp^*</math>. In this way we get <math>\iota(pup^* + qvq^*) = qup^* + pvq^*</math>. Therefore <math>(\iota(pup^* + qvq^*))^2 = quvq^* + pvup^*</math>, from which one deduces that this operator is nilpotent iff <math>uv</math> is nilpotent. It is the case since <math>u</math> is in <math>A</math> and <math>v</math> in <math>A\orth</math>.

It is interesting to note that the <math>\iota</math> thus defined is actually the internalization of the operator on <math>H\oplus H</math> given by the matrix:
: <math>\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}</math>.

We will see once the composition is defined that the <math>\iota</math> operator is the interpretation of the identity proof, as expected.

=== The execution formula, version 1: application ===

{{Definition|
Let <math>u</math> and <math>v</math> be two operators; as above denote by <math>u_{ij}</math> the external components of <math>u</math>. If <math>u_{11}v</math> is nilpotent we define the ''application of <math>u</math> to <math>v</math>'' by:
: <math>\mathrm{App}(u,v) = u_{22} + u_{21}v\sum_k(u_{11}v)^ku_{12}</math>.
}}

Note that the hypothesis that <math>u_{11}v</math> is nilpotent entails that the sum <math>\sum_k(u_{11}v)^k</math> is actually finite. It would be enough to assume that this sum converges. For simplicity we stick to the nilpotency condition, but we should mention that weak nilpotency would do as well.

{{Theorem|
If <math>u</math> and <math>v</math> are <math>p</math>-isometries such that <math>u_{11}v</math> is nilpotent, then <math>\mathrm{App}(u,v)</math> is also a <math>p</math>-isometry.
}}

{{Proof|
Let us note <math>E_k = u_{21}v(u_{11}v)^ku_{12}</math>. Recall that <math>u_{22}</math> and <math>u_{12}</math> being external components of the <math>p</math>-isometry <math>u</math>, they have disjoint domains. Thus it is also the case of <math>u_{22}</math> and <math>E_k</math>. Similarly <math>u_{22}</math> and <math>E_k</math> have disjoint codomains because <math>u_{22}</math> and <math>u_{21}</math> have disjoint codomains.

Let now <math>k</math> and <math>l</math> be two integers such that <math>k>l</math> and let us compute for example the intersection of the codomains of <math>E_k</math> and <math>E_l</math>:
: <math>
E_kE^*_kE_lE^*_l = (u_{21}v(u_{11}v)^ku_{12})(u^*_{12}(v^*u^*_{11})^kv^*u^*_{21})(u_{21}v(u_{11}v)^lu_{12})(u^*_{12}(v^*u^*_{11})^lv^*u_{21}^*)
</math>
As <math>k>l</math> we may write <math>(v^*u_{11}^*)^l = (v^*u^*_{11})^{k-l-1}v^*u^*_{11}(v^*u^*_{11})^l</math>. Let us note <math>E = u^*_{11}(v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^lu_{12}</math> so that <math>E_kE^*_kE_lE^*_l = u_{21}v(u_{11}v)^ku_{12}u^*_{12}(v^*u^*_{11})^{k-l-1}v^*Eu^*_{12}(v^*u^*_{11})^lv^*u_{21}^*</math>. We have:
: <math>\begin{align}
E &= u^*_{11}(v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^lu_{12}\\
&= (u^*_{11}u_{11}u^*_{11})(v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^lu_{12}\\
&= u^*_{11}(u_{11}u^*_{11})\bigl((v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^l\bigr)u_{12}\\
&= u^*_{11}\bigl((v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^l\bigr)(u_{11}u^*_{11})u_{12}\\
&= u^*_{11}(v^*u^*_{11})^lv^*u_{21}^*u_{21}v(u_{11}v)^lu_{11}u^*_{11}u_{12}\\
&= 0
\end{align}</math>
because <math>u_{11}</math> and <math>u_{12}</math> have disjoint codomains, thus <math>u^*_{11}u_{12} = 0</math>.

Similarly we can show that <math>E_k</math> and <math>E_l</math> have disjoint domains. Therefore we have proved that all terms of the sum <math>\mathrm{App}(u,v)</math> have disjoint domains and disjoint codomains. Consequently <math>\mathrm{App}(u,v)</math> is a <math>p</math>-isometry.
}}

{{Theorem|
Let <math>A</math> and <math>B</math> be two types and <math>u</math> a <math>p</math>-isometry. Then the two following conditions are equivalent:
# <math>u\in A\limp B</math>;
# for any <math>v\in A</math> we have:
#* <math>u_{11}v</math> is nilpotent and
#* <math>\mathrm{App}(u, v)\in B</math>.
}}

{{Proof|
Let <math>v</math> and <math>w</math> be two <math>p</math>-isometries. If we compute
: <math>(u.(pvp^* + qwq^*))^n = \bigl((pu_{11}p^* + pu_{12}q^* + qu_{21}p^* + qu_{22}q^*)(pvp^* + qwq^*)\bigr)^n</math>
we get a finite sum of monomial operators of the form:
# <math>p(u_{11}v)^{i_0}u_{12}w(u_{22}w)^{i_1}\dots u_{21}v(u_{11}v)^{i_m}p^*</math>
# <math>p(u_{11}v)^{i_0}u_{12}w(u_{22}w)^{i_1}\dots u_{12}w(u_{22}w)^{i_m}q^*</math>,
# <math>q(u_{22}w)^{i_0}u_{21}v(u_{11}v)^{i_1}\dots u_{21}v(u_{11}v)^{i_m}p^*</math> or
# <math>q(u_{22}w)^{i_0}u_{21}v(u_{11}v)^{i_1}\dots u_{12}w(u_{22}w)^{i_m}q^*</math>,
for all tuples of (nonnegative) integers <math>(i_1,\dots, i_m)</math> such that <math>i_0+\cdots+i_m+m = n</math>.

Each of these monomial is a <math>p</math>-isometry. Furthermore they have disjoint domains and disjoint codomains because their sum is the <math>p</math>-isometry <math>(u.(pvp^* + qwq^*))^n</math>. This entails that <math>(u.(pvp^* + qwq^*))^n = 0</math> iff all these monomials are null.

Suppose <math>u_{11}v</math> is nilpotent and consider:
: <math>\bigl(\mathrm{App}(u,v)w\bigr)^n = \biggl(\bigl(u_{22} + u_{21}v\sum_k(u_{11}v)^k u_{12}\bigr)w\biggr)^n</math>.
Developping we get a finite sum of monomials of the form:
: 5. <math>(u_{22}w)^{l_0}u_{21}v(u_{11}v)^{k_1}u_{12}w(u_{22}w)^{l_1}\dots u_{21}v(u_{11}v)^{k_m}u_{12}w(u_{22}w)^{l_m}</math>
for all tuples <math>(l_0, k_1, l_1,\dots, k_m, l_m)</math> such that <math>l_0\cdots l_m + m = n</math> and <math>k_i</math> is less than the degree of nilpotency of <math>u_{11}v</math> for all <math>i</math>.

Again as these monomials are <math>p</math>-isometries and their sum is the <math>p</math>-isometry <math>(\mathrm{App}(u,v)w)^n</math>, they have pairwise disjoint domains and pairwise disjoint codomains. Note that each of these monomial is equal to <math>q^*Mq</math> where <math>M</math> is a monomial of type 4 above.

As before we thus have that <math>\bigl(\mathrm{App}(u,v)w\bigr)^n = 0</math> iff all monomials of type 5 are null.

Suppose now that <math>u\in A\limp B</math> and <math>v\in A</math>. Then, since <math>0\in B\orth</math> (<math>0</math> belongs to any type) <math>u.(pvp^*) = pu_{11}vp^*</math> is nilpotent, thus <math>u_{11}v</math> is nilpotent.

Suppose further that <math>w\in B\orth</math>. Then <math>u.(pvp^*+qwq^*)</math> is nilpotent, thus there is a <math>N</math> such that <math>(u.(pvp^* + qwq^*))^n=0</math> for any <math>n\geq N</math>. This entails that all monomials of type 1 to 4 are null. Therefore all monomials appearing in the developpment of <math>(\mathrm{App}(u,v)w)^N</math> are null which proves that <math>\mathrm{App}(u,v)w</math> is nilpotent. Thus <math>\mathrm{App}(u,v)\in B</math>.

Conversely suppose for any <math>v\in A</math> and <math>w\in B\orth</math>, <math>u_{11}v</math> and <math>\mathrm{App}(u,v)w</math> are nilpotent. Let <math>P</math> and <math>N</math> be their respective degrees of nilpotency and put <math>n=N(P+1)+N</math>. Then we claim that all monomials of type 1 to 4 appearing in the development of <math>(u.(pvp^*+qwq^*))^n</math> are null.

Consider for example a monomial of type 1:
: <math>p(u_{11}v)^{i_0}u_{12}w(u_{22}w)^{i_1}\dots u_{21}v(u_{11}v)^{i_m}p^*</math>
with <math>i_0+\cdots+i_m + m = n</math>. Note that <math>m</math> must be even.

If <math>i_{2k}\geq P</math> for some <math>0\leq k\leq m/2</math> then <math>(u_{11}v)^{i_{2k}}=0</math> thus our monomial is null. Otherwise if <math>i_{2k}<P</math> for all <math>k</math> we have:
: <math>i_1+i_3+\cdots +i_{m-1} + m/2 = n - m/2 - (i_0+i_2+\cdots +i_m)</math>
thus:
: <math>i_1+i_3+\cdots +i_{m-1} + m/2\geq n - m/2 - (1+m/2)P</math>.
Now if <math>m/2\geq N</math> then <math>i_1+\cdots+i_{m-1}+m/2 \geq N</math>. Otherwise <math>1+m/2\leq N</math> thus
: <math>i_1+i_3+\cdots +i_{m-1} + m/2\geq n - N - NP = N</math>.
Since <math>N</math> is the degree of nilpotency of <math>\mathrm{App}(u,v)w</math> we have that the monomial:
: <math>(u_{22}w)^{i_1}u_{21}v(u_{11}v)^{i_2}u_{12}w\dots(u_{11}v)^{i_{m-2}}u_{12}w(u_{22}w)^{i_{m-1}}</math>
is null, thus also the monomial of type 1 we started with.
}}

{{Corollary|
If <math>A</math> and <math>B</math> are types then we have:
: <math>A\limp B = \{u\in\mathcal{P} \text{ such that }\forall v\in A: u_{11}v\in\bot\text{ and } \mathrm{App}(u, v)\in B\}</math>.
}}

As an example if we compute the application of the interpretation of the identity <math>\iota</math> in type <math>A\limp A</math> to the operator <math>v\in A</math> then we have:
: <math>\mathrm{App}(\iota, v) = \iota_{22} + \iota_{21}v\sum(\iota_{11}v)^k\iota_{12}</math>.
Now recall that <math>\iota = pq^* + qp^*</math> so that <math>\iota_{11} = \iota_{22} = 0</math> and <math>\iota_{12} = \iota_{21} = 1</math> and we thus get:
: <math>\mathrm{App}(\iota, v) = v</math>
as expected.

=== The tensor rule ===

Let now <math>A, A', B</math> and <math>B'</math> be types and consider two operators <math>u</math> and <math>u'</math> respectively in <math>A\limp B</math> and <math>A\limp B'</math>. We define an operator denoted by <math>u\tens u'</math> by:
: <math>\begin{align}
u\tens u' &= ppp^*upp^*p^* + qpq^*upp^*p^* + ppp^*uqp^*q^* + qpq^*uqp^*q^*\\
&+ pqp^*u'pq^*p^* + qqq^*u'pq^*p^* + pqp^*u'qq^*q^* + qqq^*u'qq^*q^*
\end{align}</math>

Once again the notation is motivated by linear logic syntax and is contradictory with linear algebra practice since what we denote by <math>u\tens u'</math> actually is the internalization of the direct sum <math>u\oplus u'</math>.

Indeed if we think of <math>u</math> and <math>u'</math> as the internalizations of the matrices:
: <math>
\begin{pmatrix}u_{11} & u_{12}\\
u_{21} & u_{22}
\end{pmatrix}
</math> and <math>
\begin{pmatrix}u'_{11} & u'_{12}\\
u'_{21} & u'_{22}
\end{pmatrix}</math>
then we may write:
: <math>\begin{align}
u\tens u' &= ppu_{11}p^*p^* + qpu_{21}p^*p^* + ppu_{12}p^*q^* + qpu_{22}p^*q^*\\
&+ pqu'_{11}q^*p^* + qqu'_{21}q^*p^* + pqu'_{12}q^*q^* + qqu'_{22}q^*q^*
\end{align}</math>

Thus the components of <math>u\tens u'</math> are given by:
: <math>(u\tens u')_{ij} = pu_{ij}p^* + qu'_{ij}q^*</math>.
and we see that <math>u\tens u'</math> is actually the internalization of the matrix:
: <math>
\begin{pmatrix}
u_{11} & 0 & u_{12} & 0 \\
0 & u'_{11} & 0 & u'_{12} \\
u_{21} & 0 & u_{22} & 0 \\
0 & u'_{21} & 0 & u'_{22} \\
\end{pmatrix}
</math>

We are now to show that if we suppose <math>u</math>and <math>u'</math> are in types <math>A\limp B</math> and <math>A'\limp B'</math>, then <math>u\tens u'</math> is in <math>A\tens A'\limp B\tens B'</math>. For this we consider <math>v</math> and <math>v'</math> in respectively in <math>A</math> and <math>A'</math>, so that <math>pvp^* + qv'q^*</math> is in <math>A\tens A'</math>, and we show that <math>\mathrm{App}(u\tens u', pvp^* + qv'q^*)\in B\tens B'</math>.

Since <math>u</math> and <math>u'</math> are in <math>A\limp B</math> and <math>A'\limp B'</math> we have that <math>\mathrm{App}(u, v)</math> and <math>\mathrm{App}(u', v')</math> are respectively in <math>B</math> and <math>B'</math>, thus:
: <math>p\mathrm{App}(u, v)p^* + q\mathrm{App}(u', v')q^* \in B\tens B'</math>.

We know that both <math>u_{11}v</math> and <math>u'_{11}v'</math> are nilpotent. But we have:
: <math>\begin{align}
\bigl((u\tens u')_{11}(pvp^* + qv'q^*)\bigr)^n
&= \bigl((pu_{11} + qu'_{11}q^*)(pvp^* + qv'q^*)\bigr)^n\\
&= (pu_{11}vp^* + qu'_{11}v'q^*)^n\\
&= p(u_{11}v)^np^* + q(u'_{11}v')^nq^*
\end{align}</math>

Therefore <math>(u\tens u')_{11}(pvp^* + qv'q^*)</math> is nilpotent. So we can compute <math>\mathrm{App}(u\tens u', pvp^* + qv'q^*)</math>:
: <math>\begin{align}
&\mathrm{App}(u\tens u', pvp^* + qv'q^*)\\
&= (u\tens u')_{22} + (u\tens u')_{21}(pvp^* + qv'q^*)\sum\bigl((u\tens u')_{11}(pvp^* + qv'q^*)\bigr)^k(u\tens u')_{12}\\
&= pu_{22}p^* + qu'_{22}q^* + (pu_{21}p^* + qu'_{21}q^*)(pvp^* + qv'q^*)\sum\bigl((pu_{11}p^* + qu'_{11}q^*)(pvp^* + qv'q^*)\bigr)^k(pu_{12}p^* + qu'_{12}q^*)\\
&= p\bigl(u_{22} + u_{21}v\sum(u_{11}v)^ku_{12}\bigr)p^* + q\bigl(u'_{22} + u'_{21}v'\sum(u'_{11}v')^ku'_{12}\bigr)q^*\\
&= p\mathrm{App}(u, v)p^* + q\mathrm{App}(u', v')q^*
\end{align}</math>
thus lives in <math>B\tens B'</math>.

=== Other monoidal constructions ===

==== Contraposition ====

Let <math>A</math> and <math>B</math> be some types; we have:
: <math>A\limp B = A\orth\limpinv B\orth</math>

Indeed, <math>u\in A\limp B</math> means that for any <math>v</math> and <math>w</math> in respectively <math>A</math> and <math>B\orth</math> we have <math>u.(pvp^* + qwq^*)\in\bot</math> which is exactly the definition of <math>A\orth\limpinv B\orth</math>.

We will denote <math>u\orth</math> the operator:
: <math>u\orth = pu_{22}p^* + pu_{12}q^* + qu_{12}p^* + qu_{11}q^*</math>
where <math>u_{ij}</math> is given by externalization. Therefore the externalization of <math>u\orth</math> is:
: <math>(u\orth)_{ij} = u_{\bar i\,\bar j}</math> where <math>\bar .</math> is defined by <math>\bar1 = 2, \bar2 = 1</math>.
From this we deduce that <math>u\orth\in B\orth\limp A\orth</math> and that <math>(u\orth)\orth = u</math>.

==== Commutativity ====
Let <math>\sigma</math> be the operator:
: <math>\sigma = ppq^*q^* +pqp^*q^* + qpq^*p^* + qqp^*p^*</math>.
One can check that <math>\sigma</math> is the internalization of the operator <math>S</math> on <math>H\oplus H\oplus H\oplus H</math> defined by: <math>S(x_1\oplus x_2\oplus x_3\oplus x_4) = x_4\oplus x_3\oplus x_2\oplus x_1</math>. In particular the components of <math>\sigma</math> are:
: <math>\sigma_{11} = \sigma_{22} = 0</math>;
: <math>\sigma_{12} = \sigma_{21} = pq^* + qp^*</math>.

Let <math>A</math> and <math>B</math> be types and <math>u</math> and <math>v</math> be operators in <math>A</math> and <math>B</math>. Then <math>pup^* + qvq^*</math> is in <math>A\tens B</math> and as <math>\sigma_{11}.(pup^* + qvq^*) = 0</math> we may compute:
: <math>\begin{align}
\mathrm{App}(\sigma, pup^* + qvq^*)
&= \sigma_{22} + \sigma_{21}(pup^* + qvq^*)\sum(\sigma_{11}(pup^* + qvq^*))^k\sigma_{12}\\
&= (pq^* + qp^*)(pup^* + qvq^*)(pq^* + qp^*)\\
&= pvp^* + quq^*
\end{align}</math>
But <math>pvp^* + quq^*\in B\tens A</math>, thus we have shown that:
: <math>\sigma\in (A\tens B) \limp (B\tens A)</math>.

==== Distributivity ====
We get distributivity by considering the operator:
: <math>\delta = ppp^*p^*q^* + pqpq^*p^*q^* + pqqq^*q^* + qppp^*p^* + qpqp^*q^*p^* + qqq^*q^*p^*</math>
that is similarly shown to be in type <math>A\tens(B\tens C)\limp(A\tens B)\tens C</math> for any types <math>A</math>, <math>B</math> and <math>C</math>.

==== Weak distributivity ====
We can finally get weak distributivity thanks to the operators:
: <math>\delta_1 = pppp^*q^* + ppqp^*q^*q^* + pqq^*q^*q^* + qpp^*p^*p^* + qqp q^*p^*p^* + qqq q^*p^*</math> and
: <math>\delta_2 = ppp^*p^*q^* + pqpq^*p^*q^* + pqqq^*q^* + qppp^*p^* + qpqp^*q^*p^* + qqq^*q^*p^*</math>.

Given three types <math>A</math>, <math>B</math> and <math>C</math> then one can show that:
: <math>\delta_1</math> has type <math>((A\limp B)\tens C)\limp A\limp (B\tens C)</math> and
: <math>\delta_2</math> has type <math>(A\tens(B\limp C))\limp (A\limp B)\limp C</math>.

=== Execution formula, version 2: composition ===

Let <math>A</math>, <math>B</math> and <math>C</math> be types and <math>u</math> and <math>v</math> be operators respectively in types <math>A\limp B</math> and <math>B\limp C</math>.

As usual we will denote <math>u_{ij}</math> and <math>v_{ij}</math> the operators obtained by externalization of <math>u</math> and <math>v</math>, eg, <math>u_{11} = p^*up</math>, ...

As <math>u</math> is in <math>A\limp B</math> we have that <math>\mathrm{App}(u, 0)=u_{22}\in B</math>; similarly as <math>v\in B\limp C</math>, thus <math>v\orth\in C\orth\limp B\orth</math>, we have <math>\mathrm{App}(v\orth, 0) = v_{11}\in B\orth</math>. Thus <math>u_{22}v_{11}</math> is nilpotent.

We define the operator <math>\mathrm{Comp}(u, v)</math> by:
: <math>\begin{align}
\mathrm{Comp}(u, v) &= p(u_{11} + u_{12}\sum(v_{11}u_{22})^k\,v_{11}u_{21})p^*\\
&+ p(u_{12}\sum(v_{11}u_{22})^k\,v_{12})q^*\\
&+ q(v_{21}\sum(u_{22}v_{11})^k\,u_{21})p^*\\
&+ q(v_{22} + v_{21}\sum(u_{22}v_{11})^k\,u_{22}v_{12})q^*
\end{align}</math>

This is well defined since <math>u_{11}v_{22}</math> is nilpotent. As an example let us compute the composition of <math>u</math> and <math>\iota</math> in type <math>B\limp B</math>; recall that <math>\iota_{ij} = \delta_{ij}</math>, so we get:
: <math>
\mathrm{Comp}(u, \iota) = pu_{11}p^* + pu_{12}q^* + qu_{21}p^* + qu_{22}q^* = u
</math>
Similar computation would show that <math>\mathrm{Comp}(\iota, v) = v</math> (we use <math>pp^* + qq^* = 1</math> here).

Coming back to the general case we claim that <math>\mathrm{Comp}(u, v)</math> is in <math>A\limp C</math>: let <math>a</math> be an operator in <math>A</math>. By computation we can check that:
: <math>\mathrm{App}(\mathrm{Comp}(u, v), a) = \mathrm{App}(v, \mathrm{App}(u, a))</math>.
Now since <math>u</math> is in <math>A\limp B</math>, <math>\mathrm{App}(u, a)</math> is in <math>B</math> and since <math>v</math> is in <math>B\limp C</math>, <math>\mathrm{App}(v, \mathrm{App}(u, a))</math> is in <math>C</math>.

If we now consider a type <math>D</math> and an operator <math>w</math> in <math>C\limp D</math> then we have:
: <math>\mathrm{Comp}(\mathrm{Comp}(u, v), w) = \mathrm{Comp}(u,
\mathrm{Comp}(v, w))</math>.

Putting together the results of this section we finally have:
{{Theorem|
Let GoI(H) be defined by:
* objects are types, ''ie'' sets <math>A</math> of operators satisfying: <math>A\biorth = A</math>;
* morphisms from <math>A</math> to <math>B</math> are operators in type <math>A\limp B</math>;
* composition is given by the formula above.

Then GoI(H) is a star-autonomous category.
}}

= The Geometry of Interaction as an abstract machine =

Geometry of interaction

2010-04-29T10:15:06Z

Emmanuel Beffara: typograhic fixes, references

Sequent calculus

2009-03-14T14:43:50Z

Emmanuel Beffara: Markup and a typo

This article presents the language and sequent calculus of second-order
linear logic and the basic properties of this sequent calculus.
The core of the article uses the two-sided system with negation as a proper
connective; the [[#One-sided sequent calculus|one-sided system]], often used
as the definition of linear logic, is presented at the end of the page.

== Formulas ==

Atomic formulas, written <math>\alpha,\beta,\gamma</math>, are predicates of
the form <math>p(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms
from some first-order language.
The predicate symbol <math>p</math> may be either a predicate constant or a
second-order variable.
By convention we will write first-order variables as <math>x,y,z</math>,
second-order variables as <math>X,Y,Z</math>, and <math>\xi</math> for a
variable of arbitrary order (see [[Notations]]).

Formulas, represented by capital letters <math>A</math>, <math>B</math>,
<math>C</math>, are built using the following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>A\orth</math>
| negation
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists \xi.A</math>
| there exists
| <math>\forall \xi.A</math>
| for all
| quantifiers
|}

Each line (except the first one) corresponds to a particular class of
connectives, and each class consists in a pair of connectives.
Those in the left column are called [[positive formula|positive]] and those in
the right column are called [[negative formula|negative]].
The ''tensor'' and ''with'' connectives are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally read
''of course <math>A</math>'' for <math>\oc A</math> and ''why not
<math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic, as
<math>A\limp B:=A\orth\parr B</math>.

Free and bound variables and first-order substitution are defined in the
standard way.
Formulas are always considered up to renaming of bound names.
If <math>A</math> is a formula, <math>X</math> is a second-order variable and
<math>B[x_1,\ldots,x_n]</math> is a formula with variables <math>x_i</math>,
then the formula <math>A[B/X]</math> is <math>A</math> where every atom
<math>X(t_1,\ldots,t_n)</math> is replaced by <math>B[t_1,\ldots,t_n]</math>.

== Sequents and proofs ==

A sequent is an expression <math>\Gamma\vdash\Delta</math> where
<math>\Gamma</math> and <math>\Delta</math> are finite multisets of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation
<math>\wn\Gamma</math> represents the multiset
<math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:
* Identity group: <math>
\LabelRule{\rulename{axiom}}
\NulRule{ A \vdash A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{\rulename{cut}}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>
* Negation: <math>
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\LabelRule{n_L}
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\LabelRule{n_R}
\DisplayProof
</math>
* Multiplicative group:
** tensor: <math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
</math>
** par: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma', B \vdash \Delta' }
\LabelRule{ \parr_L }
\BinRule{ \Gamma, \Gamma', A \parr B \vdash \Delta, \Delta' }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, B, \Delta }
\LabelRule{ \parr_R }
\UnaRule{ \Gamma \vdash A \parr B, \Delta }
\DisplayProof
</math>
** one: <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>
** bottom: <math>
\LabelRule{ \bot_L }
\NulRule{ \bot \vdash }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \bot_R }
\UnaRule{ \Gamma \vdash \bot, \Delta }
\DisplayProof
</math>
* Additive group:
** plus: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
</math>
** with: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ \with_{L1} }
\UnaRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \with_{L2} }
\UnaRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash, A \Delta }
\AxRule{ \Gamma \vdash, B \Delta }
\LabelRule{ \with_R }
\BinRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math>
** zero: <math>
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>
** top: <math>
\LabelRule{ \top_R }
\NulRule{ \Gamma \vdash \top, \Delta }
\DisplayProof
</math>
* Exponential group:
** of course: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>
** why not: <math>
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ d_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \wn A, \wn A, \Delta }
\LabelRule{ c_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ A \vdash \wn B_1, \ldots, \wn B_n }
\LabelRule{ \wn_L }
\UnaRule{ \wn A \vdash \wn B_1, \ldots, \wn B_n }
\DisplayProof
</math>
* Quantifier group (in the <math>\exists_L</math> and <math>\forall_R</math> rules, <math>\xi</math> must not occur free in <math>\Gamma</math> or <math>\Delta</math>):
** there exists: <math>
\AxRule{ \Gamma , A \vdash \Delta }
\LabelRule{ \exists_L }
\UnaRule{ \Gamma, \exists\xi.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A[t/x] }
\LabelRule{ \exists^1_R }
\UnaRule{ \Gamma \vdash \Delta, \exists x.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A[B/X] }
\LabelRule{ \exists^2_R }
\UnaRule{ \Gamma \vdash \Delta, \exists X.A }
\DisplayProof
</math>
** for all: <math>
\AxRule{ \Gamma, A[t/x] \vdash \Delta }
\LabelRule{ \forall^1_L }
\UnaRule{ \Gamma, \forall x.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, A[B/X] \vdash \Delta }
\LabelRule{ \forall^2_L }
\UnaRule{ \Gamma, \forall X.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A }
\LabelRule{ \forall_R }
\UnaRule{ \Gamma \vdash \Delta, \forall\xi.A }
\DisplayProof
</math>

The left rules for ''of course'' and right rules for ''why not'' are called
''dereliction'', ''weakening'' and ''contraction'' rules.
The right rule for ''of course'' and the left rule for ''why not'' are called
''promotion'' rules.
Note the fundamental fact that there are no contraction and weakening rules
for arbitrary formulas, but only for the formulas starting with the
<math>\wn</math> modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An alternative presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rules:
: <math>
\AxRule{ \Gamma_1, A, B, \Gamma_2 \vdash \Delta }
\LabelRule{\rulename{exchange}_L}
\UnaRule{ \Gamma_1, B, A, \Gamma_2 \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta_1, A, B, \Delta_2 }
\LabelRule{\rulename{exchange}_R}
\UnaRule{ \Gamma \vdash \Delta_1, B, A, \Delta_2 }
\DisplayProof
</math>

== Equivalences ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent,
written <math>A\linequiv B</math>, if both implications <math>A\limp B</math>
and <math>B\limp A</math> are provable.
Equivalently, <math>A\linequiv B</math> if both <math>A\vdash B</math> and
<math>B\vdash A</math> are provable.
Another formulation of <math>A\linequiv B</math> is that, for all
<math>\Gamma</math> and <math>\Delta</math>, <math>\Gamma\vdash\Delta,A</math>
is provable if and only if <math>\Gamma\vdash\Delta,B</math> is provable.

Two related notions are [[isomorphism]] (stronger than equivalence) and
[[equiprovability]] (weaker than equivalence).

=== De Morgan laws ===

Negation is involutive:
: <math>A\linequiv A\biorth</math>
Duality between connectives:
{|
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>\linequiv A\orth \parr B\orth </math>
|width=30|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>\linequiv A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>\linequiv \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>\linequiv \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>\linequiv A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>\linequiv A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>\linequiv \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>\linequiv \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>\linequiv \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>\linequiv \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists \xi.A )\orth </math>
| <math>\linequiv \forall \xi.( A\orth ) </math>
|
|align="right"| <math> ( \forall \xi.A )\orth </math>
| <math>\linequiv \exists \xi.( A\orth ) </math>
|}

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality:
*: <math>
A \tens (B \tens C) \linequiv (A \tens B) \tens C </math> &emsp; <math>
A \tens B \linequiv B \tens A </math> &emsp; <math>
A \tens \one \linequiv A </math>
*: <math>
A \parr (B \parr C) \linequiv (A \parr B) \parr C </math> &emsp; <math>
A \parr B \linequiv B \parr A </math> &emsp; <math>
A \parr \bot \linequiv A </math>
*: <math>
A \plus (B \plus C) \linequiv (A \plus B) \plus C </math> &emsp; <math>
A \plus B \linequiv B \plus A </math> &emsp; <math>
A \plus \zero \linequiv A </math>
*: <math>
A \with (B \with C) \linequiv (A \with B) \with C </math> &emsp; <math>
A \with B \linequiv B \with A </math> &emsp; <math>
A \with \top \linequiv A </math>
* Idempotence of additives:
*: <math>
A \plus A \linequiv A </math> &emsp; <math>
A \with A \linequiv A </math>
* Distributivity of multiplicatives over additives:
*: <math>
A \tens (B \plus C) \linequiv (A \tens B) \plus (A \tens C) </math> &emsp; <math>
A \tens \zero \linequiv \zero </math>
*: <math>
A \parr (B \with C) \linequiv (A \parr B) \with (A \parr C) </math> &emsp; <math>
A \parr \top \linequiv \top </math>
* Defining property of exponentials:
*: <math>
\oc(A \with B) \linequiv \oc A \tens \oc B </math> &emsp; <math>
\oc\top \linequiv \one </math>
*: <math>
\wn(A \plus B) \linequiv \wn A \parr \wn B </math> &emsp; <math>
\wn\zero \linequiv \bot </math>
* Monoidal structure of exponentials:
*: <math>
\oc A \tens \oc A \linequiv \oc A </math> &emsp; <math>
\oc \one \linequiv \one </math>
*: <math>
\wn A \parr \wn A \linequiv \wn A </math> &emsp; <math>
\wn \bot \linequiv \bot </math>
* Digging:
*: <math>
\oc\oc A \linequiv \oc A </math> &emsp; <math>
\wn\wn A \linequiv \wn A </math>
* Other properties of exponentials:
*: <math>
\oc\wn\oc\wn A \linequiv \oc\wn A </math> &emsp; <math>
\oc\wn \one \linequiv \one </math>
*: <math>
\wn\oc\wn\oc A \linequiv \wn\oc A </math> &emsp; <math>
\wn\oc \bot \linequiv \bot </math>
* Commutation of quantifiers (<math>\zeta</math> does not occur in <math>A</math>):
*: <math>
\exists \xi. \exists \psi. A \linequiv \exists \psi. \exists \xi. A </math> &emsp; <math>
\exists \xi.(A \plus B) \linequiv \exists \xi.A \plus \exists \xi.B </math> &emsp; <math>
\exists \zeta.(A\tens B) \linequiv A\tens\exists \zeta.B </math> &emsp; <math>
\exists \zeta.A \linequiv A </math>
*: <math>
\forall \xi. \forall \psi. A \linequiv \forall \psi. \forall \xi. A </math> &emsp; <math>
\forall \xi.(A \with B) \linequiv \forall \xi.A \with \forall \xi.B </math> &emsp; <math>
\forall \zeta.(A\parr B) \linequiv A\parr\forall \zeta.B </math> &emsp; <math>
\forall \zeta.A \linequiv A </math>

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:
* <math> \zero \linequiv \forall X.X </math>
* <math> \one \linequiv \forall X.(X \limp X) </math>
* <math> A \plus B \linequiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X) </math>
The constants <math>\top</math> and <math>\bot</math> and the connective
<math>\with</math> can be defined by duality.

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

== Properties of proofs ==

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\Gamma\vdash\Delta</math>, there is a proof of
<math>\Gamma\vdash\Delta</math> if and only if there is a proof of
<math>\Gamma\vdash\Delta</math> that does not use the cut rule.}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas of its immediate subformulas:
* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,
* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,
* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas have no immediate subformula,
* the immediate subformulas of <math>\exists x.A</math> and <math>\forall x.A</math> are all the <math>A[t/x]</math> for all first-order terms <math>t</math>,
* the immediate subformulas of <math>\exists X.A</math> and <math>\forall X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math> (with the appropriate number of parameters).}}

{{Theorem|title=subformula property|
A sequent <math>\Gamma\vdash\Delta</math> is provable if and only if it is the conclusion of a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math> and <math>\Delta</math>.}}
{{Proof|By the cut elimination theorem, if a sequent is provable, then it is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its
negation <math>A\orth</math>; it is impossible to prove <math>\zero</math> or
<math>\bot</math>.}}
{{Proof|If a sequent is provable, then it is the conclusion of a cut-free proof.
In each rule except the cut rule, there is at least one formula in conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.
The other properties are immediate consequences: if <math>\vdash A\orth</math>
and <math>\vdash A</math> are provable, then by the left negation rule
<math>A\orth\vdash</math> is provable, and by the cut rule one gets empty
conclusion, which is not possible.
As particular cases, since <math>\one</math> and <math>\top</math> are
provable, <math>\bot</math> and <math>\zero</math> are not, since they are
equivalent to <math>\one\orth</math> and <math>\top\orth</math>
respectively.}}

=== Expansion of identities ===

Let us write <math>\pi:\Gamma\vdash\Delta</math> to signify that
<math>\pi</math> is a proof with conclusion <math>\Gamma\vdash\Delta</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the
same conclusion as <math>\pi</math> in which the axiom rule is only used with
atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.}}
{{Proof|It suffices to prove that for every formula <math>A</math>, the sequent
<math>A\vdash A</math> has a cut-free proof in which the axiom rule is used
only for atomic formulas.
We prove this by induction on <math>A</math>.
* If <math>A</math> is atomic, then <math>A\vdash A</math> is an instance of the atomic axiom rule.
* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 : A_1 \vdash A_1 }
\AxRule{ \pi_2 : A_2 \vdash A_2 }
\LabelRule{ \tens_R }
\BinRule{ A_1, A_2 \vdash A_1 \tens A_2 }
\LabelRule{ \tens_L }
\UnaRule{ A_1 \tens A_2 \vdash A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* If <math>A=A_1\parr A_2</math> then we have <math>
\AxRule{ \pi_1 : A_1 \vdash A_1 }
\AxRule{ \pi_2 : A_2 \vdash A_2 }
\LabelRule{ \parr_L }
\BinRule{ A_1 \parr A_2 \vdash A_1, A_2 }
\LabelRule{ \parr_R }
\UnaRule{ A_1 \parr A_2 \vdash A_1 \parr A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* All other connectives follow the same pattern.}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if
* for every proof <math>\pi:\Gamma\vdash\Delta,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,
* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are reversible.}}
{{Proof|Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only applied to atomic formulas.
Then each top-level connective is introduced either by its associated (left or
right) rule or in an instance of the <math>\zero_L</math> or
<math>\top_R</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\Gamma\vdash\Delta,A\parr
B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr_R</math> rule (not
necessarily the last rule in <math>\pi</math>), then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction on <math>\pi</math>).
If it is introduced in the context of a <math>\zero_L</math> or
<math>\top_R</math> rule, then this rule can be changed so that
<math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the
expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a
<math>\bot_R</math> rule, then remove this rule to get a proof of
<math>\vdash\Gamma</math>, if it is introduced by a <math>\zero_L</math> or
<math>\top_R</math> rule, remove the <math>\bot</math> from this rule, then
apply the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof
<math>\pi:\Gamma\vdash\Delta,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is
applied in a context like

<math>
\AxRule{ \pi_1 \Gamma' \vdash \Delta', A }
\AxRule{ \pi_2 \Gamma' \vdash \Delta', B }
\LabelRule{ \with }
\BinRule{ \Gamma' \vdash \Delta', A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except
as context), if we replace this step by <math>\pi_1</math> in <math>\pi</math>
we finally get a proof <math>\pi'_1:\Gamma\vdash\Delta,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof
<math>\pi'_2:\Gamma\vdash\Delta,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final
<math>\with</math> rule we finally get the expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math>
rule is solved as before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as
an instance of the <math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math>, consider a proof
<math>\pi:\Gamma\vdash\Delta,\forall\xi.A</math>.
Up to renaming, we can assume that <math>\xi</math> occurs free only above the
rule that introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we
remove this rule, we can check that we get a proof of
<math>\Gamma\vdash\Delta,A</math> on which we can finally apply the
<math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math>
rule is solved as before.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the
original proof.}}

== One-sided sequent calculus ==

The sequent calculus presented above is very symmetric: for every left
introduction rule, there is a right introduction rule for the dual connective
that has the exact same structure.
Moreover, because of the involutivity of negation, a sequent
<math>\Gamma,A\vdash\Delta</math> is provable if and only if the sequent
<math>\Gamma\vdash A\orth,\Delta</math> is provable.
From these remarks, we can define an equivalent one-sided sequent calculus:
* Formulas are considered up to De Morgan duality. Equivalently, one can consider that negation is not a connective but a syntactically defined operation on formulas. In this case, negated atoms <math>\alpha\orth</math> must be considered as another kind of atomic formulas.
* Sequents have the form <math>\vdash\Gamma</math>.
The inference rules are essentially the same except that the left hand side of
sequents is kept empty:
* Identity group:
*: <math>
\LabelRule{\rulename{axiom}}
\NulRule{ \vdash A\orth, A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{\rulename{cut}}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>
* Multiplicative group:
*: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>
* Additive group:
*: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>
* Exponential group:
*: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>
* Quantifier group (in the <math>\forall</math> rule, <math>\xi</math> must not occur free in <math>\Gamma</math>):
*: <math>
\AxRule{ \vdash \Gamma, A[t/x] }
\LabelRule{ \exists^1 }
\UnaRule{ \vdash \Gamma, \exists x.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists^2 }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall \xi.A }
\DisplayProof
</math>

{{Theorem|A two-sided sequent <math>\Gamma\vdash\Delta</math> is provable if
and only if the sequent <math>\vdash\Gamma\orth,\Delta</math> is provable in
the one-sided system.}}

The one-sided system enjoys the same properties as the two-sided one,
including cut elimination, the subformula property, etc.
This formulation is often used when studying proofs because it is much lighter
than the two-sided form while keeping the same expressiveness.
In particular, [[proof-nets]] can be seen as a quotient of one-sided sequent
calculus proofs under commutation of rules.

== Variations ==

The same remarks that lead to the definition of the one-sided calculus can
lead the definition of other simplified systems:
* A one-sided variant with sequents of the form <math>\Gamma\vdash</math> could be defined.
* When considering formulas up to De Morgan duality, an equivalent system is obtained by considering only the left and right rules for positive connectives (or the ones for negative connectives only, obviously).
* [[Intuitionistic linear logic]] is the two-sided system where the right-hand side is constrained to always contain exactly one formula (with a few associated restrictions).
* Similar restrictions are used in various [[semantics]] and [[proof search]] formalisms.

Fragment

2009-03-12T16:01:23Z

Emmanuel Beffara: Updated links to 'sequent calculus'

In general, a '''fragment''' of a logical system ''S'' is a logical system obtained by restricting the language of ''S'', and by restricting the rules of ''S'' accordingly. In linear logic, the most well known fragments are obtained by combining/removing in different ways the classes of connectives present in the [[Sequent calculus|language of linear logic]] itself:

* '''Multiplicative connectives:''' the conjunction <math>\tens</math> (''tensor'') and the disjunction <math>\parr</math> (''par''), with their respective units <math>\one</math> (''one'') and <math>\bot</math> (''bottom''); these connectives are the combinatorial base of linear logic (permutations, circuits, etc.).
* '''Additive connectives:''' the conjunction <math>\with</math> (''with'') and the disjunction <math>\plus</math> (''plus''), with their respective units <math>\top</math> (''top'') and <math>\zero</math> (''zero''); the computational content of these connectives, which behave more closely to their intuitionistic counterparts (''e.g.'', <math>A\with B\limp A</math> and <math>A\with B\limp B</math> are provable), is strongly related to choice (''if...then...else'', product and sum types, etc.).
* '''Exponential connectives:''' the modalities <math>\oc</math> (''of course'') and <math>\wn</math> (''why not'') handle the structural rules in linear logic, and are necessary to recover the expressive power of intuitionistic or classical logic.
* '''Quantifiers:''' just as in classical logic, quantifiers may be added to propositional linear logic, at any order. The most frequently considered are the second order ones (in analogy with System F).

The additive and exponential connectives, if taken alone, yield fragments of limited interest, so one usually considers only fragments containing at least the multiplicative connectives (perhaps without units). It is important to observe that the [[Sequent calculus#Cut elimination and consequences|cut elimination rules]] of linear logic do not introduce connectives belonging to a different class than that of the pair of dual formulas whose cut is being reduced. Hence, any fragment defined by combining the above classes will enjoy cut elimination. Since cut elimination implies the subformula property, all of the [[Sequent calculus#Equivalences|fundamental equivalences]] provable in full linear logic remain valid within such fragments, as soon as the formulas concerned belong to the fragment itself.

Conventionally, if <math>LL</math> denotes full linear logic, its fragments are denoted by prefixing <math>LL</math> with letters corresponding to the classes of connectives being considered: <math>M</math> for multiplicative connectives, <math>A</math> for additive connectives, and <math>E</math> for exponential connectives. Additional subscripts may specify whether units and/or quantifiers are present or not, and, for quantifiers, of what order (see the article on [[notations]]).

== Motivations ==

The main interest of studying fragments of linear logic is that these are usually simpler than the whole system, so that certain properties may be first analyzed on fragments, and then extended or adapted to increasingly larger fragments. It may also be interesting to see, given a property that does not hold for full linear logic, whether it holds for a fragment, and where the "breaking point" is situated. Examples of such questions include:

* '''logical complexity:''' proving cut elimination for full linear logic with second order quantification is equivalent to proving the consistency of second order Peano arithmetic (Girard, via [[Translations of intuitionistic logic|translations of System F]] in linear logic). One may expect that smaller fragments have lower logical complexity.
* '''provability:''' the ''provability problem'' for a logical system ''S'' is defined as follows: given a formula <math>A</math> in the language of ''S'', is <math>A</math> provable in ''S''? This problem is undecidable in full linear logic with quantifiers, of whatever order (again, because [[Translations of classical logic|classical logic can be translated]] in linear logic). On what fragments does it become decidable? And if it does, what is its computational complexity?
* '''computational complexity of cut elimination:''' the ''cut elimination problem'' (Mairson-Terui) for a logical system ''S'' is defined as follows: given two proofs of <math>A</math> in ''S'', do they reduce to the same cut-free proof? Although decidable (thanks to strong normalization), this problem is not elementary recursive in full propositional linear logic (Statman, again via the above-mentioned translations). Does the problem fall into any interesting complexity class when applied to fragments?
* '''proof nets:''' the definition of proof nets, and in particular the formulation of correctness criteria and the study of their complexity, is a good example of how a methodology can be applied to a small fragment of linear logic and later adapted (more or less successfully) to wider fragments.

== Multiplicative fragments ==

Multiplicative linear logic (<math>MLL</math>) is the simplest of the well known fragments of linear logic. Its formulas are obtained by combining propositional atoms with the connectives ''tensor'' and ''par'' only. As a consequence, the [[Sequent calculus#Sequents and proofs|sequent calculus]] of <math>MLL</math> is limited to the rules <math>\rulename{axiom}</math>, <math>\rulename{cut}</math>, <math>\tens</math>, and <math>\parr</math>. These rules actually determine the multiplicative connectives: if a dual pair of connectives <math>\tens'</math> and <math>\parr'</math> is introduced, with the same rules as <math>\tens</math> and <math>\parr</math>, respectively, then one can show <math>A\tens' B</math> to be provably equivalent to <math>A\tens B</math> (and, dually, <math>A\parr'B</math> to be provably equivalent to <math>A\parr B</math>).

The cut elimination problem for <math>MLL</math> is <math>\mathbf P</math>-complete (Mairson-Terui), even though there exists a deterministic algorithm solving the problem in logarithmic space if one considers only [[Sequent calculus#Expansion of identities|eta-expanded]] proofs (Mairson-Terui). On the other hand, provability for <math>MLL</math> is <math>\mathbf{NP}</math>-complete, and it remains so even in presence of first order quantifiers.

Another multiplicative fragment, less considered in the literature, can be defined by using the units <math>\one</math> and <math>\bot</math> instead of the propositional atoms. In this fragment, denoted by <math>MLL_u</math>, one can also eliminate the <math>\rulename{axiom}</math> rule from sequent calculus, since it is redundant. <math>MLL_u</math> is even simpler than <math>MLL</math>: its provability problem is in <math>\mathbf P</math>, and, since all proofs are eta-expandend, its cut elimination problem is in <math>\mathbf L</math>.

The union of <math>MLL</math> and <math>MLL_u</math> is the full propositional multiplicative fragment of linear logic, and is denoted by <math>MLL_0</math>. It has the same properties as <math>MLL</math>, which shows that the presence/absence of propositional atoms (and of the <math>\rulename{axiom}</math> rule) has a non-trivial effect on the complexity of provability and cut elimination, ''i.e.'', the complexity is not altered iff <math>\mathbf P\subsetneq\mathbf{NP}</math> and <math>\mathbf L\subsetneq\mathbf P</math>, respectively.

If we add second order quantifiers to <math>MLL</math> (resp. <math>MLL_u</math>), we obtain a system denoted by <math>MLL_2</math> (resp. <math>MLL_{02}</math>). In <math>MLL_{02}</math> one can show that <math>\one</math> and <math>\bot</math> are provably equivalent to <math>\forall X.(X\orth\parr X)</math> and <math>\exists X.(X\orth\tens X)</math>, respectively. Hence, <math>MLL_2</math> is as expressive as <math>MLL_{02}</math>. In these second order fragments, provability is undecidable, while cut elimination is still <math>\mathbf P</math>-complete.

== Additive fragments ==

The most studied additive fragments of linear logic are defined by taking <math>MLL</math> or <math>MLL_0</math> and by enriching their language with the additive connectives, with or without units. The same can be done in presence of quantifiers. We thus obtain:

* <math>MALL</math>: formulas built from propositional atoms using <math>\tens,\parr,\with,\plus</math>;
* <math>MALL_0</math>: formulas built from propositional atoms and <math>\one,\bot,\top,\zero</math>, using <math>\tens,\parr,\with,\plus</math>;
* <math>MALL_n</math>: <math>MALL</math> with quantifiers of order <math>n</math>;
* <math>MALL_{0n}</math>: <math>MALL_0</math> with quantifiers of order <math>n</math>.

As for the multiplicative connectives, the additive connectives are also defined by their rules: adding a pair of dual connectives <math>\with',\plus'</math> to <math>MALL</math>, and giving them the same rules as <math>\with,\plus</math>, makes the new connectives provably equivalent to the old ones.

In <math>MALL_{02}</math>, the additive units <math>\top</math> and <math>\zero</math> are provably equivalent to <math>\exists X.X\orth</math> and <math>\forall X.X</math>, respectively. Since multiplicative units are also definable in terms of second order quantification, we obtain that <math>MALL_2</math> is as expressive as <math>MALL_{02}</math>.

The cut elimination problem is <math>\mathbf{coNP}</math>-complete for all of the fragments defined above (Mairson-Terui).

Provability is undecidable in any additive fragment as soon as second order quantification is considered. It is decidable, although quite complex, in the propositional and first order case: it is <math>\mathbf{PSPACE}</math>-complete in <math>MALL_0</math>, and <math>\mathbf{NEXP}</math>-complete in <math>MALL_{01}</math>. This latter result is indicative of the fact that the undecidability of predicate calculus is not ascribable to existential quantification alone, but rather to the simultaneous presence of existential quantification and contraction.

== Exponential fragments ==

The most common proper fragments of linear logic containing the exponential connectives are defined as in the case of the additive fragments, ''i.e.'', by adding the modalities on top of <math>MLL</math> and its variants:

* <math>MELL</math>: formulas built from propositional atoms using <math>\tens,\parr,\oc,\wn</math>;
* <math>MELL_0</math>: formulas built from propositional atoms and <math>\one,\bot</math>, using <math>\tens,\parr,\oc,\wn</math>;
* <math>MELL_n</math>: <math>MELL</math> with quantifiers of order <math>n</math>;
* <math>MELL_{0n}</math>: <math>MELL_0</math> with quantifiers of order <math>n</math>.

If, instead of taking <math>MLL</math>, we add the modalities to <math>MALL</math>, we obtain of course various versions of full linear logic:

* <math>LL</math>: full linear logic, without units;
* <math>LL_0</math>: full linear logic, with units;
* <math>LL_n</math>: <math>LL</math> with quantifiers of order <math>n</math>;
* <math>LL_{0n}</math>: <math>LL_0</math> with quantifiers of order <math>n</math>.

In <math>LL_{02}</math> the formulas <math>A\with B</math> and <math>A\plus B</math> are provably equivalent to <math>\exists X.(\oc{(X\orth\parr A)}\tens\oc{(X\orth\parr B)}\tens X)</math> and <math>\forall X.(\wn{(X\orth\tens A)}\parr\wn{(X\orth \tens B)}\parr X)</math>, respectively, for all <math>A,B</math>. Thanks to the second-order definability of units discussed above, we obtain that <math>MELL_2</math> is as expressive as <math>LL_{02}</math>, ''i.e.'', full propositional second order linear logic embeds in its second order multiplicative exponential fragment without units.

Girard showed how cut elimination for <math>LL_{02}</math> ''without the contraction rule'' can be proved by a simple induction up to <math>\omega</math>, ''i.e.'', in first order Peano arithmetic. This gives a huge gap between the logical complexity of full linear logic and its contraction-free subsystem: in fact, still by Girard's results, we know that cut elimination in <math>MELL_2</math> is equivalent to the consistency of second order Peano arithmetic, for which no ordinal analysis exists. There are nevertheless subsystems of <math>MELL_2</math>, the so-called [[Light linear logics|light subsystems]] of linear logic, in which the exponential connectives are weakened, whose cut elimination can be proved in seconder order Peano arithmetic even in presence of contraction.

The cut elimination problem is never elementary recursive in presence of exponential connectives: the simply typed <math>\lambda</math>-calculus with arrow types only can be encoded in <math>MELL</math>, and this is enough for Statman's lower bound to apply. However, it becomes elementary recursive in the above mentioned [[Light linear logics|light logics]].

Albeit perhaps surprisingly, provability in <math>LL</math> is already undecidable. This result, obtained by coding Minsky machines with linear logic formulas, contrasts with the situation in classical logic, whose propositional fragment is notoriously decidable. It is indicative of the fact that modalities are themselves a form of quantification, although this claim is far from being clear: as a matter of fact, the decidability of propositional provability in the absence of additives, ''i.e.'', in <math>MELL</math> alone, is still an open problem. It is known that adding first order quantification to <math>MELL</math> makes it undecidable.

=== About exponential rules ===

In this section, provability is assumed to be in <math>LL_{02}</math>, ''i.e.'', full propositional second order linear logic.

In contrast with multiplicative and additive connectives, the modalities of linear logic are not defined by their rules: one may introduce a pair of dual modalities <math>\oc',\wn'</math>, each with the same rules as <math>\oc,\wn</math>, without <math>\oc'A</math> (resp. <math>\wn'A</math>) being in general provably equivalent to <math>\oc A</math> (resp. <math>\wn A</math>).

The [[Sequent calculus#Sequents and proofs|promotion rule]] is derivable from the following two rules, called ''functorial promotion'' and ''digging'', respectively:

<center><math>
\AxRule{\vdash\Gamma,A}
\LabelRule{f\oc}
\UnaRule{\vdash\wn\Gamma,\oc A}
\DisplayProof
\qquad\qquad\qquad
\AxRule{\vdash\Gamma,\wn{\wn A}}
\LabelRule{dig}
\UnaRule{\vdash\Gamma,\wn A}
\DisplayProof
</math></center>

Functorial promotion is itself derivable from dereliction and promotion; the digging rule is also derivable, but only using the <math>\rulename{cut}</math> rule (in fact, digging does not enjoy the subformula property). It may be convenient to consider this pair of rules instead of the standard promotion rule in the context of [[categorical semantics]] of linear logic.

In presence of the digging rule, dereliction, weakening, and contraction can be derived from the following rule, called ''multiplexing'', in which <math>A^{(n)}</math> stands for the sequence <math>A,\ldots,A</math> containing <math>n</math> occurrences of <math>A</math>:

<center><math>
\AxRule{\vdash\Gamma,A^{(n)}}
\LabelRule{mux}
\UnaRule{\vdash\Gamma,\wn A}
\DisplayProof
</math></center>

Of course, multiplexing is itself derivable from dereliction, weakening, and contraction. Hence, there are several alternative but equivalent presentations of the exponential fragment of linear logic, such as
# remove promotion, and replace it with functorial promotion and digging;
# remove promotion, dereliction, weakening, and contraction, and replace them with functorial promotion, digging, and multiplexing.
Apart from their usefulness in [[categorical semantics]], these alternative formulations are of interest in the context of the so-called [[light linear logics]] mentioned above. For example, ''elementary linear logic'' is obtained by removing dereliction and digging from formulation 1, and ''soft linear logic'' is obtained by removing digging from formulation 2.

Multiplexing is invertible in certain circumstances. A sequent <math>\vdash\Gamma,\wn A</math> containing no occurrence of <math>\with</math>, <math>\oc</math>, or second order <math>\exists</math> is provable iff <math>\vdash\Gamma,A^{(n)}</math> is provable for some <math>n</math> (this is easily checked by induction on cut-free proofs). To see that this does not hold in general, take for instance <math>A=X\orth</math> and <math>\Gamma=X\with\one</math>, or <math>\Gamma=\oc X</math>. The restriction on the presence of additive conjunction can be removed by slightly changing the statement: a sequent <math>\vdash\Gamma,\wn A</math> containing no occurrence of <math>\oc</math> or second order <math>\exists</math> is provable iff <math>\vdash\Gamma,(A\plus\bot)^{(n)}</math> is provable for some <math>n</math>.

The latter result can be generalized as follows. If <math>A</math> is a formula, <math>\oc_nA</math> stands for the formula <math>(A\with\one)\tens\cdots\tens(A\with\one)</math> (<math>n</math> times) and <math>\wn_nA</math> for the formula <math>(A\plus\bot)\parr\cdots\parr(A\plus\bot)</math> (<math>n</math> times). Then, we have

{{Theorem|title=Approximation Theorem|
Let <math>\vdash\Gamma</math> be a provable sequent containing <math>p</math> occurrences of <math>\oc</math>, <math>q</math> occurrences of <math>\wn</math>, and no occurrence of second order <math>\exists</math>. Then, for all <math>m_1,\ldots,m_p\in\mathbb N</math>, there are <math>n_1,\ldots,n_q\in\mathbb N</math> such that the sequent obtained from <math>\vdash\Gamma</math> by replacing the <math>p</math> occurrences of <math>\oc</math> with <math>\oc_{m_1},\ldots,\oc_{m_p}</math> and the <math>q</math> occurrences of <math>\wn</math> with <math>\wn_{n_1},\ldots,\wn_{n_q}</math> is provable.}}

A ''structural formula'' is a formula <math>C</math> such that <math>C\limp C\tens C</math> and <math>C\limp\one</math> are provable. Obviously, any formula of the form <math>\wn B</math> is structural. However, the promotion rule cannot be extended to arbitrary structural formulas, ''i.e.'', the following rule is ''not'' admissible:

<center>
<math>
\AxRule{C\vdash A}
\AxRule{C\vdash C\tens C}
\AxRule{C\vdash\one}
\TriRule{C\vdash\oc A}
\DisplayProof
</math>
</center>

For instance, if <math>A=C=\alpha\tens\oc{(\alpha\limp\alpha\tens\alpha)}\tens\oc{(\alpha\limp\one)}</math>, the three premises are provable but not the conclusion.

The following rule, called ''absorption'', is derivable in the standard sequent calculus:
<center>
<math>
\AxRule{\vdash\Gamma,\wn A,A}
\UnaRule{\vdash\Gamma,\wn A}
\DisplayProof
</math>
</center>
The absorption rule is useful in the context of proof search in linear logic.

== The provability problem ==

It is well known that the decidability of the provability problem is connected to the [[Phase semantics|finite model property]]: if a fragment of a logic with a truth semantics enjoys the finite model property, then the provability in that fragment is decidable. Note of course that the converse may fail.

In this section, we summarize the known results about the validity of the final model property and the decidability of provability, with its complexity, for the various fragments of linear logic introduced above. Question marks in the tables below denote open problems. For brevity, all fragments are assumed to have units and propositional atoms, ''e.g.'', <math>MLL</math> actually denotes what we called <math>MLL_0</math> above.

=== The finite model property ===
{| border="1"
|-
| <math>MLL</math>
| <math>MALL</math>
| <math>MELL</math>
| <math>LL</math>
|-
| yes
| yes
| no
| no
|}

=== Provability ===
{| border="1"
|-
|
| <math>MLL</math>
| <math>MALL</math>
| <math>MELL</math>
| <math>LL</math>
|-
| propositional case
| <math>\mathbf{NP}</math>-complete
| <math>\mathbf{PSPACE}</math>-complete
| ?
| undecidable
|-
| first order case
| <math>\mathbf{NP}</math>-complete
| <math>\mathbf{NEXP}</math>-complete
| undecidable
| undecidable
|-
| second order case
| undecidable
| undecidable
| undecidable
| undecidable
|}

== The cut elimination problem ==

In this section, we summarize the known results about the complexity of the cut elimination problem for the various fragments of linear logic introduced above, plus some [[light linear logics]]. All fragments are assumed to be propositional; the results do not change in presence of quantification of any order.

{| border="1"
|-
| <math>MLL_u</math>
| <math>MLL</math>
| <math>MALL</math>
| <math>MSLL</math>
| <math>MLLL</math>
| <math>MELL</math>
|-
| <math>\mathbf L</math>
| <math>\mathbf{P}</math>-complete
| <math>\mathbf{coNP}</math>-complete
| <math>\mathbf{EXP}</math>-complete
| <math>\mathbf{2EXP}</math>-complete
| not elementary recursive
|}

Notations used in the above table:
* <math>MSLL</math>: multiplicative soft linear logic;
* <math>MLLL</math>: multiplicative light linear logic.

Positive formula

2009-03-12T15:53:49Z

Emmanuel Beffara: Updated the 'equivalent' link

A ''positive formula'' is a formula <math>P</math> such that <math>P\limp\oc P</math> (thus a [[Wikipedia:F-coalgebra|coalgebra]] for the [[Wikipedia:Comonad|comonad]] <math>\oc</math>). As a consequence <math>P</math> and <math>\oc P</math> are [[Sequent calculus#Equivalences|equivalent]].

== Positive connectives ==

A connective <math>c</math> of arity <math>n</math> is ''positive'' if for any positive formulas <math>P_1</math>,...,<math>P_n</math>, <math>c(P_1,\dots,P_n)</math> is positive.

{{Proposition|title=Positive connectives|
<math>\tens</math>, <math>\one</math>, <math>\plus</math>, <math>\zero</math>, <math>\oc</math> and <math>\exists</math> are positive connectives.
}}

{{Proof|
<math>
\AxRule{P_2\vdash\oc{P_2}}
\AxRule{P_1\vdash\oc{P_1}}
\LabelRule{\rulename{ax}}
\NulRule{P_1\vdash P_1}
\LabelRule{\rulename{ax}}
\NulRule{P_2\vdash P_2}
\LabelRule{\tens R}
\BinRule{P_1,P_2\vdash P_1\tens P_2}
\LabelRule{\oc d L}
\UnaRule{\oc{P_1},P_2\vdash P_1\tens P_2}
\LabelRule{\oc d L}
\UnaRule{\oc{P_1},\oc{P_2}\vdash P_1\tens P_2}
\LabelRule{\oc R}
\UnaRule{\oc{P_1},\oc{P_2}\vdash\oc{(P_1\tens P_2)}}
\LabelRule{\rulename{cut}}
\BinRule{P_1,\oc{P_2}\vdash\oc{(P_1\tens P_2)}}
\LabelRule{\rulename{cut}}
\BinRule{P_1,P_2\vdash\oc{(P_1\tens P_2)}}
\LabelRule{\tens L}
\UnaRule{P_1\tens P_2\vdash\oc{(P_1\tens P_2)}}
\DisplayProof
</math>

 

<math>
\LabelRule{\one R}
\NulRule{\vdash\one}
\LabelRule{\oc R}
\UnaRule{\vdash\oc{\one}}
\LabelRule{\one L}
\UnaRule{\one\vdash\oc{\one}}
\DisplayProof
</math>

 

<math>
\AxRule{P_1\vdash\oc{P_1}}
\LabelRule{\rulename{ax}}
\NulRule{P_1\vdash P_1}
\LabelRule{\plus_1 R}
\UnaRule{P_1\vdash P_1\plus P_2}
\LabelRule{\oc d L}
\UnaRule{\oc{P_1}\vdash P_1\plus P_2}
\LabelRule{\oc R}
\UnaRule{\oc{P_1}\vdash\oc{(P_1\plus P_2)}}
\LabelRule{\rulename{cut}}
\BinRule{P_1\vdash\oc{(P_1\plus P_2)}}
\AxRule{P_2\vdash\oc{P_2}}
\LabelRule{\rulename{ax}}
\NulRule{P_2\vdash P_2}
\LabelRule{\plus_2 R}
\UnaRule{P_2\vdash P_1\plus P_2}
\LabelRule{\oc d L}
\UnaRule{\oc{P_2}\vdash P_1\plus P_2}
\LabelRule{\oc R}
\UnaRule{\oc{P_2}\vdash\oc{(P_1\plus P_2)}}
\LabelRule{\rulename{cut}}
\BinRule{P_2\vdash\oc{(P_1\plus P_2)}}
\LabelRule{\plus L}
\BinRule{P_1\plus P_2\vdash\oc{(P_1\plus P_2)}}
\DisplayProof
</math>

 

<math>
\LabelRule{\zero L}
\NulRule{\zero\vdash\oc{\zero}}
\DisplayProof
</math>

 

<math>
\LabelRule{\rulename{ax}}
\NulRule{\oc{P}\vdash\oc{P}}
\LabelRule{\oc R}
\UnaRule{\oc{P}\vdash\oc{\oc{P}}}
\DisplayProof
</math>

 

<math>
\AxRule{P\vdash\oc{P}}
\LabelRule{\rulename{ax}}
\NulRule{P\vdash P}
\LabelRule{\exists R}
\UnaRule{P\vdash \exists\xi P}
\LabelRule{\oc d L}
\UnaRule{\oc{P}\vdash \exists\xi P}
\LabelRule{\oc R}
\UnaRule{\oc{P}\vdash\oc{\exists\xi P}}
\LabelRule{\rulename{cut}}
\BinRule{P\vdash\oc{\exists\xi P}}
\LabelRule{\exists L}
\UnaRule{\exists\xi P\vdash\oc{\exists\xi P}}
\DisplayProof
</math>
}}

More generally, <math>\oc A</math> is positive for any formula <math>A</math>.

The notion of positive connective is related with but different from the notion of [[asynchronous connective]].

== Generalized structural rules ==

Positive formulas admit generalized left structural rules corresponding to a structure of <math>\tens</math>-comonoid: <math>P\limp P\tens P</math> and <math>P\limp\one</math>. The following rule is derivable:

<math>
\AxRule{\Gamma,P,P\vdash\Delta}
\LabelRule{+ c L}
\UnaRule{\Gamma,P\vdash\Delta}
\DisplayProof
\qquad
\AxRule{\Gamma\vdash\Delta}
\LabelRule{+ w L}
\UnaRule{\Gamma,P\vdash\Delta}
\DisplayProof
</math>

{{Proof|
<math>
\AxRule{P\vdash\oc{P}}
\AxRule{\Gamma,P,P\vdash\Delta}
\LabelRule{\oc L}
\UnaRule{\Gamma,P,\oc P\vdash\Delta}
\LabelRule{\oc L}
\UnaRule{\Gamma,\oc P,\oc P\vdash\Delta}
\LabelRule{\oc c L}
\UnaRule{\Gamma,\oc P\vdash\Delta}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma,P\vdash\Delta}
\DisplayProof
</math>

 

<math>
\AxRule{P\vdash\oc{P}}
\AxRule{\Gamma\vdash\Delta}
\LabelRule{\oc w L}
\UnaRule{\Gamma,\oc P\vdash\Delta}
\LabelRule{\rulename{cut}}
\BinRule{\Gamma,P\vdash\Delta}
\DisplayProof
</math>
}}

Positive formulas are also acceptable in the left-hand side context of the promotion rule. The following rule is derivable:

<math>
\AxRule{\oc\Gamma,P_1,\dots,P_n\vdash A,\wn\Delta}
\LabelRule{+ \oc R}
\UnaRule{\oc\Gamma,P_1,\dots,P_n\vdash \oc{A},\wn\Delta}
\DisplayProof
</math>

{{Proof|
<math>
\AxRule{P_1\vdash\oc{P_1}}
\AxRule{P_n\vdash\oc{P_n}}
\AxRule{\oc\Gamma,P_1,\dots,P_n\vdash A,\wn\Delta}
\LabelRule{\oc L}
\UnaRule{\oc\Gamma,P_1,\dots,\oc{P_n}\vdash A,\wn\Delta}
\UnaRule{\vdots}
\LabelRule{\oc L}
\UnaRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_n}\vdash A,\wn\Delta}
\LabelRule{\oc R}
\UnaRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_n}\vdash \oc{A},\wn\Delta}
\LabelRule{\rulename{cut}}
\BinRule{\oc\Gamma,\oc{P_1},\dots,P_n\vdash \oc{A},\wn\Delta}
\UnaRule{\vdots}
\LabelRule{\rulename{cut}}
\BinRule{\oc\Gamma,P_1,\dots,P_n\vdash \oc{A},\wn\Delta}
\DisplayProof
</math>
}}

Template:Proof

2009-03-10T15:46:02Z

Emmanuel Beffara:

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Proof.&emsp;{{{1}}}
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Template:Proof

2009-03-10T15:42:44Z

Emmanuel Beffara: Trying some visual formatting

<div style="margin-left: 1.5em; border-left: double 4px #ccc; padding-left: .5em">Proof.&emsp;{{{1}}}</div>

Sequent calculus

2009-03-10T15:30:23Z

Emmanuel Beffara: Made the two-sided system the reference one, presenting the one-sided system as a simplification

This article presents the language and sequent calculus of second-order
linear logic and the basic properties of this sequent calculus.
The core of the article uses the two-sided system with negation as a proper
connective; the [[#One-sided sequent calculus|one-sided system]], often used
as the definition of linear logic, is presented at the end of the page.

== Formulas ==

Atomic formulas, written <math>\alpha,\beta,\gamma</math>, are predicates of
the form <math>p(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms
from some first-order language.
The predicate symbol <math>p</math> may be either a predicate constant or a
second-order variable.
By convention we will write first-order variables as <math>x,y,z</math>,
second-order variables as <math>X,Y,Z</math>, and <math>\xi</math> for a
variable of arbitrary order (see [[Notations]]).

Formulas, represented by capital letters <math>A</math>, <math>B</math>,
<math>C</math>, are built using the following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>A\orth</math>
| negation
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists \xi.A</math>
| there exists
| <math>\forall \xi.A</math>
| for all
| quantifiers
|}

Each line (except the first one) corresponds to a particular class of
connectives, and each class consists in a pair of connectives.
Those in the left column are called [[positive formula|positive]] and those in
the right column are called [[negative formula|negative]].
The ''tensor'' and ''with'' connectives are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally read
''of course <math>A</math>'' for <math>\oc A</math> and ''why not
<math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic, as
<math>A\limp B:=A\orth\parr B</math>.

Free and bound variables and first-order substitution are defined in the
standard way.
Formulas are always considered up to renaming of bound names.
If <math>A</math> is a formula, <math>X</math> is a second-order variable and
<math>B[x_1,\ldots,x_n]</math> is a formula with variables <math>x_i</math>,
then the formula <math>A[B/X]</math> is <math>A</math> where every atom
<math>X(t_1,\ldots,t_n)</math> is replaced by <math>B[t_1,\ldots,t_n]</math>.

== Sequents and proofs ==

A sequent is an expression <math>\Gamma\vdash\Delta</math> where
<math>\Gamma</math> and <math>\Delta</math> are finite multisets of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation
<math>\wn\Gamma</math> represents the multiset
<math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:
* Identity group: <math>
\LabelRule{\rulename{axiom}}
\NulRule{ A \vdash A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{\rulename{cut}}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>
* Negation: <math>
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\LabelRule{n_L}
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\LabelRule{n_R}
\DisplayProof
</math>
* Multiplicative group:
** tensor: <math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
</math>
** par: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma', B \vdash \Delta' }
\LabelRule{ \parr_L }
\BinRule{ \Gamma, \Gamma', A \parr B \vdash \Delta, \Delta' }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, B, \Delta }
\LabelRule{ \parr_R }
\UnaRule{ \Gamma \vdash A \parr B, \Delta }
\DisplayProof
</math>
** one: <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>
** bottom: <math>
\LabelRule{ \bot_L }
\NulRule{ \bot \vdash }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \bot_R }
\UnaRule{ \Gamma \vdash \bot, \Delta }
\DisplayProof
</math>
* Additive group:
** plus: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
</math>
** with: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ \with_{L1} }
\UnaRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \with_{L2} }
\UnaRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash, A \Delta }
\AxRule{ \Gamma \vdash, B \Delta }
\LabelRule{ \with_R }
\BinRule{ \Gamma, A \with B \vdash \Delta }
\DisplayProof
</math>
** zero: <math>
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>
** top: <math>
\LabelRule{ \top_R }
\NulRule{ \Gamma \vdash \top, \Delta }
\DisplayProof
</math>
* Exponential group:
** of course: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c_L }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>
** why not: <math>
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ d_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \wn A, \wn A, \Delta }
\LabelRule{ c_R }
\UnaRule{ \Gamma \vdash \wn A, \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ A \vdash \wn B_1, \ldots, \wn B_n }
\LabelRule{ \wn_L }
\UnaRule{ \wn A \vdash \wn B_1, \ldots, \wn B_n }
\DisplayProof
</math>
* Quantifier group (in the <math>\exists_L</math> and <math>\forall_R</math> rules, <math>\xi</math> must not occur free in <math>\Gamma</math> or <math>\Delta</math>):
** there exists: <math>
\AxRule{ \Gamma , A \vdash \Delta }
\LabelRule{ \exists_L }
\UnaRule{ \Gamma, \exists\xi.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A[t/x] }
\LabelRule{ \exists^1_R }
\UnaRule{ \Gamma \vdash \Delta, \exists x.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A[B/X] }
\LabelRule{ \exists^2_R }
\UnaRule{ \Gamma \vdash \Delta, \exists X.A }
\DisplayProof
</math>
** for all: <math>
\AxRule{ \Gamma, A[t/x] \vdash \Delta }
\LabelRule{ \forall^1_L }
\UnaRule{ \Gamma, \forall x.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma, A[B/X] \vdash \Delta }
\LabelRule{ \forall^2_L }
\UnaRule{ \Gamma, \forall X.A \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta, A }
\LabelRule{ \forall_R }
\UnaRule{ \Gamma \vdash \Delta, \forall\xi.A }
\DisplayProof
</math>

The left rules for ''of course'' and right rules for ''why not'' are called
''dereliction'', ''weakening'' and ''contraction'' rules.
The right rule for ''of course'' and the left rule for ''why not'' are called
''promotion'' rules.
Note the fundamental fact that there are no contraction and weakening rules
for arbitrary formulas, but only for the formulas starting with the
<math>\wn</math> modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An alternative presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rules:

<math>
\AxRule{ \Gamma_1, A, B, \Gamma_2 \vdash \Delta }
\LabelRule{\rulename{exchange}_L}
\UnaRule{ \Gamma_1, B, A, \Gamma_2 \vdash \Delta }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \Gamma \vdash \Delta_1, A, B, \Delta_2 }
\LabelRule{\rulename{exchange}_R}
\UnaRule{ \Gamma \vdash \Delta_1, B, A, \Delta_2 }
\DisplayProof
</math>

== Equivalences ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent,
written <math>A\linequiv B</math>, if both implications <math>A\limp B</math>
and <math>B\limp A</math> are provable.
Equivalently, <math>A\linequiv B</math> if both <math>A\vdash B</math> and
<math>B\vdash A</math> are provable.
Another formulation of <math>A\linequiv B</math> is that, for all
<math>\Gamma</math> and <math>\Delta</math>, <math>\Gamma\vdash\Delta,A</math>
is provable if and only if <math>\Gamma\vdash\Delta,B</math> is provable.

Two related notions are [[isomorphism]] (stronger than equivalence) and
[[equiprovability]] (weaker than equivalence).

=== De Morgan laws ===

Negation is involutive:

<math>A\linequiv A\biorth</math>

Duality between connectives:

{|
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>\linequiv A\orth \parr B\orth </math>
|width=30|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>\linequiv A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>\linequiv \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>\linequiv \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>\linequiv A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>\linequiv A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>\linequiv \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>\linequiv \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>\linequiv \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>\linequiv \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists \xi.A )\orth </math>
| <math>\linequiv \forall \xi.( A\orth ) </math>
|
|align="right"| <math> ( \forall \xi.A )\orth </math>
| <math>\linequiv \exists \xi.( A\orth ) </math>
|}

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality: <math>
A \tens (B \tens C) \linequiv (A \tens B) \tens C </math> &emsp; <math>
A \tens B \linequiv B \tens A </math> &emsp; <math>
A \tens \one \linequiv A </math> <math>
A \parr (B \parr C) \linequiv (A \parr B) \parr C </math> &emsp; <math>
A \parr B \linequiv B \parr A </math> &emsp; <math>
A \parr \bot \linequiv A </math> <math>
A \plus (B \plus C) \linequiv (A \plus B) \plus C </math> &emsp; <math>
A \plus B \linequiv B \plus A </math> &emsp; <math>
A \plus \zero \linequiv A </math> <math>
A \with (B \with C) \linequiv (A \with B) \with C </math> &emsp; <math>
A \with B \linequiv B \with A </math> &emsp; <math>
A \with \top \linequiv A </math>
* Idempotence of additives: <math>
A \plus A \linequiv A </math> &emsp; <math>
A \with A \linequiv A </math>
* Distributivity of multiplicatives over additives: <math>
A \tens (B \plus C) \linequiv (A \tens B) \plus (A \tens C) </math> &emsp; <math>
A \tens \zero \linequiv \zero </math> <math>
A \parr (B \with C) \linequiv (A \parr B) \with (A \parr C) </math> &emsp; <math>
A \parr \top \linequiv \top </math>
* Defining property of exponentials: <math>
\oc(A \with B) \linequiv \oc A \tens \oc B </math> &emsp; <math>
\oc\top \linequiv \one </math>
* Monoidal structure of exponentials: <math>
\oc A \tens \oc A \linequiv \oc A </math> &emsp; <math>
\oc \one \linequiv \one </math> <math>
\wn A \parr \wn A \linequiv \wn A </math> &emsp; <math>
\wn \bot \linequiv \bot </math>
* Digging: <math>
\oc\oc A \linequiv \oc A </math> &emsp; <math>
\wn\wn A \linequiv \oc A </math>
* Other properties of exponentials: <math>
\oc\wn\oc\wn A \linequiv \oc\wn A </math> &emsp; <math>
\oc\wn \one \linequiv \one </math>
* Commutation of quantifiers (<math>\zeta</math> does not occur in <math>A</math>): <math>
\exists \xi. \exists \psi. A \linequiv \exists \psi. \exists \xi. A </math> &emsp; <math>
\exists \xi.(A \plus B) \linequiv \exists \xi.A \plus \exists \xi.B </math> &emsp; <math>
\exists \zeta.(A\tens B) \linequiv A\tens\exists \zeta.B </math> &emsp; <math>
\exists \zeta.A \linequiv A </math> <math>
\forall \xi. \forall \psi. A \linequiv \forall \psi. \forall \xi. A </math> &emsp; <math>
\forall \xi.(A \with B) \linequiv \forall \xi.A \with \forall \xi.B </math> &emsp; <math>
\forall \zeta.(A\parr B) \linequiv A\parr\forall \zeta.B </math> &emsp; <math>
\forall \zeta.A \linequiv A </math>

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:
* <math> \zero \linequiv \forall X.X </math>
* <math> \one \linequiv \forall X.(X \limp X) </math>
* <math> A \plus B \linequiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X) </math>
The constants <math>\top</math> and <math>\bot</math> and the connective
<math>\with</math> can be defined by duality.

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.



== Properties of proofs ==

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\Gamma\vdash\Delta</math>, there is a proof of
<math>\Gamma\vdash\Delta</math> if and only if there is a proof of
<math>\Gamma\vdash\Delta</math> that does not use the cut rule.}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas of its immediate subformulas:
* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,
* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,
* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas have no immediate subformula,
* the immediate subformulas of <math>\exists x.A</math> and <math>\forall x.A</math> are all the <math>A[t/x]</math> for all first-order terms <math>t</math>,
* the immediate subformulas of <math>\exists X.A</math> and <math>\forall X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math> (with the appropriate number of parameters).}}

{{Theorem|title=subformula property|
A sequent <math>\Gamma\vdash\Delta</math> is provable if and only if it is the conclusion of a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math> and <math>\Delta</math>.}}
{{Proof|By the cut elimination theorem, if a sequent is provable, then it is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its
negation <math>A\orth</math>; it is impossible to prove <math>\zero</math> or
<math>\bot</math>.}}
{{Proof|If a sequent is provable, then it is the conclusion of a cut-free proof.
In each rule except the cut rule, there is at least one formula in conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.
The other properties are immediate consequences: if <math>\vdash A\orth</math>
and <math>\vdash A</math> are provable, then by the left negation rule
<math>A\orth\vdash</math> is provable, and by the cut rule one gets empty
conclusion, which is not possible.
As particular cases, since <math>\one</math> and <math>\top</math> are
provable, <math>\bot</math> and <math>\zero</math> are not, since they are
equivalent to <math>\one\orth</math> and <math>\top\orth</math>
respectively.}}

=== Expansion of identities ===

Let us write <math>\pi:\Gamma\vdash\Delta</math> to signify that
<math>\pi</math> is a proof with conclusion <math>\Gamma\vdash\Delta</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the
same conclusion as <math>\pi</math> in which the axiom rule is only used with
atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.}}
{{Proof|It suffices to prove that for every formula <math>A</math>, the sequent
<math>A\vdash A</math> has a cut-free proof in which the axiom rule is used
only for atomic formulas.
We prove this by induction on <math>A</math>.
* If <math>A</math> is atomic, then <math>A\vdash A</math> is an instance of the atomic axiom rule.
* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 : A_1 \vdash A_1 }
\AxRule{ \pi_2 : A_2 \vdash A_2 }
\LabelRule{ \tens_R }
\BinRule{ A_1, A_2 \vdash A_1 \tens A_2 }
\LabelRule{ \tens_L }
\UnaRule{ A_1 \tens A_2 \vdash A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* If <math>A=A_1\parr A_2</math> then we have <math>
\AxRule{ \pi_1 : A_1 \vdash A_1 }
\AxRule{ \pi_2 : A_2 \vdash A_2 }
\LabelRule{ \parr_L }
\BinRule{ A_1 \parr A_2 \vdash A_1, A_2 }
\LabelRule{ \parr_R }
\UnaRule{ A_1 \parr A_2 \vdash A_1 \parr A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* All other connectives follow the same pattern.}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if
* for every proof <math>\pi:\Gamma\vdash\Delta,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,
* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are reversible.}}
{{Proof|Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only applied to atomic formulas.
Then each top-level connective is introduced either by its associated (left or
right) rule or in an instance of the <math>\zero_L</math> or
<math>\top_R</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\Gamma\vdash\Delta,A\parr
B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr_R</math> rule (not
necessarily the last rule in <math>\pi</math>), then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction on <math>\pi</math>).
If it is introduced in the context of a <math>\zero_L</math> or
<math>\top_R</math> rule, then this rule can be changed so that
<math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the
expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a
<math>\bot_R</math> rule, then remove this rule to get a proof of
<math>\vdash\Gamma</math>, if it is introduced by a <math>\zero_L</math> or
<math>\top_R</math> rule, remove the <math>\bot</math> from this rule, then
apply the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof
<math>\pi:\Gamma\vdash\Delta,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is
applied in a context like

<math>
\AxRule{ \pi_1 \Gamma' \vdash \Delta', A }
\AxRule{ \pi_2 \Gamma' \vdash \Delta', B }
\LabelRule{ \with }
\BinRule{ \Gamma' \vdash \Delta', A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except
as context), if we replace this step by <math>\pi_1</math> in <math>\pi</math>
we finally get a proof <math>\pi'_1:\Gamma\vdash\Delta,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof
<math>\pi'_2:\Gamma\vdash\Delta,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final
<math>\with</math> rule we finally get the expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math>
rule is solved as before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as
an instance of the <math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math>, consider a proof
<math>\pi:\Gamma\vdash\Delta,\forall\xi.A</math>.
Up to renaming, we can assume that <math>\xi</math> occurs free only above the
rule that introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we
remove this rule, we can check that we get a proof of
<math>\Gamma\vdash\Delta,A</math> on which we can finally apply the
<math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math>
rule is solved as before.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the
original proof.}}

== One-sided sequent calculus ==

The sequent calculus presented above is very symmetric: for every left
introduction rule, there is a right introduction rule for the dual connective
that has the exact same structure.
Moreover, because of the involutivity of negation, a sequent
<math>\Gamma,A\vdash\Delta</math> is provable if and only if the sequent
<math>\Gamma\vdash A\orth,\Delta</math> is provable.
From these remarks, we can define an equivalent one-sided sequent calculus:
* Formulas are considered up to De Morgan duality. Equivalently, one can consider that negation is not a connective but a syntactically defined operation on formulas. In this case, negated atoms <math>\alpha\orth</math> must be considered as another kind of atomic formulas.
* Sequents have the form <math>\vdash\Gamma</math>.
The inference rules are essentially the same except that the left hand side of
sequents is kept empty:
* Identity group: <math>
\LabelRule{\rulename{axiom}}
\NulRule{ \vdash A\orth, A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{\rulename{cut}}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>
* Multiplicative group: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>
* Additive group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
</math> &emsp; <math>
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>
* Exponential group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>
* Quantifier group (in the <math>\forall</math> rule, <math>\xi</math> must not occur free in <math>\Gamma</math>): <math>
\AxRule{ \vdash \Gamma, A[t/x] }
\LabelRule{ \exists^1 }
\UnaRule{ \vdash \Gamma, \exists x.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists^2 }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
</math> &emsp; <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall \xi.A }
\DisplayProof
</math>

{{Theorem|A two-sided sequent <math>\Gamma\vdash\Delta</math> is provable if
and only if the sequent <math>\vdash\Gamma\orth,\Delta</math> is provable in
the one-sided system.}}

The one-sided system enjoys the same properties as the two-sided one,
including cut elimination, the subformula property, etc.
This formulation is often used when studying proofs because it is much lighter
than the two-sided form while keeping the same expressiveness.
In particular, [[proof-nets]] can be seen as a quotient of one-sided sequent
calculus proofs under commutation of rules.

== Variations ==

The same remarks that lead to the definition of the one-sided calculus can
lead the definition of other simplified systems:
* A one-sided variant with sequents of the form <math>\Gamma\vdash</math> could be defined.
* When considering formulas up to De Morgan duality, an equivalent system is obtained by considering only the left and right rules for positive connectives (or the ones for negative connectives only, obviously).
* [[Intuitionistic linear logic]] is the two-sided system where the right-hand side is constrained to always contain exactly one formula (with a few associated restrictions).
* Similar restrictions are used in various [[semantics]] and [[proof search]] formalisms.

Sequent calculus

2009-02-07T12:05:26Z

Emmanuel Beffara: minor typos, markup improvement

This article presents the language and sequent calculus of second-order
propositional linear logic and the basic properties of this sequent calculus.

== Formulas ==

Atomic formulas, written <math>\alpha,\beta,\gamma</math>, are predicates of the form <math>p(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms from some first-order language.
The predicate symbol <math>p</math> may be either a predicate constant or a second-order variable.
By convention we will write first-order variables as <math>x,y,z</math>, second-order variables as <math>X,Y,Z</math>, and <math>\xi</math> for a variable of arbitrary order (see [[Notations]]).

Formulas, represented by capital letters <math>A</math>, <math>B</math>, <math>C</math>, are built using the following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>\alpha\orth</math>
| negated atom
| atoms
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists \xi.A</math>
| there exists
| <math>\forall \xi.A</math>
| for all
| quantifiers
|}

Each line corresponds to a particular class of connectives, and each class
consists in a pair of connectives.
Those in the left column are called positive and those in the right column are
called negative.
The ''tensor'' and ''with'' are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally
read ''of course <math>A</math>'' for <math>\oc A</math> and ''why not <math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

Given a formula <math>A</math>, its linear negation, also called ''orthogonal'' and
written <math>A\orth</math>, is obtained by exchanging each positive connective with the
negative one of the same class and vice versa, in a way analogous to de Morgan
laws in classical logic.
Formally, the definition of linear negation is

{|
|-
|align="right"| <math> ( \alpha )\orth </math>
| <math>:= \alpha\orth </math>
|width=30|
|align="right"| <math> ( \alpha\orth )\orth </math>
| <math>:= \alpha </math>
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>:= A\orth \parr B\orth </math>
|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>:= A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>:= \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>:= \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>:= A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>:= A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>:= \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>:= \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>:= \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>:= \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists \xi.A )\orth </math>
| <math>:= \forall \xi.( A\orth ) </math>
|
|align="right"| <math> ( \forall \xi.A )\orth </math>
| <math>:= \exists \xi.( A\orth ) </math>
|}

Note that this operation is defined syntactically, hence negation is not a
connective, the only place in formulas where the symbol <math>(\cdot)\orth</math> occurs
is for negated atoms <math>\alpha\orth</math>.
Note also that, by construction, negation is involutive: for any formula <math>A</math>,
it holds that <math>A\biorth=A</math>.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic, as
<math>A\limp B:=A\orth\parr B</math>.

Free and bound variables and first-order substitution are defined in the standard way.
Formulas are always considered up to renaming of bound names.
If <math>A</math> is a formula, <math>X</math> is a second-order variable and <math>B[x_1,\ldots,x_n]</math> is a formula with variables <math>x_i</math>, then the formula
<math>A[B/X]</math> is <math>A</math> where every atom <math>X(t_1,\ldots,t_n)</math> is replaced by <math>B[t_1,\ldots,t_n]</math> and every atom <math>X(t_1,\ldots,t_n)\orth</math> is replaced by <math>B[t_1,\ldots,t_n]\orth</math>.

== Sequents and proofs ==

A sequent is an expression <math>\vdash\Gamma</math> where <math>\Gamma</math> is a finite multiset
of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation <math>\wn\Gamma</math> represents
the multiset <math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:
* Identity group: <math>
\LabelRule{axiom}
\NulRule{ \vdash A, A\orth }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{cut}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>
* Multiplicative group: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
\qquad
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>
* Additive group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
\qquad
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>
* Exponential group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>
* Quantifier group (in the <math>\forall</math> rule, <math>\xi</math> must not occur free in <math>\Gamma</math>): <math>
\AxRule{ \vdash \Gamma, A[t/x] }
\LabelRule{ \exists^1 }
\UnaRule{ \vdash \Gamma, \exists x.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists^2 }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall \xi.A }
\DisplayProof
</math>

The rules for exponentials are called ''dereliction'', ''weakening'',
''contraction'' and ''promotion'', respectively.
Note the fundamental fact that there are no contraction and weakening rules
for arbitrary formulas, but only for the formulas starting with the <math>\wn</math>
modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Note that this system contains only introduction rules and no elimination
rule.
Moreover, there is no introduction rule for the additive unit <math>\zero</math>, the
only ways to introduce it at top level are the axiom rule and the <math>\top</math> rule.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An equivalent presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rule:

<math>
\AxRule{ \vdash \Gamma, A, B, \Delta }
\LabelRule{exchange}
\UnaRule{ \vdash \Gamma, B, A, \Delta }
\DisplayProof
</math>

== Equivalences and definability ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent, written <math>A\linequiv B</math>, if
both implications <math>A\limp B</math> and <math>B\limp A</math> are provable.
Equivalently, <math>A\linequiv B</math> if both <math>\vdash A\orth,B</math> and <math>\vdash B\orth,A</math>
are provable.
Another formulation of <math>A\linequiv B</math> is that, for all <math>\Gamma</math>,
<math>\vdash\Gamma,A</math> is provable if and only if <math>\vdash\Gamma,B</math> is provable.
Note that, because of the definition of negation, an equivalence
<math>A\linequiv B</math> holds if and only if the dual equivalence
<math>A\orth\linequiv B\orth</math> holds.

Two related notions are [[isomorphism]] (stronger than equivalence) and [[equiprovability]] (weaker than equivalence).

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality: <math>
A \tens (B \tens C) \linequiv (A \tens B) \tens C </math> &emsp; <math>
A \tens B \linequiv B \tens A </math> &emsp; <math>
A \tens \one \linequiv A </math> <math>
A \plus (B \plus C) \linequiv (A \plus B) \plus C </math> &emsp; <math>
A \plus B \linequiv B \plus A </math> &emsp; <math>
A \plus \zero \linequiv A </math>
* Idempotence of additives: <math>
A \plus A \linequiv A </math>
* Distributivity of multiplicatives over additives: <math>
A \tens (B \plus C) \linequiv (A \tens B) \plus (A \tens C) </math> &emsp; <math>
A \tens \zero \linequiv \zero </math>
* Defining property of exponentials: <math>
\oc(A \with B) \linequiv \oc A \tens \oc B </math> &emsp; <math>
\oc\top \linequiv \one </math>
* Monoidal structure of exponentials, digging:
** <math> \oc A \otimes \oc A \linequiv \oc A </math>
** <math> \oc \one \linequiv \one </math>
** <math> \oc\oc A \linequiv \oc A </math>
* Commutation of quantifiers (<math>\zeta</math> does not occur in <math>A</math>): <math>
\exists \xi. \exists \psi. A \linequiv \exists \psi. \exists \xi. A </math> &emsp; <math>
\exists \xi.(A \plus B) \linequiv \exists \xi.A \plus \exists \xi.B </math> &emsp; <math>
\exists \zeta.(A\tens B) \linequiv A\tens\exists \zeta.B </math> &emsp; <math>
\exists \zeta.A \linequiv A </math>

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:
* <math> \zero \linequiv \forall X.X </math>
* <math> \one \linequiv \forall X.(X \limp X) </math>
* <math> A \plus B \linequiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X) </math>

=== Additional equivalences ===

<math> \oc\wn\oc\wn A \linequiv \oc\wn A </math> &emsp;
<math> \oc\wn \one \linequiv \one </math>

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

=== Positive/negative commutation ===

<math>\exists\forall\limp\forall\exists</math>,
<math>A\tens(B\parr C)\limp(A\tens B)\parr C</math>

== Properties of proofs ==

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\vdash\Gamma</math>, there is a proof of <math>\vdash\Gamma</math> if and only if there is a proof of <math>\vdash\Gamma</math> that does not use the cut rule.}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas of its immediate subformulas:
* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,
* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,
* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas <math>\alpha</math> and <math>\alpha\orth</math> have no immediate subformula,
* the immediate subformulas of <math>\exists x.A</math> are all the <math>A[t/x]</math> for all first-order terms <math>t</math>,
* the immediate subformulas of <math>\exists X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math>,
* the only immediate subformula of <math>\forall\xi.A</math> is <math>A</math>.}}

{{Theorem|title=subformula property|
A sequent <math>\vdash\Gamma</math> is provable if and only if it is the conclusion of a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math>.}}
{{Proof|By the cut elimination theorem, if a sequent is provable, then it is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its negation <math>A\orth</math>; it is impossible to prove <math>\zero</math> or <math>\bot</math>.}}
{{Proof|If <math>\vdash\Gamma</math> is a provable sequent, then it is the conclusion of a cut-free proof.
In each rule except the cut rule, there is at least one formula in conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.

The other properties are immediate consequences: if <math>A</math> and <math>A\orth</math> were provable, then by a cut rule one would get an empty conclusion, which is not possible.
As particular cases, since <math>\one</math> and <math>\top</math> are provable, their negations <math>\bot</math> and <math>\zero</math> are not.}}

=== Expansion of identities ===

Let us write <math>\pi\vdash\Gamma</math> to signify that <math>\pi</math> is a proof with
conclusion <math>\vdash\Gamma</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the same conclusion as <math>\pi</math> in which the axiom rule is only used with atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.}}
{{Proof|It suffices to prove that for every formula <math>A</math>, the sequent
<math>\vdash A\orth,A</math> has a cut-free proof in which the axiom rule is used only
for atomic formulas.
We prove this by induction on <math>A</math>.
Not that there is a case for each pair of dual connectives.
* If <math>A</math> is atomic, then <math>\vdash A\orth,A</math> is an instance of the atomic axiom rule.
* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \tens }
\BinRule{ \vdash A_1\orth, A_2\orth, A_1 \tens A_2 }
\LabelRule{ \parr }
\UnaRule{ \vdash A_1\orth \parr A_2\orth, A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* If <math>A=\one</math> or <math>A=\bot</math> then we have <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\LabelRule{ \bot }
\UnaRule{ \vdash \one, \bot }
\DisplayProof
</math>
* If <math>A=A_1\plus A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash A_1\orth, A_1 \plus A_2 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash A_2\orth, A_1 \plus A_2 }
\LabelRule{ \with }
\BinRule{ \vdash A_1\orth \with A_2\orth, A_1 \plus A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.
* If <math>A=\zero</math> or <math>A=\top</math>, we have <math>
\LabelRule{ \top }
\NulRule{ \vdash \top, \zero }
\DisplayProof
</math>
* If <math>A=\oc B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ d }
\UnaRule{ \pi \vdash \wn B\orth, B }
\LabelRule{ \oc }
\UnaRule{ \pi \vdash \wn B\orth, \oc B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.
* If <math>A=\exists X.B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ \exists }
\UnaRule{ \vdash B\orth, \exists X.B }
\LabelRule{ \forall }
\UnaRule{ \vdash \forall X.B\orth, \exists X.B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.
* First-order quantification works like second-order quantification.}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if
* for every proof <math>\pi\vdash\Gamma,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,
* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are reversible.}}
{{Proof|Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only applied to atomic formulas.
Then each top-level connective is introduced either by its associated rule
or in an instance of the <math>\top</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\vdash\Gamma,A\parr B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr</math> rule, then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction).
If it is introduced in the contect of a <math>\top</math> rule, then this rule can be
changed so that <math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a <math>\bot</math>
rule, then remove this rule to get a proof of <math>\vdash\Gamma</math>, if it is
introduced by a <math>\top</math> rule, remove the <math>\bot</math> from this rule, then apply
the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof <math>\pi\vdash\Gamma,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is applied
in a context like

<math>
\AxRule{ \pi_1 \vdash \Delta, A }
\AxRule{ \pi_2 \vdash \Delta, B }
\LabelRule{ \with }
\BinRule{ \vdash \Delta, A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except as
context), if we replace this step by <math>\pi_1</math> in <math>\pi</math> we finally get a proof
<math>\pi'_1\vdash\Gamma,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof <math>\pi'_2\vdash\Gamma,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final <math>\with</math> rule we finally get the
expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math> rule is solved as
before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as an instance of the
<math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math> at second order, consider a proof <math>\pi\vdash\Gamma,\forall X.A</math>.
Up to renaming, we can assume that <math>X</math> occurs free only above the rule that
introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we remove this
rule, we can check that we get a proof of <math>\vdash\Gamma,A</math> on which we can
finally apply the <math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math> rule is solved as
before.
First-order quantification is similar.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the original
proof.}}

== Variations ==

=== Two-sided sequent calculus ===

The sequent calculus of linear logic can also be presented using two-sided
sequents <math>\Gamma\vdash\Delta</math>, with any number of formulas on the left and
right.
In this case, it is customary to provide rules only for the positive
connectives, then there are left and right introduction rules and a negation
rule that moves formulas between the left and right sides:
* Negation group: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\DisplayProof
</math>
* Identity group: <math>
\LabelRule{axiom}
\NulRule{ A \vdash A }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{cut}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>
* Multiplicative group: <math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
\qquad
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>
* Additive group: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>
* Exponential group: <math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>

Sequent calculus

2009-02-07T11:32:49Z

Emmanuel Beffara: /* Formulas */ proper definition of atoms and substitution

This article presents the language and sequent calculus of second-order
propositional linear logic and the basic properties of this sequent calculus.

== Formulas ==

Atomic formulas, written <math>\alpha,\beta,\gamma</math>, are predicates of the form <math>p(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms from some first-order language.
The predicate symbol <math>p</math> may be either a predicate constant or a second-order variable.
By convention we will write first-order variables as <math>x,y,z</math>, second-order variables as <math>X,Y,Z</math>, and <math>\xi</math> for a variable of arbitrary order (see [[Notations]]).

Formulas, represented by capital letters <math>A</math>, <math>B</math>, <math>C</math>, are built using the following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>\alpha\orth</math>
| negated atom
| atoms
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists \xi.A</math>
| there exists
| <math>\forall \xi.A</math>
| for all
| quantifiers
|}

Each line corresponds to a particular class of connectives, and each class
consists in a pair of connectives.
Those in the left column are called positive and those in the right column are
called negative.
The ''tensor'' and ''with'' are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally
read ''of course <math>A</math>'' for <math>\oc A</math> and ''why not <math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

Given a formula <math>A</math>, its linear negation, also called ''orthogonal'' and
written <math>A\orth</math>, is obtained by exchanging each positive connective with the
negative one of the same class and vice versa, in a way analogous to de Morgan
laws in classical logic.
Formally, the definition of linear negation is

{|
|-
|align="right"| <math> ( \alpha )\orth </math>
| <math>:= \alpha\orth </math>
|width=30|
|align="right"| <math> ( \alpha\orth )\orth </math>
| <math>:= \alpha </math>
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>:= A\orth \parr B\orth </math>
|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>:= A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>:= \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>:= \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>:= A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>:= A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>:= \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>:= \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>:= \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>:= \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists \xi.A )\orth </math>
| <math>:= \forall \xi.( A\orth ) </math>
|
|align="right"| <math> ( \forall \xi.A )\orth </math>
| <math>:= \exists \xi.( A\orth ) </math>
|}

Note that this operation is defined syntactically, hence negation is not a
connective, the only place in formulas where the symbol <math>(\cdot)\orth</math> occurs
is for negated atoms <math>\alpha\orth</math>.
Note also that, by construction, negation is involutive: for any formula <math>A</math>,
it holds that <math>A\biorth=A</math>.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic, as
<math>A\limp B:=A\orth\parr B</math>.

Free and bound variables and first-order substitution are defined in the standard way.
Formulas are always considered up to renaming of bound names.
If <math>A</math> is a formula, <math>X</math> is a second-order variable and <math>B[x_1,\ldots,x_n]</math> is a formula with variables <math>x_i</math>, then the formula
<math>A[B/X]</math> is <math>A</math> where every atom <math>X(t_1,\ldots,t_n)</math> is replaced by <math>B[t_1,\ldots,t_n]</math> and every atom <math>X(t_1,\ldots,t_n)\orth</math> is replaced by <math>B[t_1,\ldots,t_n]\orth</math>.

== Sequents and proofs ==

A sequent is an expression <math>\vdash\Gamma</math> where <math>\Gamma</math> is a finite multiset
of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation <math>\wn\Gamma</math> represents
the multiset <math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:

* Identity group: <math>
\LabelRule{axiom}
\NulRule{ \vdash A, A\orth }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{cut}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>

* Multiplicative group: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
\qquad
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>

* Additive group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
\qquad
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>

* Exponential group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>

* Quantifier group (in the <math>\forall</math> rule, <math>\xi</math> must not occur free in <math>\Gamma</math>): <math>
\AxRule{ \vdash \Gamma, A[t/x] }
\LabelRule{ \exists^1 }
\UnaRule{ \vdash \Gamma, \exists x.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists^2 }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall \xi.A }
\DisplayProof
</math>

The rules for exponentials are called ''dereliction'', ''weakening'',
''contraction'' and ''promotion'', respectively.
Note the fundamental fact that there are no contraction and weakening rules
for arbitrary formulas, but only for the formulas starting with the <math>\wn</math>
modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Note that this system contains only introduction rules and no elimination
rule.
Moreover, there is no introduction rule for the additive unit <math>\zero</math>, the
only ways to introduce it at top level are the axiom rule and the <math>\top</math> rule.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An equivalent presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rule:

<math>
\AxRule{ \vdash \Gamma, A, B, \Delta }
\LabelRule{exchange}
\UnaRule{ \vdash \Gamma, B, A, \Delta }
\DisplayProof
</math>

== Equivalences and definability ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent, written <math>A\linequiv B</math>, if
both implications <math>A\limp B</math> and <math>B\limp A</math> are provable.
Equivalently, <math>A\linequiv B</math> if both <math>\vdash A\orth,B</math> and <math>\vdash B\orth,A</math>
are provable.
Another formulation of <math>A\linequiv B</math> is that, for all <math>\Gamma</math>,
<math>\vdash\Gamma,A</math> is provable if and only if <math>\vdash\Gamma,B</math> is provable.
Note that, because of the definition of negation, an equivalence
<math>A\linequiv B</math> holds if and only if the dual equivalence
<math>A\orth\linequiv B\orth</math> holds.

Two related notions are [[isomorphism]] (stronger than equivalence) and [[equiprovability]] (weaker than equivalence).

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality: <math>
A \tens (B \tens C) \linequiv (A \tens B) \tens C </math> &emsp; <math>
A \tens B \linequiv B \tens A </math> &emsp; <math>
A \tens \one \linequiv A </math> <math>
A \plus (B \plus C) \linequiv (A \plus B) \plus C </math> &emsp; <math>
A \plus B \linequiv B \plus A </math> &emsp; <math>
A \plus \zero \linequiv A </math>

* Idempotence of additives: <math>
A \plus A \linequiv A </math>

* Distributivity of multiplicatives over additives: <math>
A \tens (B \plus C) \linequiv (A \tens B) \plus (A \tens C) </math> &emsp; <math>
A \tens \zero \linequiv \zero </math>

* Defining property of exponentials: <math>
\oc(A \with B) \linequiv \oc A \tens \oc B </math> &emsp; <math>
\oc\top \linequiv \one </math>

* Monoidal structure of exponentials, digging: <math>
\oc A \otimes \oc A \linequiv \oc A </math> &emsp; <math>
\oc \one \linequiv \one </math> &emsp; <math>
\oc\oc A \linequiv \oc A </math>

* Commutation of quantifiers (<math>\zeta</math> does not occur in <math>A</math>): <math>
\exists \xi. \exists \psi. A \linequiv \exists \psi. \exists \xi. A </math> &emsp; <math>
\exists \xi.(A \plus B) \linequiv \exists \xi.A \plus \exists \xi.B </math> &emsp; <math>
\exists \zeta.(A\tens B) \linequiv A\tens\exists \zeta.B </math> &emsp; <math>
\exists \zeta.A \linequiv A </math>

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:

<math> \zero \linequiv \forall X.X </math> &emsp;
<math> \one \linequiv \forall X.(X \limp X) </math> &emsp;
<math> A \plus B \linequiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X) </math>

=== Additional equivalences ===

<math> \oc\wn\oc\wn A \linequiv \oc\wn A </math> &emsp;
<math> \oc\wn \one \linequiv \one </math>

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

=== Positive/negative commutation ===

<math>\exists\forall\limp\forall\exists</math>,
<math>A\tens(B\parr C)\limp(A\tens B)\parr C</math>

== Properties of proofs ==

The fundamental property of the sequent calculus of linear logic is the cut
elimination property, which states that the cut rule is useless as far as
provability is concerned.
This property is exposed in the following section, together with a sketch of
proof.

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\vdash\Gamma</math>, there is a proof of <math>\vdash\Gamma</math> if and only if there is a proof of <math>\vdash\Gamma</math> that does not use the cut rule.
}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas of its immediate subformulas:
* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,
* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,
* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas <math>\alpha</math> and <math>\alpha\orth</math> have no immediate subformula,
* the immediate subformulas of <math>\exists x.A</math> are all the <math>A[t/x]</math> for all first-order terms <math>t</math>,
* the immediate subformulas of <math>\exists X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math>,
* the only immediate subformula of <math>\forall\xi.A</math> is <math>A</math>.
}}

{{Theorem|title=subformula property|
A sequent <math>\vdash\Gamma</math> is provable if and only if it is the conclusion of
a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math>.
}}
{{Proof|By the cut elimination theorem, if a sequent <math>\vdash</math> is provable, then it
is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.
}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its negation
<math>A\orth</math>; it is impossible to prove <math>\zero</math> or <math>\bot</math>.
}}
{{Proof|
If <math>\vdash\Gamma</math> is a provable sequent, then it is the conclusion of a
cut-free proof.
In each rule except the cut rule, there is at least one formula in
conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.

The other properties are immediate consequences: if <math>A</math> and <math>A\orth</math> were
provable, then by a cut rule one would get an empty conclusion, which is not
possible.
As particular cases, since <math>\one</math> and <math>\top</math> are provable, their negations
<math>\bot</math> and <math>\zero</math> are not.
}}

=== Expansion of identities ===

Let us write <math>\pi\vdash\Gamma</math> to signify that <math>\pi</math> is a proof with
conclusion <math>\vdash\Gamma</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the same conclusion as
<math>\pi</math> in which the axiom rule is only used with atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.
}}
{{Proof|
It suffices to prove that for every formula <math>A</math>, the sequent
<math>\vdash A\orth,A</math> has a cut-free proof in which the axiom rule is used only
for atomic formulas.
We prove this by induction on <math>A</math>.
Not that there is a case for each pair of dual connectives.

* If <math>A</math> is atomic, then <math>\vdash A\orth,A</math> is an instance of the atomic axiom rule.

* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \tens }
\BinRule{ \vdash A_1\orth, A_2\orth, A_1 \tens A_2 }
\LabelRule{ \parr }
\UnaRule{ \vdash A_1\orth \parr A_2\orth, A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\one</math> or <math>A=\bot</math> then we have <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\LabelRule{ \bot }
\UnaRule{ \vdash \one, \bot }
\DisplayProof
</math>

* If <math>A=A_1\plus A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash A_1\orth, A_1 \plus A_2 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash A_2\orth, A_1 \plus A_2 }
\LabelRule{ \with }
\BinRule{ \vdash A_1\orth \with A_2\orth, A_1 \plus A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\zero</math> or <math>A=\top</math>, we have <math>
\LabelRule{ \top }
\NulRule{ \vdash \top, \zero }
\DisplayProof
</math>

* If <math>A=\oc B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ d }
\UnaRule{ \pi \vdash \wn B\orth, B }
\LabelRule{ \oc }
\UnaRule{ \pi \vdash \wn B\orth, \oc B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* If <math>A=\exists X.B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ \exists }
\UnaRule{ \vdash B\orth, \exists X.B }
\LabelRule{ \forall }
\UnaRule{ \vdash \forall X.B\orth, \exists X.B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* First-order quantification works like second-order quantification.
}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if

* for every proof <math>\pi\vdash\Gamma,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,

* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.
}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are
reversible.
}}
{{Proof|
Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only
applied to atomic formulas.
Then each top-level connective is introduced either by its associated rule
or in an instance of the <math>\top</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\vdash\Gamma,A\parr B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr</math> rule, then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction).
If it is introduced in the contect of a <math>\top</math> rule, then this rule can be
changed so that <math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a <math>\bot</math>
rule, then remove this rule to get a proof of <math>\vdash\Gamma</math>, if it is
introduced by a <math>\top</math> rule, remove the <math>\bot</math> from this rule, then apply
the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof <math>\pi\vdash\Gamma,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is applied
in a context like

<math>
\AxRule{ \pi_1 \vdash \Delta, A }
\AxRule{ \pi_2 \vdash \Delta, B }
\LabelRule{ \with }
\BinRule{ \vdash \Delta, A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except as
context), if we replace this step by <math>\pi_1</math> in <math>\pi</math> we finally get a proof
<math>\pi'_1\vdash\Gamma,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof <math>\pi'_2\vdash\Gamma,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final <math>\with</math> rule we finally get the
expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math> rule is solved as
before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as an instance of the
<math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math> at second order, consider a proof <math>\pi\vdash\Gamma,\forall X.A</math>.
Up to renaming, we can assume that <math>X</math> occurs free only above the rule that
introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we remove this
rule, we can check that we get a proof of <math>\vdash\Gamma,A</math> on which we can
finally apply the <math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math> rule is solved as
before.
First-order quantification is similar.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the original
proof.
}}

== Variations ==

=== Two-sided sequent calculus ===

The sequent calculus of linear logic can also be presented using two-sided
sequents <math>\Gamma\vdash\Delta</math>, with any number of formulas on the left and
right.
In this case, it is customary to provide rules only for the positive
connectives, then there are left and right introduction rules and a negation
rule that moves formulas between the left and right sides:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\DisplayProof
</math>

Identity group:

<math>
\LabelRule{axiom}
\NulRule{ A \vdash A }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{cut}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>

Multiplicative group:

<math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
\qquad
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>

Additive group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>

Exponential group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>

Sequent calculus

2009-01-22T09:30:22Z

Emmanuel Beffara: /* Equivalences and definability */ use \linequiv instead of \equiv

This article presents the language and sequent calculus of second-order
propositional linear logic and the basic properties of this sequent calculus.

== Formulas ==

Formulas are built on a set of atoms, written <math>\alpha,\beta,\ldots</math>, that can
be either propositional variables <math>X,Y,Z\ldots</math> or atomic formulas
<math>p(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms from some first-order language and <math>p</math> is a predicate symbol.
Formulas, represented by capital letters <math>A</math>, <math>B</math>, <math>C</math>, are built using the
following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>\alpha\orth</math>
| negated atom
| atoms
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists \xi.A</math>
| there exists
| <math>\forall \xi.A</math>
| for all
| quantifiers (<math>\xi</math> is a first or second order variable)
|}

Each line corresponds to a particular class of connectives, and each class
consists in a pair of connectives.
Those in the left column are called positive and those in the right column are
called negative.
The ''tensor'' and ''with'' are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally
read ''of course <math>A</math>'' for <math>\oc A</math> and ''why not <math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

Given a formula <math>A</math>, its linear negation, also called ''orthogonal'' and
written <math>A\orth</math>, is obtained by exchanging each positive connective with the
negative one of the same class and vice versa, in a way analogous to de Morgan
laws in classical logic.
Formally, the definition of linear negation is

{|
|-
|align="right"| <math> ( \alpha )\orth </math>
| <math>:= \alpha\orth </math>
|width=30|
|align="right"| <math> ( \alpha\orth )\orth </math>
| <math>:= \alpha </math>
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>:= A\orth \parr B\orth </math>
|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>:= A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>:= \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>:= \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>:= A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>:= A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>:= \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>:= \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>:= \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>:= \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists \xi.A )\orth </math>
| <math>:= \forall \xi.( A\orth ) </math>
|
|align="right"| <math> ( \forall \xi.A )\orth </math>
| <math>:= \exists \xi.( A\orth ) </math>
|}

Note that this operation is defined syntactically, hence negation is not a
connective, the only place in formulas where the symbol <math>(\cdot)\orth</math> occurs
is for negated atoms <math>\alpha\orth</math>.
Note also that, by construction, negation is involutive: for any formula <math>A</math>,
it holds that <math>A\biorth=A</math>.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic:

<math>
A \limp B := A\orth \parr B
</math>

Free and bound variables are defined in the standard way, as well as
substitution.
Formulas are always considered up to renaming of bound names.
If <math>A</math> and <math>B</math> are formulas and <math>X</math> is a propositional variable, the formula
<math>A[B/X]</math> is <math>A</math> where all atoms <math>X</math> are replaced (without capture) by <math>B</math> and
all atoms <math>X\orth</math> are replaced by the formula <math>B\orth</math>.

== Sequents and proofs ==

A sequent is an expression <math>\vdash\Gamma</math> where <math>\Gamma</math> is a finite multiset
of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation <math>\wn\Gamma</math> represents
the multiset <math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:

* Identity group: <math>
\LabelRule{axiom}
\NulRule{ \vdash A, A\orth }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{cut}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>

* Multiplicative group: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
\qquad
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>

* Additive group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
\qquad
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>

* Exponential group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>

* Quantifier group (in the <math>\forall</math> rule, <math>\xi</math> must not occur free in <math>\Gamma</math>): <math>
\AxRule{ \vdash \Gamma, A[t/x] }
\LabelRule{ \exists^1 }
\UnaRule{ \vdash \Gamma, \exists x.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists^2 }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall \xi.A }
\DisplayProof
</math>

The rules for exponentials are called ''dereliction'', ''weakening'',
''contraction'' and ''promotion'', respectively.
Note the fundamental fact that there are no contraction and weakening rules
for arbitrary formulas, but only for the formulas starting with the <math>\wn</math>
modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Note that this system contains only introduction rules and no elimination
rule.
Moreover, there is no introduction rule for the additive unit <math>\zero</math>, the
only ways to introduce it at top level are the axiom rule and the <math>\top</math> rule.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An equivalent presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rule:

<math>
\AxRule{ \vdash \Gamma, A, B, \Delta }
\LabelRule{exchange}
\UnaRule{ \vdash \Gamma, B, A, \Delta }
\DisplayProof
</math>

== Equivalences and definability ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent, written <math>A\linequiv B</math>, if
both implications <math>A\limp B</math> and <math>B\limp A</math> are provable.
Equivalently, <math>A\linequiv B</math> if both <math>\vdash A\orth,B</math> and <math>\vdash B\orth,A</math>
are provable.
Another formulation of <math>A\linequiv B</math> is that, for all <math>\Gamma</math>,
<math>\vdash\Gamma,A</math> is provable if and only if <math>\vdash\Gamma,B</math> is provable.
Note that, because of the definition of negation, an equivalence
<math>A\linequiv B</math> holds if and only if the dual equivalence
<math>A\orth\linequiv B\orth</math> holds.

Two related notions are [[isomorphism]] (stronger than equivalence) and [[equiprovability]] (weaker than equivalence).

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality: <math>
A \tens (B \tens C) \linequiv (A \tens B) \tens C </math> &emsp; <math>
A \tens B \linequiv B \tens A </math> &emsp; <math>
A \tens \one \linequiv A </math> <math>
A \plus (B \plus C) \linequiv (A \plus B) \plus C </math> &emsp; <math>
A \plus B \linequiv B \plus A </math> &emsp; <math>
A \plus \zero \linequiv A </math>

* Idempotence of additives: <math>
A \plus A \linequiv A </math>

* Distributivity of multiplicatives over additives: <math>
A \tens (B \plus C) \linequiv (A \tens B) \plus (A \tens C) </math> &emsp; <math>
A \tens \zero \linequiv \zero </math>

* Defining property of exponentials: <math>
\oc(A \with B) \linequiv \oc A \tens \oc B </math> &emsp; <math>
\oc\top \linequiv \one </math>

* Monoidal structure of exponentials, digging: <math>
\oc A \otimes \oc A \linequiv \oc A </math> &emsp; <math>
\oc \one \linequiv \one </math> &emsp; <math>
\oc\oc A \linequiv \oc A </math>

* Commutation of quantifiers (<math>\zeta</math> does not occur in <math>A</math>): <math>
\exists \xi. \exists \psi. A \linequiv \exists \psi. \exists \xi. A </math> &emsp; <math>
\exists \xi.(A \plus B) \linequiv \exists \xi.A \plus \exists \xi.B </math> &emsp; <math>
\exists \zeta.(A\tens B) \linequiv A\tens\exists \zeta.B </math> &emsp; <math>
\exists \zeta.A \linequiv A </math>

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:

<math> \zero \linequiv \forall X.X </math> &emsp;
<math> \one \linequiv \forall X.(X \limp X) </math> &emsp;
<math> A \plus B \linequiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X) </math>

=== Additional equivalences ===

<math> \oc\wn\oc\wn A \linequiv \oc\wn A </math> &emsp;
<math> \oc\wn \one \linequiv \one </math>

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

=== Positive/negative commutation ===

<math>\exists\forall\limp\forall\exists</math>,
<math>A\tens(B\parr C)\limp(A\tens B)\parr C</math>

== Properties of proofs ==

The fundamental property of the sequent calculus of linear logic is the cut
elimination property, which states that the cut rule is useless as far as
provability is concerned.
This property is exposed in the following section, together with a sketch of
proof.

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\vdash\Gamma</math>, there is a proof of <math>\vdash\Gamma</math> if and only if there is a proof of <math>\vdash\Gamma</math> that does not use the cut rule.
}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas of its immediate subformulas:
* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,
* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,
* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas <math>\alpha</math> and <math>\alpha\orth</math> have no immediate subformula,
* the immediate subformulas of <math>\exists x.A</math> are all the <math>A[t/x]</math> for all first-order terms <math>t</math>,
* the immediate subformulas of <math>\exists X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math>,
* the only immediate subformula of <math>\forall\xi.A</math> is <math>A</math>.
}}

{{Theorem|title=subformula property|
A sequent <math>\vdash\Gamma</math> is provable if and only if it is the conclusion of
a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math>.
}}
{{Proof|By the cut elimination theorem, if a sequent <math>\vdash</math> is provable, then it
is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.
}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its negation
<math>A\orth</math>; it is impossible to prove <math>\zero</math> or <math>\bot</math>.
}}
{{Proof|
If <math>\vdash\Gamma</math> is a provable sequent, then it is the conclusion of a
cut-free proof.
In each rule except the cut rule, there is at least one formula in
conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.

The other properties are immediate consequences: if <math>A</math> and <math>A\orth</math> were
provable, then by a cut rule one would get an empty conclusion, which is not
possible.
As particular cases, since <math>\one</math> and <math>\top</math> are provable, their negations
<math>\bot</math> and <math>\zero</math> are not.
}}

=== Expansion of identities ===

Let us write <math>\pi\vdash\Gamma</math> to signify that <math>\pi</math> is a proof with
conclusion <math>\vdash\Gamma</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the same conclusion as
<math>\pi</math> in which the axiom rule is only used with atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.
}}
{{Proof|
It suffices to prove that for every formula <math>A</math>, the sequent
<math>\vdash A\orth,A</math> has a cut-free proof in which the axiom rule is used only
for atomic formulas.
We prove this by induction on <math>A</math>.
Not that there is a case for each pair of dual connectives.

* If <math>A</math> is atomic, then <math>\vdash A\orth,A</math> is an instance of the atomic axiom rule.

* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \tens }
\BinRule{ \vdash A_1\orth, A_2\orth, A_1 \tens A_2 }
\LabelRule{ \parr }
\UnaRule{ \vdash A_1\orth \parr A_2\orth, A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\one</math> or <math>A=\bot</math> then we have <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\LabelRule{ \bot }
\UnaRule{ \vdash \one, \bot }
\DisplayProof
</math>

* If <math>A=A_1\plus A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash A_1\orth, A_1 \plus A_2 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash A_2\orth, A_1 \plus A_2 }
\LabelRule{ \with }
\BinRule{ \vdash A_1\orth \with A_2\orth, A_1 \plus A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\zero</math> or <math>A=\top</math>, we have <math>
\LabelRule{ \top }
\NulRule{ \vdash \top, \zero }
\DisplayProof
</math>

* If <math>A=\oc B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ d }
\UnaRule{ \pi \vdash \wn B\orth, B }
\LabelRule{ \oc }
\UnaRule{ \pi \vdash \wn B\orth, \oc B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* If <math>A=\exists X.B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ \exists }
\UnaRule{ \vdash B\orth, \exists X.B }
\LabelRule{ \forall }
\UnaRule{ \vdash \forall X.B\orth, \exists X.B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* First-order quantification works like second-order quantification.
}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if

* for every proof <math>\pi\vdash\Gamma,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,

* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.
}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are
reversible.
}}
{{Proof|
Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only
applied to atomic formulas.
Then each top-level connective is introduced either by its associated rule
or in an instance of the <math>\top</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\vdash\Gamma,A\parr B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr</math> rule, then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction).
If it is introduced in the contect of a <math>\top</math> rule, then this rule can be
changed so that <math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a <math>\bot</math>
rule, then remove this rule to get a proof of <math>\vdash\Gamma</math>, if it is
introduced by a <math>\top</math> rule, remove the <math>\bot</math> from this rule, then apply
the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof <math>\pi\vdash\Gamma,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is applied
in a context like

<math>
\AxRule{ \pi_1 \vdash \Delta, A }
\AxRule{ \pi_2 \vdash \Delta, B }
\LabelRule{ \with }
\BinRule{ \vdash \Delta, A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except as
context), if we replace this step by <math>\pi_1</math> in <math>\pi</math> we finally get a proof
<math>\pi'_1\vdash\Gamma,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof <math>\pi'_2\vdash\Gamma,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final <math>\with</math> rule we finally get the
expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math> rule is solved as
before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as an instance of the
<math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math> at second order, consider a proof <math>\pi\vdash\Gamma,\forall X.A</math>.
Up to renaming, we can assume that <math>X</math> occurs free only above the rule that
introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we remove this
rule, we can check that we get a proof of <math>\vdash\Gamma,A</math> on which we can
finally apply the <math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math> rule is solved as
before.
First-order quantification is similar.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the original
proof.
}}

== Variations ==

=== Two-sided sequent calculus ===

The sequent calculus of linear logic can also be presented using two-sided
sequents <math>\Gamma\vdash\Delta</math>, with any number of formulas on the left and
right.
In this case, it is customary to provide rules only for the positive
connectives, then there are left and right introduction rules and a negation
rule that moves formulas between the left and right sides:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\DisplayProof
</math>

Identity group:

<math>
\LabelRule{axiom}
\NulRule{ A \vdash A }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{cut}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>

Multiplicative group:

<math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
\qquad
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>

Additive group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>

Exponential group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>

User:Emmanuel Beffara

2009-01-21T16:19:25Z

Emmanuel Beffara: New page: Wanna see my [http://iml.univ-mrs.fr/~beffara/ home page]?

Wanna see my [http://iml.univ-mrs.fr/~beffara/ home page]?

Talk:Sequent calculus

2009-01-21T16:07:12Z

Emmanuel Beffara: /* Quantifiers */

== Quantifiers ==

The presentation does not seem to be completely uniform concerning quantifiers: are first-order quantifiers taken into account? It would be nice.

A few related points:
* Why a distinction between atomic formulas and propositional variables?
* Some mixing between <math>\forall x A</math> and <math>\forall X A</math>. I tried to propose a [[notations#formulas|convention]] on that point, but it does not match here with the use of <math>\alpha</math> for atoms.
* Define immediate subformula of <math>\forall X A</math> as <math>A</math>?
-- [[User:Olivier Laurent|Olivier Laurent]] 18:37, 14 January 2009 (UTC)

:I improved the uniformity for quantifiers: the full system with first and second order quantification is described, only predicate variables with first-order arguments are not described.
:The distinction between atomic formulas and propositional variables is because there are systems with atomic formulas that are not propositional variables but fixed predicates like equalities.
:I found <math>\alpha</math> to be a used notation for atomic formulas in other texts, so I used <math>\xi,\psi,\zeta</math> instead for arbitrary variables.

:Using substitution in the definition of subformulas is questionable, but if the only immediate subformula of <math>\forall\xi.A</math> is <math>A</math>, then the ''subformula'' property does not hold.

:''Edit:'' Well... my bad. The subformula property does hold if the only immediate subformula of <math>\forall\xi.A</math> is <math>A</math>, substitution is necessary only for <math>\exists\xi.A</math>. I changed it.

:-- [[User:Emmanuel Beffara|Emmanuel Beffara]]

== Two-sided sequent calculus ==

I think the terminology "two-sided sequent calculus" should be used for the system where all the connectives are involved and all the rules are duplicated (with respect to the one-sided version) and negation is a connective.

In this way, we obtain the one-sided version from the two-sided one by:
* quotient the formulas by de Morgan laws and get negation only on atoms, negation is defined for compound formulas (not a connective)
* fold all the rules by <math>\Gamma\vdash\Delta \mapsto {}\vdash\Gamma\orth,\Delta</math>
* remove useless rules (negation rules become identities, almost all the rules appear twice)

A possible name for the two-sided system presented here could be "two-sided positive sequent calculus".

-- [[User:Olivier Laurent|Olivier Laurent]] 21:34, 15 January 2009 (UTC)

Sequent calculus

2009-01-21T16:06:20Z

Emmanuel Beffara: /* Cut elimination and consequences */ definition of subformula

This article presents the language and sequent calculus of second-order
propositional linear logic and the basic properties of this sequent calculus.

== Formulas ==

Formulas are built on a set of atoms, written <math>\alpha,\beta,\ldots</math>, that can
be either propositional variables <math>X,Y,Z\ldots</math> or atomic formulas
<math>p(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms from some first-order language and <math>p</math> is a predicate symbol.
Formulas, represented by capital letters <math>A</math>, <math>B</math>, <math>C</math>, are built using the
following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>\alpha\orth</math>
| negated atom
| atoms
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists \xi.A</math>
| there exists
| <math>\forall \xi.A</math>
| for all
| quantifiers (<math>\xi</math> is a first or second order variable)
|}

Each line corresponds to a particular class of connectives, and each class
consists in a pair of connectives.
Those in the left column are called positive and those in the right column are
called negative.
The ''tensor'' and ''with'' are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally
read ''of course <math>A</math>'' for <math>\oc A</math> and ''why not <math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

Given a formula <math>A</math>, its linear negation, also called ''orthogonal'' and
written <math>A\orth</math>, is obtained by exchanging each positive connective with the
negative one of the same class and vice versa, in a way analogous to de Morgan
laws in classical logic.
Formally, the definition of linear negation is

{|
|-
|align="right"| <math> ( \alpha )\orth </math>
| <math>:= \alpha\orth </math>
|width=30|
|align="right"| <math> ( \alpha\orth )\orth </math>
| <math>:= \alpha </math>
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>:= A\orth \parr B\orth </math>
|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>:= A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>:= \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>:= \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>:= A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>:= A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>:= \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>:= \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>:= \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>:= \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists \xi.A )\orth </math>
| <math>:= \forall \xi.( A\orth ) </math>
|
|align="right"| <math> ( \forall \xi.A )\orth </math>
| <math>:= \exists \xi.( A\orth ) </math>
|}

Note that this operation is defined syntactically, hence negation is not a
connective, the only place in formulas where the symbol <math>(\cdot)\orth</math> occurs
is for negated atoms <math>\alpha\orth</math>.
Note also that, by construction, negation is involutive: for any formula <math>A</math>,
it holds that <math>A\biorth=A</math>.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic:

<math>
A \limp B := A\orth \parr B
</math>

Free and bound variables are defined in the standard way, as well as
substitution.
Formulas are always considered up to renaming of bound names.
If <math>A</math> and <math>B</math> are formulas and <math>X</math> is a propositional variable, the formula
<math>A[B/X]</math> is <math>A</math> where all atoms <math>X</math> are replaced (without capture) by <math>B</math> and
all atoms <math>X\orth</math> are replaced by the formula <math>B\orth</math>.

== Sequents and proofs ==

A sequent is an expression <math>\vdash\Gamma</math> where <math>\Gamma</math> is a finite multiset
of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation <math>\wn\Gamma</math> represents
the multiset <math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:

* Identity group: <math>
\LabelRule{axiom}
\NulRule{ \vdash A, A\orth }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{cut}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>

* Multiplicative group: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
\qquad
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>

* Additive group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
\qquad
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>

* Exponential group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>

* Quantifier group (in the <math>\forall</math> rule, <math>\xi</math> must not occur free in <math>\Gamma</math>): <math>
\AxRule{ \vdash \Gamma, A[t/x] }
\LabelRule{ \exists^1 }
\UnaRule{ \vdash \Gamma, \exists x.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists^2 }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall \xi.A }
\DisplayProof
</math>

The rules for exponentials are called ''dereliction'', ''weakening'',
''contraction'' and ''promotion'', respectively.
Note the fundamental fact that there are no contraction and weakening rules
for arbitrary formulas, but only for the formulas starting with the <math>\wn</math>
modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Note that this system contains only introduction rules and no elimination
rule.
Moreover, there is no introduction rule for the additive unit <math>\zero</math>, the
only ways to introduce it at top level are the axiom rule and the <math>\top</math> rule.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An equivalent presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rule:

<math>
\AxRule{ \vdash \Gamma, A, B, \Delta }
\LabelRule{exchange}
\UnaRule{ \vdash \Gamma, B, A, \Delta }
\DisplayProof
</math>

== Equivalences and definability ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent, written <math>A\equiv B</math>, if
both implications <math>A\limp B</math> and <math>B\limp A</math> are provable.
Equivalently, <math>A\equiv B</math> if both <math>\vdash A\orth,B</math> and <math>\vdash B\orth,A</math>
are provable.
Another formulation of <math>A\equiv B</math> is that, for all <math>\Gamma</math>,
<math>\vdash\Gamma,A</math> is provable if and only if <math>\vdash\Gamma,B</math> is provable.
Note that, because of the definition of negation, an equivalence
<math>A\equiv B</math> holds if and only if the dual equivalence
<math>A\orth\equiv B\orth</math> holds.

Two related notions are [[isomorphism]] (stronger than equivalence) and [[equiprovability]] (weaker than equivalence).

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality: <math>
A \tens (B \tens C) \equiv (A \tens B) \tens C </math> &emsp; <math>
A \tens B \equiv B \tens A </math> &emsp; <math>
A \tens \one \equiv A </math> <math>
A \plus (B \plus C) \equiv (A \plus B) \plus C </math> &emsp; <math>
A \plus B \equiv B \plus A </math> &emsp; <math>
A \plus \zero \equiv A </math>

* Idempotence of additives: <math>
A \plus A \equiv A </math>

* Distributivity of multiplicatives over additives: <math>
A \tens (B \plus C) \equiv (A \tens B) \plus (A \tens C) </math> &emsp; <math>
A \tens \zero \equiv \zero </math>

* Defining property of exponentials: <math>
\oc(A \with B) \equiv \oc A \tens \oc B </math> &emsp; <math>
\oc\top \equiv \one </math>

* Monoidal structure of exponentials, digging: <math>
\oc A \otimes \oc A \equiv \oc A </math> &emsp; <math>
\oc \one \equiv \one </math> &emsp; <math>
\oc\oc A \equiv \oc A </math>

* Commutation of quantifiers (<math>\zeta</math> does not occur in <math>A</math>): <math>
\exists \xi. \exists \psi. A \equiv \exists \psi. \exists \xi. A </math> &emsp; <math>
\exists \xi.(A \plus B) \equiv \exists \xi.A \plus \exists \xi.B </math> &emsp; <math>
\exists \zeta.(A\tens B) \equiv A\tens\exists \zeta.B </math> &emsp; <math>
\exists \zeta.A \equiv A </math>

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:

<math> \zero \equiv \forall X.X </math> &emsp;
<math> \one \equiv \forall X.(X \limp X) </math> &emsp;
<math> A \plus B \equiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X) </math>

=== Additional equivalences ===

<math> \oc\wn\oc\wn A \equiv \oc\wn A </math> &emsp;
<math> \oc\wn \one \equiv \one </math>

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

=== Positive/negative commutation ===

<math>\exists\forall\limp\forall\exists</math>,
<math>A\tens(B\parr C)\limp(A\tens B)\parr C</math>

== Properties of proofs ==

The fundamental property of the sequent calculus of linear logic is the cut
elimination property, which states that the cut rule is useless as far as
provability is concerned.
This property is exposed in the following section, together with a sketch of
proof.

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\vdash\Gamma</math>, there is a proof of <math>\vdash\Gamma</math> if and only if there is a proof of <math>\vdash\Gamma</math> that does not use the cut rule.
}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas of its immediate subformulas:
* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,
* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,
* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas <math>\alpha</math> and <math>\alpha\orth</math> have no immediate subformula,
* the immediate subformulas of <math>\exists x.A</math> are all the <math>A[t/x]</math> for all first-order terms <math>t</math>,
* the immediate subformulas of <math>\exists X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math>,
* the only immediate subformula of <math>\forall\xi.A</math> is <math>A</math>.
}}

{{Theorem|title=subformula property|
A sequent <math>\vdash\Gamma</math> is provable if and only if it is the conclusion of
a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math>.
}}
{{Proof|By the cut elimination theorem, if a sequent <math>\vdash</math> is provable, then it
is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.
}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its negation
<math>A\orth</math>; it is impossible to prove <math>\zero</math> or <math>\bot</math>.
}}
{{Proof|
If <math>\vdash\Gamma</math> is a provable sequent, then it is the conclusion of a
cut-free proof.
In each rule except the cut rule, there is at least one formula in
conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.

The other properties are immediate consequences: if <math>A</math> and <math>A\orth</math> were
provable, then by a cut rule one would get an empty conclusion, which is not
possible.
As particular cases, since <math>\one</math> and <math>\top</math> are provable, their negations
<math>\bot</math> and <math>\zero</math> are not.
}}

=== Expansion of identities ===

Let us write <math>\pi\vdash\Gamma</math> to signify that <math>\pi</math> is a proof with
conclusion <math>\vdash\Gamma</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the same conclusion as
<math>\pi</math> in which the axiom rule is only used with atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.
}}
{{Proof|
It suffices to prove that for every formula <math>A</math>, the sequent
<math>\vdash A\orth,A</math> has a cut-free proof in which the axiom rule is used only
for atomic formulas.
We prove this by induction on <math>A</math>.
Not that there is a case for each pair of dual connectives.

* If <math>A</math> is atomic, then <math>\vdash A\orth,A</math> is an instance of the atomic axiom rule.

* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \tens }
\BinRule{ \vdash A_1\orth, A_2\orth, A_1 \tens A_2 }
\LabelRule{ \parr }
\UnaRule{ \vdash A_1\orth \parr A_2\orth, A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\one</math> or <math>A=\bot</math> then we have <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\LabelRule{ \bot }
\UnaRule{ \vdash \one, \bot }
\DisplayProof
</math>

* If <math>A=A_1\plus A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash A_1\orth, A_1 \plus A_2 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash A_2\orth, A_1 \plus A_2 }
\LabelRule{ \with }
\BinRule{ \vdash A_1\orth \with A_2\orth, A_1 \plus A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\zero</math> or <math>A=\top</math>, we have <math>
\LabelRule{ \top }
\NulRule{ \vdash \top, \zero }
\DisplayProof
</math>

* If <math>A=\oc B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ d }
\UnaRule{ \pi \vdash \wn B\orth, B }
\LabelRule{ \oc }
\UnaRule{ \pi \vdash \wn B\orth, \oc B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* If <math>A=\exists X.B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ \exists }
\UnaRule{ \vdash B\orth, \exists X.B }
\LabelRule{ \forall }
\UnaRule{ \vdash \forall X.B\orth, \exists X.B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* First-order quantification works like second-order quantification.
}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if

* for every proof <math>\pi\vdash\Gamma,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,

* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.
}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are
reversible.
}}
{{Proof|
Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only
applied to atomic formulas.
Then each top-level connective is introduced either by its associated rule
or in an instance of the <math>\top</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\vdash\Gamma,A\parr B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr</math> rule, then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction).
If it is introduced in the contect of a <math>\top</math> rule, then this rule can be
changed so that <math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a <math>\bot</math>
rule, then remove this rule to get a proof of <math>\vdash\Gamma</math>, if it is
introduced by a <math>\top</math> rule, remove the <math>\bot</math> from this rule, then apply
the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof <math>\pi\vdash\Gamma,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is applied
in a context like

<math>
\AxRule{ \pi_1 \vdash \Delta, A }
\AxRule{ \pi_2 \vdash \Delta, B }
\LabelRule{ \with }
\BinRule{ \vdash \Delta, A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except as
context), if we replace this step by <math>\pi_1</math> in <math>\pi</math> we finally get a proof
<math>\pi'_1\vdash\Gamma,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof <math>\pi'_2\vdash\Gamma,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final <math>\with</math> rule we finally get the
expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math> rule is solved as
before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as an instance of the
<math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math> at second order, consider a proof <math>\pi\vdash\Gamma,\forall X.A</math>.
Up to renaming, we can assume that <math>X</math> occurs free only above the rule that
introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we remove this
rule, we can check that we get a proof of <math>\vdash\Gamma,A</math> on which we can
finally apply the <math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math> rule is solved as
before.
First-order quantification is similar.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the original
proof.
}}

== Variations ==

=== Two-sided sequent calculus ===

The sequent calculus of linear logic can also be presented using two-sided
sequents <math>\Gamma\vdash\Delta</math>, with any number of formulas on the left and
right.
In this case, it is customary to provide rules only for the positive
connectives, then there are left and right introduction rules and a negation
rule that moves formulas between the left and right sides:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\DisplayProof
</math>

Identity group:

<math>
\LabelRule{axiom}
\NulRule{ A \vdash A }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{cut}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>

Multiplicative group:

<math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
\qquad
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>

Additive group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>

Exponential group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>

Notations

2009-01-17T16:12:19Z

Emmanuel Beffara: /* Formulas */

== Logical systems ==

For a given logical system such as <math>MLL</math> (for multiplicative linear logic), we consider the following variations:
{|
|-
! Notation
! Meaning
! Connectives
|-
| <math>MLL</math>
| propositional without units
| <math>X,{\tens},{\parr}</math>
|-
| <math>MLL_u</math>
| propositional with units only
| <math>\one,\bot,{\tens},{\parr}</math>
|-
| <math>MLL_0</math>
| propositional with units and variables
| <math>X,\one,\bot,{\tens},{\parr}</math>
|-
| <math>MLL_1</math>
| first-order without units
| <math>X\vec{t},{\tens},{\parr},\forall x A,\exists x A</math>
|-
| <math>MLL_{01}</math>
| first-order with units
| <math>X\vec{t},\one,\bot,{\tens},{\parr},\forall x A,\exists x A</math>
|-
| <math>MLL_2</math>
| second-order propositional without units
| <math>X,{\tens},{\parr},\forall X A,\exists X A</math>
|-
| <math>MLL_{02}</math>
| second-order propositional with units
| <math>X,\one,\bot,{\tens},{\parr},\forall X A,\exists X A</math>
|-
| <math>MLL_{12}</math>
| first-order and second-order without units
| <math>X\vec{t},{\tens},{\parr},\forall x A,\exists x A,\forall X A,\exists X A</math>
|-
| <math>MLL_{012}</math>
| first-order and second-order with units
| <math>X\vec{t},\one,\bot,{\tens},{\parr},\forall x A,\exists x A,\forall X A,\exists X A</math>
|-
|}

== Formulas and proof trees ==

=== Formulas ===

* First order quantification: <math>\forall x A</math> with substitution <math>A[t/x]</math>
* Second order quantification: <math>\forall X A</math> with substitution <math>A[B/X]</math>
* Quantification of arbitrary order (mainly first or second): <math>\forall\xi A</math> with substitution <math>A[\tau/\xi]</math>

=== Rule names ===

Name of the connective, followed by some additional information if required, followed by "L" for a left rule or "R" for a right rule. This is for a two-sided system, "R" is implicit for one-sided systems.
For example: <math>\wedge_1 \text{add} L</math>.

Talk:Sequent calculus

2009-01-17T16:10:10Z

Emmanuel Beffara: /* Quantifiers */

== Quantifiers ==

:The presentation does not seem to be completely uniform concerning quantifiers: are first-order quantifiers taken into account? It would be nice.
:
:A few related points:
:* Why a distinction between atomic formulas and propositional variables?
:* Some mixing between <math>\forall x A</math> and <math>\forall X A</math>. I tried to propose a [[notations#formulas|convention]] on that point, but it does not match here with the use of <math>\alpha</math> for atoms.
:* Define immediate subformula of <math>\forall X A</math> as <math>A</math>?
:-- [[User:Olivier Laurent|Olivier Laurent]] 18:37, 14 January 2009 (UTC)

I improved the uniformity for quantifiers: the full system with first and second order quantification is described, only predicate variables with first-order arguments are not described.
The distinction between atomic formulas and propositional variables is because there are systems with atomic formulas that are not propositional variables but fixed predicates like equalities.
I found <math>\alpha</math> to be a used notation for atomic formulas in other texts, so I used <math>\xi,\psi,\zeta</math> instead for arbitrary variables.

Using substitution in the definition of subformulas is questionable, but if the only immediate subformula of <math>\forall\xi.A</math> is <math>A</math>, then the ''subformula'' property does not hold.

-- [[User:Emmanuel Beffara|Emmanuel Beffara]]

== Two-sided sequent calculus ==

I think the terminology "two-sided sequent calculus" should be used for the system where all the connectives are involved and all the rules are duplicated (with respect to the one-sided version) and negation is a connective.

In this way, we obtain the one-sided version from the two-sided one by:
* quotient the formulas by de Morgan laws and get negation only on atoms, negation is defined for compound formulas (not a connective)
* fold all the rules by <math>\Gamma\vdash\Delta \mapsto {}\vdash\Gamma\orth,\Delta</math>
* remove useless rules (negation rules become identities, almost all the rules appear twice)

A possible name for the two-sided system presented here could be "two-sided positive sequent calculus".

-- [[User:Olivier Laurent|Olivier Laurent]] 21:34, 15 January 2009 (UTC)

Sequent calculus

2009-01-17T16:10:04Z

Emmanuel Beffara: conventions for quantifiers

This article presents the language and sequent calculus of second-order
propositional linear logic and the basic properties of this sequent calculus.

== Formulas ==

Formulas are built on a set of atoms, written <math>\alpha,\beta,\ldots</math>, that can
be either propositional variables <math>X,Y,Z\ldots</math> or atomic formulas
<math>p(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms from some first-order language and <math>p</math> is a predicate symbol.
Formulas, represented by capital letters <math>A</math>, <math>B</math>, <math>C</math>, are built using the
following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>\alpha\orth</math>
| negated atom
| atoms
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists \xi.A</math>
| there exists
| <math>\forall \xi.A</math>
| for all
| quantifiers (<math>\xi</math> is a first or second order variable)
|}

Each line corresponds to a particular class of connectives, and each class
consists in a pair of connectives.
Those in the left column are called positive and those in the right column are
called negative.
The ''tensor'' and ''with'' are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally
read ''of course <math>A</math>'' for <math>\oc A</math> and ''why not <math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

Given a formula <math>A</math>, its linear negation, also called ''orthogonal'' and
written <math>A\orth</math>, is obtained by exchanging each positive connective with the
negative one of the same class and vice versa, in a way analogous to de Morgan
laws in classical logic.
Formally, the definition of linear negation is

{|
|-
|align="right"| <math> ( \alpha )\orth </math>
| <math>:= \alpha\orth </math>
|width=30|
|align="right"| <math> ( \alpha\orth )\orth </math>
| <math>:= \alpha </math>
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>:= A\orth \parr B\orth </math>
|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>:= A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>:= \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>:= \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>:= A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>:= A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>:= \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>:= \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>:= \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>:= \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists \xi.A )\orth </math>
| <math>:= \forall \xi.( A\orth ) </math>
|
|align="right"| <math> ( \forall \xi.A )\orth </math>
| <math>:= \exists \xi.( A\orth ) </math>
|}

Note that this operation is defined syntactically, hence negation is not a
connective, the only place in formulas where the symbol <math>(\cdot)\orth</math> occurs
is for negated atoms <math>\alpha\orth</math>.
Note also that, by construction, negation is involutive: for any formula <math>A</math>,
it holds that <math>A\biorth=A</math>.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic:

<math>
A \limp B := A\orth \parr B
</math>

Free and bound variables are defined in the standard way, as well as
substitution.
Formulas are always considered up to renaming of bound names.
If <math>A</math> and <math>B</math> are formulas and <math>X</math> is a propositional variable, the formula
<math>A[B/X]</math> is <math>A</math> where all atoms <math>X</math> are replaced (without capture) by <math>B</math> and
all atoms <math>X\orth</math> are replaced by the formula <math>B\orth</math>.

== Sequents and proofs ==

A sequent is an expression <math>\vdash\Gamma</math> where <math>\Gamma</math> is a finite multiset
of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation <math>\wn\Gamma</math> represents
the multiset <math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:

* Identity group: <math>
\LabelRule{axiom}
\NulRule{ \vdash A, A\orth }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{cut}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>

* Multiplicative group: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
\qquad
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>

* Additive group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
\qquad
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>

* Exponential group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>

* Quantifier group (in the <math>\forall</math> rule, <math>\xi</math> must not occur free in <math>\Gamma</math>): <math>
\AxRule{ \vdash \Gamma, A[t/x] }
\LabelRule{ \exists^1 }
\UnaRule{ \vdash \Gamma, \exists x.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists^2 }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall \xi.A }
\DisplayProof
</math>

The rules for exponentials are called ''dereliction'', ''weakening'',
''contraction'' and ''promotion'', respectively.
Note the fundamental fact that there are no contraction and weakening rules
for aribtrary formulas, but only for the formulas starting with the <math>\wn</math>
modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Note that this system contains only introduction rules and no elimination
rule.
Moreover, there is no introduction rule for the additive unit <math>\zero</math>, the
only ways to introduce it at top level are the axiom rule and the <math>\top</math> rule.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An equivalent presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rule:

<math>
\AxRule{ \vdash \Gamma, A, B, \Delta }
\LabelRule{exchange}
\UnaRule{ \vdash \Gamma, B, A, \Delta }
\DisplayProof
</math>

== Equivalences and definability ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent, written <math>A\equiv B</math>, if
both implications <math>A\limp B</math> and <math>B\limp A</math> are provable.
Equivalently, <math>A\equiv B</math> if both <math>\vdash A\orth,B</math> and <math>\vdash B\orth,A</math>
are provable.
Another formulation of <math>A\equiv B</math> is that, for all <math>\Gamma</math>,
<math>\vdash\Gamma,A</math> is provable if and only if <math>\vdash\Gamma,B</math> is provable.
Note that, because of the definition of negation, an equivalence
<math>A\equiv B</math> holds if and only if the dual equivalence
<math>A\orth\equiv B\orth</math> holds.

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality: <math>
A \tens (B \tens C) \equiv (A \tens B) \tens C </math> &emsp; <math>
A \tens B \equiv B \tens A </math> &emsp; <math>
A \tens \one \equiv A </math> <math>
A \plus (B \plus C) \equiv (A \plus B) \plus C </math> &emsp; <math>
A \plus B \equiv B \plus A </math> &emsp; <math>
A \plus \zero \equiv A </math>

* Idempotence of additives: <math>
A \plus A \equiv A </math>

* Distributivity of multiplicatives over additives: <math>
A \tens (B \plus C) \equiv (A \tens B) \plus (A \tens C) </math> &emsp; <math>
A \tens \zero \equiv \zero </math>

* Defining property of exponentials: <math>
\oc(A \with B) \equiv \oc A \tens \oc B </math> &emsp; <math>
\oc\top \equiv \one </math>

* Monoidal structure of exponentials, digging: <math>
\oc A \otimes \oc A \equiv \oc A </math> &emsp; <math>
\oc \one \equiv \one </math> &emsp; <math>
\oc\oc A \equiv \oc A </math>

* Commutation of quantifiers (<math>\zeta</math> does not occur in <math>A</math>): <math>
\exists \xi. \exists \psi. A \equiv \exists \psi. \exists \xi. A </math> &emsp; <math>
\exists \xi.(A \plus B) \equiv \exists \xi.A \plus \exists \xi.B </math> &emsp; <math>
\exists \zeta.(A\tens B) \equiv A\tens\exists \zeta.B </math> &emsp; <math>
\exists \zeta.A \equiv A </math>

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:

<math> \zero \equiv \forall X.X </math> &emsp;
<math> \one \equiv \forall X.(X \limp X) </math> &emsp;
<math> A \plus B \equiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X) </math>

=== Additional equivalences ===

<math> \oc\wn\oc\wn A \equiv \oc\wn A </math> &emsp;
<math> \oc\wn \one \equiv \one </math>

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

=== Positive/negative commutation ===

<math>\exists\forall\limp\forall\exists</math>,
<math>A\tens(B\parr C)\limp(A\tens B)\parr C</math>

== Properties of proofs ==

The fundamental property of the sequent calculus of linear logic is the cut
elimination property, which states that the cut rule is useless as far as
provability is concerned.
This property is exposed in the following section, together with a sketch of
proof.

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\vdash\Gamma</math>, there is a proof of <math>\vdash\Gamma</math> if and only if there is a proof of <math>\vdash\Gamma</math> that does not use the cut rule.
}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas of its immediate subformulas:
* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,
* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,
* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas <math>\alpha</math> and <math>\alpha\orth</math> have no immediate subformula,
* the immediate subformulas of <math>\exists x.A</math> and <math>\forall x.A</math> are all the <math>A[t/x]</math> for all first-order terms <math>t</math>.
* the immediate subformulas of <math>\exists X.A</math> and <math>\forall X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math>.
}}

{{Theorem|title=subformula property|
A sequent <math>\vdash\Gamma</math> is provable if and only if it is the conclusion of
a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math>.
}}
{{Proof|By the cut elimination theorem, if a sequent <math>\vdash</math> is provable, then it
is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.
}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its negation
<math>A\orth</math>; it is impossible to prove <math>\zero</math> or <math>\bot</math>.
}}
{{Proof|
If <math>\vdash\Gamma</math> is a provable sequent, then it is the conclusion of a
cut-free proof.
In each rule except the cut rule, there is at least one formula in
conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.

The other properties are immediate consequences: if <math>A</math> and <math>A\orth</math> were
provable, then by a cut rule one would get an empty conclusion, which is not
possible.
As particular cases, since <math>\one</math> and <math>\top</math> are provable, their negations
<math>\bot</math> and <math>\zero</math> are not.
}}

=== Expansion of identities ===

Let us write <math>\pi\vdash\Gamma</math> to signify that <math>\pi</math> is a proof with
conclusion <math>\vdash\Gamma</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the same conclusion as
<math>\pi</math> in which the axiom rule is only used with atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.
}}
{{Proof|
It suffices to prove that for every formula <math>A</math>, the sequent
<math>\vdash A\orth,A</math> has a cut-free proof in which the axiom rule is used only
for atomic formulas.
We prove this by induction on <math>A</math>.
Not that there is a case for each pair of dual connectives.

* If <math>A</math> is atomic, then <math>\vdash A\orth,A</math> is an instance of the atomic axiom rule.

* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \tens }
\BinRule{ \vdash A_1\orth, A_2\orth, A_1 \tens A_2 }
\LabelRule{ \parr }
\UnaRule{ \vdash A_1\orth \parr A_2\orth, A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\one</math> or <math>A=\bot</math> then we have <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\LabelRule{ \bot }
\UnaRule{ \vdash \one, \bot }
\DisplayProof
</math>

* If <math>A=A_1\plus A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash A_1\orth, A_1 \plus A_2 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash A_2\orth, A_1 \plus A_2 }
\LabelRule{ \with }
\BinRule{ \vdash A_1\orth \with A_2\orth, A_1 \plus A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\zero</math> or <math>A=\top</math>, we have <math>
\LabelRule{ \top }
\NulRule{ \vdash \top, \zero }
\DisplayProof
</math>

* If <math>A=\oc B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ d }
\UnaRule{ \pi \vdash \wn B\orth, B }
\LabelRule{ \oc }
\UnaRule{ \pi \vdash \wn B\orth, \oc B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* If <math>A=\exists X.B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ \exists }
\UnaRule{ \vdash B\orth, \exists X.B }
\LabelRule{ \forall }
\UnaRule{ \vdash \forall X.B\orth, \exists X.B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* First-order quantification works like second-order quantification.
}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if

* for every proof <math>\pi\vdash\Gamma,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,

* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.
}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are
reversible.
}}
{{Proof|
Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only
applied to atomic formulas.
Then each top-level connective is introduced either by its associated rule
or in an instance of the <math>\top</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\vdash\Gamma,A\parr B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr</math> rule, then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction).
If it is introduced in the contect of a <math>\top</math> rule, then this rule can be
changed so that <math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a <math>\bot</math>
rule, then remove this rule to get a proof of <math>\vdash\Gamma</math>, if it is
introduced by a <math>\top</math> rule, remove the <math>\bot</math> from this rule, then apply
the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof <math>\pi\vdash\Gamma,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is applied
in a context like

<math>
\AxRule{ \pi_1 \vdash \Delta, A }
\AxRule{ \pi_2 \vdash \Delta, B }
\LabelRule{ \with }
\BinRule{ \vdash \Delta, A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except as
context), if we replace this step by <math>\pi_1</math> in <math>\pi</math> we finally get a proof
<math>\pi'_1\vdash\Gamma,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof <math>\pi'_2\vdash\Gamma,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final <math>\with</math> rule we finally get the
expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math> rule is solved as
before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as an instance of the
<math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math> at second order, consider a proof <math>\pi\vdash\Gamma,\forall X.A</math>.
Up to renaming, we can assume that <math>X</math> occurs free only above the rule that
introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we remove this
rule, we can check that we get a proof of <math>\vdash\Gamma,A</math> on which we can
finally apply the <math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math> rule is solved as
before.
First-order quantification is similar.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the original
proof.
}}

== Variations ==

=== Two-sided sequent calculus ===

The sequent calculus of linear logic can also be presented using two-sided
sequents <math>\Gamma\vdash\Delta</math>, with any number of formulas on the left and
right.
In this case, it is customary to provide rules only for the positive
connectives, then there are left and right introduction rules and a negation
rule that moves formulas between the left and right sides:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\DisplayProof
</math>

Identity group:

<math>
\LabelRule{axiom}
\NulRule{ A \vdash A }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{cut}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>

Multiplicative group:

<math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
\qquad
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>

Additive group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>

Exponential group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>

Sequent calculus

2009-01-17T14:50:08Z

Emmanuel Beffara: /* Formulas */ table markup

This article presents the language and sequent calculus of second-order
propositional linear logic and the basic properties of this sequent calculus.

== Formulas ==

Formulas are built on a set of atoms, written <math>\alpha,\beta,\ldots</math>, that can
be either propositional variables <math>X,Y,Z\ldots</math> or atomic formulas
<math>X(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms from some first-order language.
Formulas, represented by capital letters <math>A</math>, <math>B</math>, <math>C</math>, are built using the
following connectives:

{| style="border-spacing: 2em 0"
|-
| <math>\alpha</math>
| atom
| <math>\alpha\orth</math>
| negated atom
| atoms
|-
| <math>A \tens B</math>
| tensor
| <math>A \parr B</math>
| par
| multiplicatives
|-
| <math>\one</math>
| one
| <math>\bot</math>
| bottom
| multiplicative units
|-
| <math>A \plus B</math>
| plus
| <math>A \with B</math>
| with
| additives
|-
| <math>\zero</math>
| zero
| <math>\top</math>
| top
| additive units
|-
| <math>\oc A</math>
| of course
| <math>\wn A</math>
| why not
| exponentials
|-
| <math>\exists x.A</math>
| there exists
| <math>\forall x.A</math>
| for all
| quantifiers
|}

Each line corresponds to a particular class of connectives, and each class
consists in a pair of connectives.
Those in the left column are called positive and those in the right column are
called negative.
The ''tensor'' and ''with'' are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally
read ''of course <math>A</math>'' for <math>\oc A</math> and ''why not <math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

Given a formula <math>A</math>, its linear negation, also called ''orthogonal'' and
written <math>A\orth</math>, is obtained by exchanging each positive connective with the
negative one of the same class and vice versa, in a way analogous to de Morgan
laws in classical logic.
Formally, the definition of linear negation is

{|
|-
|align="right"| <math> ( \alpha )\orth </math>
| <math>:= \alpha\orth </math>
|width=30|
|align="right"| <math> ( \alpha\orth )\orth </math>
| <math>:= \alpha </math>
|-
|align="right"| <math> ( A \tens B )\orth </math>
| <math>:= A\orth \parr B\orth </math>
|
|align="right"| <math> ( A \parr B )\orth </math>
| <math>:= A\orth \tens B\orth </math>
|-
|align="right"| <math> \one\orth </math>
| <math>:= \bot </math>
|
|align="right"| <math> \bot\orth </math>
| <math>:= \one </math>
|-
|align="right"| <math> ( A \plus B )\orth </math>
| <math>:= A\orth \with B\orth </math>
|
|align="right"| <math> ( A \with B )\orth </math>
| <math>:= A\orth \plus B\orth </math>
|-
|align="right"| <math> \zero\orth </math>
| <math>:= \top </math>
|
|align="right"| <math> \top\orth </math>
| <math>:= \zero </math>
|-
|align="right"| <math> ( \oc A )\orth </math>
| <math>:= \wn ( A\orth ) </math>
|
|align="right"| <math> ( \wn A )\orth </math>
| <math>:= \oc ( A\orth ) </math>
|-
|align="right"| <math> ( \exists X.A )\orth </math>
| <math>:= \forall X.( A\orth ) </math>
|
|align="right"| <math> ( \forall X.A )\orth </math>
| <math>:= \exists X.( A\orth ) </math>
|}

Note that this operation is defined syntactically, hence negation is not a
connective, the only place in formulas where the symbol <math>(\cdot)\orth</math> occurs
is for negated atoms <math>\alpha\orth</math>.
Note also that, by construction, negation is involutive: for any formula <math>A</math>,
it holds that <math>A\biorth=A</math>.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic:

<math>
A \limp B := A\orth \parr B
</math>

Free and bound variables are defined in the standard way, as well as
substitution.
Formulas are always considered up to renaming of bound names.
If <math>A</math> and <math>B</math> are formulas and <math>X</math> is a propositional variable, the formula
<math>A[B/X]</math> is <math>A</math> where all atoms <math>X</math> are replaced (without capture) by <math>B</math> and
all atoms <math>X\orth</math> are replaced by the formula <math>B\orth</math>.

== Sequents and proofs ==

A sequent is an expression <math>\vdash\Gamma</math> where <math>\Gamma</math> is a finite multiset
of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation <math>\wn\Gamma</math> represents
the multiset <math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:

* Identity group: <math>
\LabelRule{axiom}
\NulRule{ \vdash A, A\orth }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{cut}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>

* Multiplicative group: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
\qquad
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>

* Additive group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
\qquad
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>

* Exponential group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>

* Quantifier group (in the <math>\forall</math> rule, <math>X</math> must not occur in <math>\Gamma</math>): <math>
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall X.A }
\DisplayProof
</math>

The rules for exponentials are called ''dereliction'', ''weakening'',
''contraction'' and ''promotion'', respectively.
Note the fundamental fact that there are no contraction and weakening rules
for aribtrary formulas, but only for the formulas starting with the <math>\wn</math>
modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Note that this system contains only introduction rules and no elimination
rule.
Moreover, there is no introduction rule for the additive unit <math>\zero</math>, the
only ways to introduce it at top level are the axiom rule and the <math>\top</math> rule.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An equivalent presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rule:

<math>
\AxRule{ \vdash \Gamma, A, B, \Delta }
\LabelRule{exchange}
\UnaRule{ \vdash \Gamma, B, A, \Delta }
\DisplayProof
</math>

== Equivalences and definability ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent, written <math>A\equiv B</math>, if
both implications <math>A\limp B</math> and <math>B\limp A</math> are provable.
Equivalently, <math>A\equiv B</math> if both <math>\vdash A\orth,B</math> and <math>\vdash B\orth,A</math>
are provable.
Another formulation of <math>A\equiv B</math> is that, for all <math>\Gamma</math>,
<math>\vdash\Gamma,A</math> is provable if and only if <math>\vdash\Gamma,B</math> is provable.
Note that, because of the definition of negation, an equivalence
<math>A\equiv B</math> holds if and only if the dual equivalence
<math>A\orth\equiv B\orth</math> holds.

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality:
{|
|-
|align="right"| &emsp; <math>
A \tens (B \tens C) </math>
| <math>\equiv (A \tens B) \tens C </math>
|align="right"| &emsp; <math>
A \tens B </math>
| <math>\equiv B \tens A </math>
|align="right"| &emsp; <math>
A \tens \one </math>
| <math>\equiv A </math>
|-
|align="right"| &emsp; <math>
A \plus (B \plus C) </math>
| <math>\equiv (A \plus B) \plus C </math>
|align="right"| &emsp; <math>
A \plus B </math>
| <math>\equiv B \plus A </math>
|align="right"| &emsp; <math>
A \plus \zero </math>
| <math>\equiv A
</math>
|}

* Idempotence of additives:
{|
|-
|align="right"| &emsp; <math>
A \plus A </math>
| <math>\equiv A
</math>
|}

* Distributivity of multiplicatives over additives:
{|
|-
|align="right"| &emsp; <math>
A \tens (B \plus C) </math>
| <math>\equiv (A \tens B) \plus (A \tens C) </math>
|align="right"| &emsp; <math>
A \tens \zero </math>
| <math>\equiv \zero
</math>
|}

* Defining property of exponentials:
{|
|-
|align="right"| &emsp; <math>
\oc(A \with B) </math>
| <math>\equiv \oc A \tens \oc B </math>
|align="right"| &emsp; <math>
\oc\top </math>
| <math>\equiv \one
</math>
|}

* Monoidal structure of exponentials, digging:
{|
|-
|align="right"| &emsp; <math>
\oc A \otimes \oc A </math>
| <math>\equiv \oc A </math>
|align="right"| &emsp; <math>
\oc \one </math>
| <math>\equiv \one </math>
|align="right"| &emsp; <math>
\oc\oc A </math>
| <math>\equiv \oc A
</math>
|}

* Commutation of quantifiers (<math>z</math> does not occur in <math>A</math>):
{|
|-
|align="right"| &emsp; <math>
\exists x. \exists y. A </math>
| <math>\equiv \exists y. \exists x. A </math>
|align="right"| &emsp; <math>
\exists x.(A \plus B) </math>
| <math>\equiv \exists x.A \plus \exists x.B </math>
|align="right"| &emsp; <math>
\exists z.(A\tens B) </math>
| <math>\equiv A\tens\exists z.B
</math>
|align="right"| &emsp; <math>
\exists z.A </math>
| <math>\equiv A
</math>
|}

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:

{|
|-
|align="right"| &emsp; <math>
\zero </math>
| <math>\equiv \forall X.X </math>
|align="right"| &emsp; <math>
\one </math>
| <math>\equiv \forall X.(X \limp X) </math>
|align="right"| &emsp; <math>
A \plus B </math>
| <math>\equiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X)
</math>
|}

=== Additional equivalences ===

{|
|-
|align="right"| &emsp; <math>
\oc\wn\oc\wn A </math>
| <math>\equiv \oc\wn A
</math>
|align="right"| &emsp; <math>
\oc\wn \one</math>
| <math>\equiv \one
</math>
|}

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

=== Positive/negative commutation ===

<math>\exists\forall\limp\forall\exists</math>,
<math>A\tens(B\parr C)\limp(A\tens B)\parr C</math>

== Properties of proofs ==

The fundamental property of the sequent calculus of linear logic is the cut
elimination property, which states that the cut rule is useless as far as
provability is concerned.
This property is exposed in the following section, together with a sketch of
proof.

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\vdash\Gamma</math>, there is a proof of <math>\vdash\Gamma</math> if and
only if there is a proof of <math>\vdash\Gamma</math> that does not use the cut rule.
}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas
of its immediate subformulas:

* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,

* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,

* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas <math>\alpha</math> and <math>\alpha\orth</math> have no immediate subformula,

* the immediate subformulas of <math>\exists X.A</math> and <math>\forall X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math>.
}}

{{Theorem|title=subformula property|
A sequent <math>\vdash\Gamma</math> is provable if and only if it is the conclusion of
a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math>.
}}
{{Proof|
By the cut elimination theorem, if a sequent <math>\vdash</math> is provable, then it
is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.
}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its negation
<math>A\orth</math>; it is impossible to prove <math>\zero</math> or <math>\bot</math>.
}}
{{Proof|
If <math>\vdash\Gamma</math> is a provable sequent, then it is the conclusion of a
cut-free proof.
In each rule except the cut rule, there is at least one formula in
conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.

The other properties are immediate consequences: if <math>A</math> and <math>A\orth</math> were
provable, then by a cut rule one would get an empty conclusion, which is not
possible.
As particular cases, since <math>\one</math> and <math>\top</math> are provable, their negations
<math>\bot</math> and <math>\zero</math> are not.
}}

=== Expansion of identities ===

Let us write <math>\pi\vdash\Gamma</math> to signify that <math>\pi</math> is a proof with
conclusion <math>\vdash\Gamma</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the same conclusion as
<math>\pi</math> in which the axiom rule is only used with atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.
}}
{{Proof|
It suffices to prove that for every formula <math>A</math>, the sequent
<math>\vdash A\orth,A</math> has a cut-free proof in which the axiom rule is used only
for atomic formulas.
We prove this by induction on <math>A</math>.
Not that there is a case for each pair of dual connectives.

* If <math>A</math> is atomic, then <math>\vdash A\orth,A</math> is an instance of the atomic axiom rule.

* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \tens }
\BinRule{ \vdash A_1\orth, A_2\orth, A_1 \tens A_2 }
\LabelRule{ \parr }
\UnaRule{ \vdash A_1\orth \parr A_2\orth, A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\one</math> or <math>A=\bot</math> then we have <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\LabelRule{ \bot }
\UnaRule{ \vdash \one, \bot }
\DisplayProof
</math>

* If <math>A=A_1\plus A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash A_1\orth, A_1 \plus A_2 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash A_2\orth, A_1 \plus A_2 }
\LabelRule{ \with }
\BinRule{ \vdash A_1\orth \with A_2\orth, A_1 \plus A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\zero</math> or <math>A=\top</math>, we have <math>
\LabelRule{ \top }
\NulRule{ \vdash \top, \zero }
\DisplayProof
</math>

* If <math>A=\oc B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ d }
\UnaRule{ \pi \vdash \wn B\orth, B }
\LabelRule{ \oc }
\UnaRule{ \pi \vdash \wn B\orth, \oc B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* If <math>A=\exists X.B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ \exists }
\UnaRule{ \vdash B\orth, \exists X.B }
\LabelRule{ \forall }
\UnaRule{ \vdash \forall X.B\orth, \exists X.B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if

* for every proof <math>\pi\vdash\Gamma,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,

* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.

}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are
reversible.
}}
{{Proof|
Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only
applied to atomic formulas.
Then each top-level connective is introduced either by its associated rule
or in an instance of the <math>\top</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\vdash\Gamma,A\parr B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr</math> rule, then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction).
If it is introduced in the contect of a <math>\top</math> rule, then this rule can be
changed so that <math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a <math>\bot</math>
rule, then remove this rule to get a proof of <math>\vdash\Gamma</math>, if it is
introduced by a <math>\top</math> rule, remove the <math>\bot</math> from this rule, then apply
the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof <math>\pi\vdash\Gamma,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is applied
in a context like

<math>
\AxRule{ \pi_1 \vdash \Delta, A }
\AxRule{ \pi_2 \vdash \Delta, B }
\LabelRule{ \with }
\BinRule{ \vdash \Delta, A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except as
context), if we replace this step by <math>\pi_1</math> in <math>\pi</math> we finally get a proof
<math>\pi'_1\vdash\Gamma,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof <math>\pi'_2\vdash\Gamma,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final <math>\with</math> rule we finally get the
expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math> rule is solved as
before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as an instance of the
<math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math>, consider a proof <math>\pi\vdash\Gamma,\forall X.A</math>.
Up to renaming, we can assume that <math>X</math> occurs free only above the rule that
introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we remove this
rule, we can check that we get a proof of <math>\vdash\Gamma,A</math> on which we can
finally apply the <math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math> rule is solved as
before.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the original
proof.
}}

== Variations ==

=== Two-sided sequent calculus ===

The sequent calculus of linear logic can also be presented using two-sided
sequents <math>\Gamma\vdash\Delta</math>, with any number of formulas on the left and
right.
In this case, it is customary to provide rules only for the positive
connectives, then there are left and right introduction rules and a negation
rule that moves formulas between the left and right sides:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\DisplayProof
</math>

Identity group:

<math>
\LabelRule{axiom}
\NulRule{ A \vdash A }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{cut}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>

Multiplicative group:

<math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
\qquad
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>

Additive group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>

Exponential group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>

Sequent calculus

2009-01-17T14:25:16Z

Emmanuel Beffara: markup fixes

This article presents the language and sequent calculus of second-order
propositional linear logic and the basic properties of this sequent calculus.

== Formulas ==

Formulas are built on a set of atoms, written <math>\alpha,\beta,\ldots</math>, that can
be either propositional variables <math>X,Y,Z\ldots</math> or atomic formulas
<math>X(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms from some first-order language.
Formulas, represented by capital letters <math>A</math>, <math>B</math>, <math>C</math>, are built using the
following connectives:

<math>
\begin{array}{clcll}
\alpha & \text{atom} &
\alpha\orth & \text{negated atom} & \text{atoms} \\
A \tens B & \text{tensor} &
A \parr B & \text{par} & \text{multiplicatives} \\
\one & \text{one} &
\bot & \text{bottom} & \text{multiplicative units} \\
A \plus B & \text{plus} &
A \with B & \text{with} & \text{additives} \\
\zero & \text{zero} &
\top & \text{top} & \text{additive units} \\
\oc A & \text{of course} &
\wn A & \text{why not} & \text{exponentials} \\
\exists x.A & \text{there exists} &
\forall x.A & \text{for all} & \text{quantifiers}
\end{array}
</math>

Each line corresponds to a particular class of connectives, and each class
consists in a pair of connectives.
Those in the left column are called positive and those in the right column are
called negative.
The ''tensor'' and ''with'' are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally
read ''of course <math>A</math>'' for <math>\oc A</math> and ''why not <math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

Given a formula <math>A</math>, its linear negation, also called ''orthogonal'' and
written <math>A\orth</math>, is obtained by exchanging each positive connective with the
negative one of the same class and vice versa, in a way analogous to de Morgan
laws in classical logic.
Formally, the definition of linear negation is

{|
|-
|align="right"| &emsp; <math>
( \alpha )\orth </math>
| <math>:= \alpha\orth </math>
|align="right"| &emsp; <math>
( \alpha\orth )\orth </math>
| <math>:= \alpha </math>
|-
|align="right"| &emsp; <math>
( A \tens B )\orth </math>
| <math>:= A\orth \parr B\orth </math>
|align="right"| &emsp; <math>
( A \parr B )\orth </math>
| <math>:= A\orth \tens B\orth </math>
|-
|align="right"| &emsp; <math>
\one\orth </math>
| <math>:= \bot </math>
|align="right"| &emsp; <math>
\bot\orth </math>
| <math>:= \one </math>
|-
|align="right"| &emsp; <math>
( A \plus B )\orth </math>
| <math>:= A\orth \with B\orth </math>
|align="right"| &emsp; <math>
( A \with B )\orth </math>
| <math>:= A\orth \plus B\orth </math>
|-
|align="right"| &emsp; <math>
\zero\orth </math>
| <math>:= \top </math>
|align="right"| &emsp; <math>
\top\orth </math>
| <math>:= \zero </math>
|-
|align="right"| &emsp; <math>
( \oc A )\orth </math>
| <math>:= \wn ( A\orth ) </math>
|align="right"| &emsp; <math>
( \wn A )\orth </math>
| <math>:= \oc ( A\orth ) </math>
|-
|align="right"| &emsp; <math>
( \exists X.A )\orth </math>
| <math>:= \forall X.( A\orth ) </math>
|align="right"| &emsp; <math>
( \forall X.A )\orth </math>
| <math>:= \exists X.( A\orth )
</math>
|}

Note that this operation is defined syntactically, hence negation is not a
connective, the only place in formulas where the symbol <math>(\cdot)\orth</math> occurs
is for negated atoms <math>\alpha\orth</math>.
Note also that, by construction, negation is involutive: for any formula <math>A</math>,
it holds that <math>A\biorth=A</math>.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic:

<math>
A \limp B := A\orth \parr B
</math>

Free and bound variables are defined in the standard way, as well as
substitution.
Formulas are always considered up to renaming of bound names.
If <math>A</math> and <math>B</math> are formulas and <math>X</math> is a propositional variable, the formula
<math>A[B/X]</math> is <math>A</math> where all atoms <math>X</math> are replaced (without capture) by <math>B</math> and
all atoms <math>X\orth</math> are replaced by the formula <math>B\orth</math>.

== Sequents and proofs ==

A sequent is an expression <math>\vdash\Gamma</math> where <math>\Gamma</math> is a finite multiset
of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation <math>\wn\Gamma</math> represents
the multiset <math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:

* Identity group: <math>
\LabelRule{axiom}
\NulRule{ \vdash A, A\orth }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{cut}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>

* Multiplicative group: <math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
\qquad
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>

* Additive group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
\qquad
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>

* Exponential group: <math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>

* Quantifier group (in the <math>\forall</math> rule, <math>X</math> must not occur in <math>\Gamma</math>): <math>
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall X.A }
\DisplayProof
</math>

The rules for exponentials are called ''dereliction'', ''weakening'',
''contraction'' and ''promotion'', respectively.
Note the fundamental fact that there are no contraction and weakening rules
for aribtrary formulas, but only for the formulas starting with the <math>\wn</math>
modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Note that this system contains only introduction rules and no elimination
rule.
Moreover, there is no introduction rule for the additive unit <math>\zero</math>, the
only ways to introduce it at top level are the axiom rule and the <math>\top</math> rule.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An equivalent presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rule:

<math>
\AxRule{ \vdash \Gamma, A, B, \Delta }
\LabelRule{exchange}
\UnaRule{ \vdash \Gamma, B, A, \Delta }
\DisplayProof
</math>

== Equivalences and definability ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent, written <math>A\equiv B</math>, if
both implications <math>A\limp B</math> and <math>B\limp A</math> are provable.
Equivalently, <math>A\equiv B</math> if both <math>\vdash A\orth,B</math> and <math>\vdash B\orth,A</math>
are provable.
Another formulation of <math>A\equiv B</math> is that, for all <math>\Gamma</math>,
<math>\vdash\Gamma,A</math> is provable if and only if <math>\vdash\Gamma,B</math> is provable.
Note that, because of the definition of negation, an equivalence
<math>A\equiv B</math> holds if and only if the dual equivalence
<math>A\orth\equiv B\orth</math> holds.

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality:
{|
|-
|align="right"| &emsp; <math>
A \tens (B \tens C) </math>
| <math>\equiv (A \tens B) \tens C </math>
|align="right"| &emsp; <math>
A \tens B </math>
| <math>\equiv B \tens A </math>
|align="right"| &emsp; <math>
A \tens \one </math>
| <math>\equiv A </math>
|-
|align="right"| &emsp; <math>
A \plus (B \plus C) </math>
| <math>\equiv (A \plus B) \plus C </math>
|align="right"| &emsp; <math>
A \plus B </math>
| <math>\equiv B \plus A </math>
|align="right"| &emsp; <math>
A \plus \zero </math>
| <math>\equiv A
</math>
|}

* Idempotence of additives:
{|
|-
|align="right"| &emsp; <math>
A \plus A </math>
| <math>\equiv A
</math>
|}

* Distributivity of multiplicatives over additives:
{|
|-
|align="right"| &emsp; <math>
A \tens (B \plus C) </math>
| <math>\equiv (A \tens B) \plus (A \tens C) </math>
|align="right"| &emsp; <math>
A \tens \zero </math>
| <math>\equiv \zero
</math>
|}

* Defining property of exponentials:
{|
|-
|align="right"| &emsp; <math>
\oc(A \with B) </math>
| <math>\equiv \oc A \tens \oc B </math>
|align="right"| &emsp; <math>
\oc\top </math>
| <math>\equiv \one
</math>
|}

* Monoidal structure of exponentials, digging:
{|
|-
|align="right"| &emsp; <math>
\oc A \otimes \oc A </math>
| <math>\equiv \oc A </math>
|align="right"| &emsp; <math>
\oc \one </math>
| <math>\equiv \one </math>
|align="right"| &emsp; <math>
\oc\oc A </math>
| <math>\equiv \oc A
</math>
|}

* Commutation of quantifiers (<math>z</math> does not occur in <math>A</math>):
{|
|-
|align="right"| &emsp; <math>
\exists x. \exists y. A </math>
| <math>\equiv \exists y. \exists x. A </math>
|align="right"| &emsp; <math>
\exists x.(A \plus B) </math>
| <math>\equiv \exists x.A \plus \exists x.B </math>
|align="right"| &emsp; <math>
\exists z.(A\tens B) </math>
| <math>\equiv A\tens\exists z.B
</math>
|align="right"| &emsp; <math>
\exists z.A </math>
| <math>\equiv A
</math>
|}

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:

{|
|-
|align="right"| &emsp; <math>
\zero </math>
| <math>\equiv \forall X.X </math>
|align="right"| &emsp; <math>
\one </math>
| <math>\equiv \forall X.(X \limp X) </math>
|align="right"| &emsp; <math>
A \plus B </math>
| <math>\equiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X)
</math>
|}

=== Additional equivalences ===

{|
|-
|align="right"| &emsp; <math>
\oc\wn\oc\wn A </math>
| <math>\equiv \oc\wn A
</math>
|align="right"| &emsp; <math>
\oc\wn \one</math>
| <math>\equiv \one
</math>
|}

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

=== Positive/negative commutation ===

<math>\exists\forall\limp\forall\exists</math>,
<math>A\tens(B\parr C)\limp(A\tens B)\parr C</math>

== Properties of proofs ==

The fundamental property of the sequent calculus of linear logic is the cut
elimination property, which states that the cut rule is useless as far as
provability is concerned.
This property is exposed in the following section, together with a sketch of
proof.

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\vdash\Gamma</math>, there is a proof of <math>\vdash\Gamma</math> if and
only if there is a proof of <math>\vdash\Gamma</math> that does not use the cut rule.
}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:

{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas
of its immediate subformulas:

* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>, <math>A\with B</math> are <math>A</math> and <math>B</math>,

* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,

* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas <math>\alpha</math> and <math>\alpha\orth</math> have no immediate subformula,

* the immediate subformulas of <math>\exists X.A</math> and <math>\forall X.A</math> are all the <math>A[B/X]</math> for all formulas <math>B</math>.
}}

{{Theorem|title=subformula property|
A sequent <math>\vdash\Gamma</math> is provable if and only if it is the conclusion of
a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math>.
}}
{{Proof|
By the cut elimination theorem, if a sequent <math>\vdash</math> is provable, then it
is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.
}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its negation
<math>A\orth</math>; it is impossible to prove <math>\zero</math> or <math>\bot</math>.
}}
{{Proof|
If <math>\vdash\Gamma</math> is a provable sequent, then it is the conclusion of a
cut-free proof.
In each rule except the cut rule, there is at least one formula in
conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.

The other properties are immediate consequences: if <math>A</math> and <math>A\orth</math> were
provable, then by a cut rule one would get an empty conclusion, which is not
possible.
As particular cases, since <math>\one</math> and <math>\top</math> are provable, their negations
<math>\bot</math> and <math>\zero</math> are not.
}}

=== Expansion of identities ===

Let us write <math>\pi\vdash\Gamma</math> to signify that <math>\pi</math> is a proof with
conclusion <math>\vdash\Gamma</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the same conclusion as
<math>\pi</math> in which the axiom rule is only used with atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.
}}
{{Proof|
It suffices to prove that for every formula <math>A</math>, the sequent
<math>\vdash A\orth,A</math> has a cut-free proof in which the axiom rule is used only
for atomic formulas.
We prove this by induction on <math>A</math>.
Not that there is a case for each pair of dual connectives.

* If <math>A</math> is atomic, then <math>\vdash A\orth,A</math> is an instance of the atomic axiom rule.

* If <math>A=A_1\tens A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \tens }
\BinRule{ \vdash A_1\orth, A_2\orth, A_1 \tens A_2 }
\LabelRule{ \parr }
\UnaRule{ \vdash A_1\orth \parr A_2\orth, A_1 \tens A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\one</math> or <math>A=\bot</math> then we have <math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\LabelRule{ \bot }
\UnaRule{ \vdash \one, \bot }
\DisplayProof
</math>

* If <math>A=A_1\plus A_2</math> then we have <math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash A_1\orth, A_1 \plus A_2 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash A_2\orth, A_1 \plus A_2 }
\LabelRule{ \with }
\BinRule{ \vdash A_1\orth \with A_2\orth, A_1 \plus A_2 }
\DisplayProof
</math> where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\zero</math> or <math>A=\top</math>, we have <math>
\LabelRule{ \top }
\NulRule{ \vdash \top, \zero }
\DisplayProof
</math>

* If <math>A=\oc B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ d }
\UnaRule{ \pi \vdash \wn B\orth, B }
\LabelRule{ \oc }
\UnaRule{ \pi \vdash \wn B\orth, \oc B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

* If <math>A=\exists X.B</math> then we have <math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ \exists }
\UnaRule{ \vdash B\orth, \exists X.B }
\LabelRule{ \forall }
\UnaRule{ \vdash \forall X.B\orth, \exists X.B }
\DisplayProof
</math> where <math>\pi</math> exists by induction hypothesis.

}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if

* for every proof <math>\pi\vdash\Gamma,c(A_1,\ldots,A_n)</math>, there is a proof <math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced by the last rule,

* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.

}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are
reversible.
}}
{{Proof|
Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only
applied to atomic formulas.
Then each top-level connective is introduced either by its associated rule
or in an instance of the <math>\top</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\vdash\Gamma,A\parr B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr</math> rule, then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction).
If it is introduced in the contect of a <math>\top</math> rule, then this rule can be
changed so that <math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a <math>\bot</math>
rule, then remove this rule to get a proof of <math>\vdash\Gamma</math>, if it is
introduced by a <math>\top</math> rule, remove the <math>\bot</math> from this rule, then apply
the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof <math>\pi\vdash\Gamma,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is applied
in a context like

<math>
\AxRule{ \pi_1 \vdash \Delta, A }
\AxRule{ \pi_2 \vdash \Delta, B }
\LabelRule{ \with }
\BinRule{ \vdash \Delta, A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except as
context), if we replace this step by <math>\pi_1</math> in <math>\pi</math> we finally get a proof
<math>\pi'_1\vdash\Gamma,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof <math>\pi'_2\vdash\Gamma,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final <math>\with</math> rule we finally get the
expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math> rule is solved as
before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as an instance of the
<math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math>, consider a proof <math>\pi\vdash\Gamma,\forall X.A</math>.
Up to renaming, we can assume that <math>X</math> occurs free only above the rule that
introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we remove this
rule, we can check that we get a proof of <math>\vdash\Gamma,A</math> on which we can
finally apply the <math>\forall</math> rule.
The case when the <math>\forall</math> was introduced in a <math>\top</math> rule is solved as
before.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the original
proof.
}}

== Variations ==

=== Two-sided sequent calculus ===

The sequent calculus of linear logic can also be presented using two-sided
sequents <math>\Gamma\vdash\Delta</math>, with any number of formulas on the left and
right.
In this case, it is customary to provide rules only for the positive
connectives, then there are left and right introduction rules and a negation
rule that moves formulas between the left and right sides:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\DisplayProof
</math>

Identity group:

<math>
\LabelRule{axiom}
\NulRule{ A \vdash A }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{cut}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>

Multiplicative group:

<math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
\qquad
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>

Additive group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>

Exponential group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>

Sequent calculus

2009-01-14T11:36:55Z

Emmanuel Beffara: New page: This article presents the language and sequent calculus of second-order propositional linear logic and the basic properties of this sequent calculus. == Formulas == Formulas are built ...

This article presents the language and sequent calculus of second-order
propositional linear logic and the basic properties of this sequent calculus.

== Formulas ==

Formulas are built on a set of atoms, written <math>\alpha,\beta,\ldots</math>, that can
be either propositional variables <math>X,Y,Z\ldots</math> or atomic formulas
<math>X(t_1,\ldots,t_n)</math>, where the <math>t_i</math> are terms from some first-order language.
Formulas, represented by capital letters <math>A</math>, <math>B</math>, <math>C</math>, are built using the
following connectives:

<math>
\begin{array}{clcll}
\alpha & \text{atom} &
\alpha\orth & \text{negated atom} & \text{atoms} \\
A \tens B & \text{tensor} &
A \parr B & \text{par} & \text{multiplicatives} \\
\one & \text{one} &
\bot & \text{bottom} & \text{multiplicative units} \\
A \plus B & \text{plus} &
A \with B & \text{with} & \text{additives} \\
\zero & \text{zero} &
\top & \text{top} & \text{additive units} \\
\oc A & \text{of course} &
\wn A & \text{why not} & \text{exponentials} \\
\exists x.A & \text{there exists} &
\forall x.A & \text{for all} & \text{quantifiers}
\end{array}
</math>

Each line corresponds to a particular class of connectives, and each class
consists in a pair of connectives.
Those in the left column are called positive and those in the right column are
called negative.
The ''tensor'' and ''with'' are conjunctions while ''par'' and
''plus'' are disjunctions.
The exponential connectives are called ''modalities'', and traditionally
read ''of course <math>A</math>'' for <math>\oc A</math> and ''why not <math>A</math>'' for <math>\wn A</math>.
Quantifiers may apply to first- or second-order variables.

Given a formula <math>A</math>, its linear negation, also called ''orthogonal'' and
written <math>A\orth</math>, is obtained by exchanging each positive connective with the
negative one of the same class and vice versa, in a way analogous to de Morgan
laws in classical logic.
Formally, the definition of linear negation is

{|
|-
|align="right"| &emsp; <math>
( \alpha )\orth </math>
| <math>:= \alpha\orth </math>
|align="right"| &emsp; <math>
( \alpha\orth )\orth </math>
| <math>:= \alpha </math>
|-
|align="right"| &emsp; <math>
( A \tens B )\orth </math>
| <math>:= A\orth \parr B\orth </math>
|align="right"| &emsp; <math>
( A \parr B )\orth </math>
| <math>:= A\orth \tens B\orth </math>
|-
|align="right"| &emsp; <math>
\one\orth </math>
| <math>:= \bot </math>
|align="right"| &emsp; <math>
\bot\orth </math>
| <math>:= \one </math>
|-
|align="right"| &emsp; <math>
( A \plus B )\orth </math>
| <math>:= A\orth \with B\orth </math>
|align="right"| &emsp; <math>
( A \with B )\orth </math>
| <math>:= A\orth \plus B\orth </math>
|-
|align="right"| &emsp; <math>
\zero\orth </math>
| <math>:= \top </math>
|align="right"| &emsp; <math>
\top\orth </math>
| <math>:= \zero </math>
|-
|align="right"| &emsp; <math>
( \oc A )\orth </math>
| <math>:= \wn ( A\orth ) </math>
|align="right"| &emsp; <math>
( \wn A )\orth </math>
| <math>:= \oc ( A\orth ) </math>
|-
|align="right"| &emsp; <math>
( \exists X.A )\orth </math>
| <math>:= \forall X.( A\orth ) </math>
|align="right"| &emsp; <math>
( \forall X.A )\orth </math>
| <math>:= \exists X.( A\orth )
</math>
|}

Note that this operation is defined syntactically, hence negation is not a
connective, the only place in formulas where the symbol <math>(\cdot)\orth</math> occurs
is for negated atoms <math>\alpha\orth</math>.
Note also that, by construction, negation is involutive: for any formula <math>A</math>,
it holds that <math>A\biorth=A</math>.

There is no connective for implication in the syntax of standard linear logic.
Instead, a ''linear implication'' is defined similarly to the decomposition
<math>A\imp B=\neg A\vee B</math> in classical logic:

<math>
A \limp B := A\orth \parr B
</math>

Free and bound variables are defined in the standard way, as well as
substitution.
Formulas are always considered up to renaming of bound names.
If <math>A</math> and <math>B</math> are formulas and <math>X</math> is a propositional variable, the formula
<math>A[B/X]</math> is <math>A</math> where all atoms <math>X</math> are replaced (without capture) by <math>B</math> and
all atoms <math>X\orth</math> are replaced by the formula <math>B\orth</math>.

== Sequents and proofs ==

A sequent is an expression <math>\vdash\Gamma</math> where <math>\Gamma</math> is a finite multiset
of formulas.
For a multiset <math>\Gamma=A_1,\ldots,A_n</math>, the notation <math>\wn\Gamma</math> represents
the multiset <math>\wn A_1,\ldots,\wn A_n</math>.
Proofs are labelled trees of sequents, built using the following inference
rules:

* Identity group:

<math>
\LabelRule{axiom}
\NulRule{ \vdash A, A\orth }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, A\orth }
\LabelRule{cut}
\BinRule{ \vdash \Gamma, \Delta }
\DisplayProof
</math>

* Multiplicative group:

<math>
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Delta, B }
\LabelRule{ \tens }
\BinRule{ \vdash \Gamma, \Delta, A \tens B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A, B }
\LabelRule{ \parr }
\UnaRule{ \vdash \Gamma, A \parr B }
\DisplayProof
\qquad
\LabelRule{ \one }
\NulRule{ \vdash \one }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ \bot }
\UnaRule{ \vdash \Gamma, \bot }
\DisplayProof
</math>

* Additive group:

<math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash \Gamma, A \plus B }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\AxRule{ \vdash \Gamma, B }
\LabelRule{ \with }
\BinRule{ \vdash, \Gamma, A \with B }
\DisplayProof
\qquad
\LabelRule{ \top }
\NulRule{ \vdash \Gamma, \top }
\DisplayProof
</math>

* Exponential group:

<math>
\AxRule{ \vdash \Gamma, A }
\LabelRule{ d }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma }
\LabelRule{ w }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, \wn A, \wn A }
\LabelRule{ c }
\UnaRule{ \vdash \Gamma, \wn A }
\DisplayProof
\qquad
\AxRule{ \vdash \wn\Gamma, B }
\LabelRule{ \oc }
\UnaRule{ \vdash \wn\Gamma, \oc B }
\DisplayProof
</math>

* Quantifier group (in the <math>\forall</math> rule, <math>X</math> must not occur in <math>\Gamma</math>):

<math>
\AxRule{ \vdash \Gamma, A[B/X] }
\LabelRule{ \exists }
\UnaRule{ \vdash \Gamma, \exists X.A }
\DisplayProof
\qquad
\AxRule{ \vdash \Gamma, A }
\LabelRule{ \forall }
\UnaRule{ \vdash \Gamma, \forall X.A }
\DisplayProof
</math>

The rules for exponentials are called ''dereliction'', ''weakening'',
''contraction'' and ''promotion'', respectively.
Note the fundamental fact that there are no contraction and weakening rules
for aribtrary formulas, but only for the formulas starting with the <math>\wn</math>
modality.
This is what distinguishes linear logic from classical logic: if weakening and
contraction were allowed for arbitrary formulas, then <math>\tens</math> and <math>\with</math>
would be identified, as well as <math>\plus</math> and <math>\parr</math>, <math>\one</math> and <math>\top</math>,
<math>\zero</math> and <math>\bot</math>.
By ''identified'', we mean here that replacing a <math>\tens</math> with a <math>\with</math> or
vice versa would preserve provability.

Note that this system contains only introduction rules and no elimination
rule.
Moreover, there is no introduction rule for the additive unit <math>\zero</math>, the
only ways to introduce it at top level are the axiom rule and the <math>\top</math> rule.

Sequents are considered as multisets, in other words as sequences up to
permutation.
An equivalent presentation would be to define a sequent as a finite sequence
of formulas and to add the exchange rule:

<math>
\AxRule{ \vdash \Gamma, A, B, \Delta }
\LabelRule{exchange}
\UnaRule{ \vdash \Gamma, B, A, \Delta }
\DisplayProof
</math>

== Equivalences and definability ==

Two formulas <math>A</math> and <math>B</math> are (linearly) equivalent, written <math>A\equiv B</math>, if
both implications <math>A\limp B</math> and <math>B\limp A</math> are provable.
Equivalently, <math>A\equiv B</math> if both <math>\vdash A\orth,B</math> and <math>\vdash B\orth,A</math>
are provable.
Another formulation of <math>A\equiv B</math> is that, for all <math>\Gamma</math>,
<math>\vdash\Gamma,A</math> is provable if and only if <math>\vdash\Gamma,B</math> is provable.
Note that, beacause of the definition of negation, an equivalence
<math>A\equiv B</math> holds if and only if the dual equivalence
<math>A\orth\equiv B\orth</math> holds.

=== Fundamental equivalences ===

* Associativity, commutativity, neutrality:

{|
|-
|align="right"| &emsp; <math>
A \tens (B \tens C) </math>
| <math>\equiv (A \tens B) \tens C </math>
|align="right"| &emsp; <math>
A \tens B </math>
| <math>\equiv B \tens A </math>
|align="right"| &emsp; <math>
A \tens \one </math>
| <math>\equiv A </math>
|-
|align="right"| &emsp; <math>
A \plus (B \plus C) </math>
| <math>\equiv (A \plus B) \plus C </math>
|align="right"| &emsp; <math>
A \plus B </math>
| <math>\equiv B \plus A </math>
|align="right"| &emsp; <math>
A \plus \zero </math>
| <math>\equiv A
</math>
|}

* Idempotence of additives:

{|
|-
|align="right"| &emsp; <math>
A \plus A </math>
| <math>\equiv A
</math>
|}

* Distributivity of multiplicatives over additives:

{|
|-
|align="right"| &emsp; <math>
A \tens (B \plus C) </math>
| <math>\equiv (A \tens B) \plus (A \tens C) </math>
|align="right"| &emsp; <math>
A \tens \zero </math>
| <math>\equiv \zero
</math>
|}

* Defining property of exponentials:

{|
|-
|align="right"| &emsp; <math>
\oc(A \with B) </math>
| <math>\equiv \oc A \tens \oc B </math>
|align="right"| &emsp; <math>
\oc\top </math>
| <math>\equiv \one
</math>
|}

* Monoidal structure of exponentials, digging:

{|
|-
|align="right"| &emsp; <math>
\oc A \otimes \oc A </math>
| <math>\equiv \oc A </math>
|align="right"| &emsp; <math>
\oc \one </math>
| <math>\equiv \one </math>
|align="right"| &emsp; <math>
\oc\oc A </math>
| <math>\equiv \oc A
</math>
|}

* Commutation of quantifiers (<math>z</math> does not occur in <math>A</math>):

{|
|-
|align="right"| &emsp; <math>
\exists x. \exists y. A </math>
| <math>\equiv \exists y. \exists x. A </math>
|align="right"| &emsp; <math>
\exists x.(A \plus B) </math>
| <math>\equiv \exists x.A \plus \exists x.B </math>
|align="right"| &emsp; <math>
\exists z.A </math>
| <math>\equiv A
</math>
|}

=== Definability ===

The units and the additive connectives can be defined using second-order
quantification and exponentials, indeed the following equivalences hold:

{|
|-
|align="right"| &emsp; <math>
\zero </math>
| <math>\equiv \forall X.X </math>
|align="right"| &emsp; <math>
\one </math>
| <math>\equiv \forall X.(X \limp X) </math>
|align="right"| &emsp; <math>
A \plus B </math>
| <math>\equiv \forall X.(\oc(A \limp X) \limp \oc(B \limp X) \limp X)
</math>
|}

Any pair of connectives that has the same rules as <math>\tens/\parr</math> is
equivalent to it, the same holds for additives, but not for exponentials.

=== Positive/negative commutation ===

<math>\exists\forall\limp\forall\exists</math>,
<math>A\tens(B\parr C)\limp(A\tens B)\parr C</math>

== Properties of proofs ==

The fundamental property of the sequent calculus of linear logic is the cut
elimination property, which states that the cut rule is useless as far as
provability is concerned.
This property is exposed in the following section, together with a sketch of
proof.

=== Cut elimination and consequences ===

{{Theorem|title=cut elimination|
For every sequent <math>\vdash\Gamma</math>, there is a proof of <math>\vdash\Gamma</math> if and
only if there is a proof of <math>\vdash\Gamma</math> that does not use the cut rule.
}}

This property is proved using a set of rewriting rules on proofs, using
appropriate termination arguments (see the specific articles on
[[cut elimination]] for detailed proofs), it is the core of the proof/program
correspondence.

It has several important consequences:
{{Definition|title=subformula|
The subformulas of a formula <math>A</math> are <math>A</math> and, inductively, the subformulas
of its immediate subformulas:

* the immediate subformulas of <math>A\tens B</math>, <math>A\parr B</math>, <math>A\plus B</math>,
<math>A\with B</math> are <math>A</math> and <math>B</math>,

* the only immediate subformula of <math>\oc A</math> and <math>\wn A</math> is <math>A</math>,

* <math>\one</math>, <math>\bot</math>, <math>\zero</math>, <math>\top</math> and atomic formulas <math>\alpha</math> and
<math>\alpha\orth</math> have no immediate subformula,

* the immediate subformulas of <math>\exists X.A</math> and <math>\exists X.A</math> are all
the <math>A[B/X]</math> for all formulas <math>B</math>.

}}
{{Theorem|title=subformula property|
A sequent <math>\vdash\Gamma</math> is provable if and only if it is the conclusion of
a proof in which each intermediate conclusion is made of subformulas of the
formulas of <math>\Gamma</math>.
}}
{{Proof|
By the cut elimination theorem, if a sequent <math>\vdash</math> is provable, then it
is provable by a cut-free proof.
In each rule except the cut rule, all formulas of the premisses are either
formulas of the conclusion, or immediate subformulas of it, therefore
cut-free proofs have the subformula property.
}}

The subformula property means essentially nothing in the second-order system,
since any formula is a subformula of a quantified formula where the quantified
variable occurs.
However, the property is very meaningful if the sequent <math>\Gamma</math> does not use
second-order quantification, as it puts a strong restriction on the set of
potential proofs of a given sequent.
In particular, it implies that the first-order fragment without quantifiers is
decidable.

{{Theorem|title=consistency|
The empty sequent <math>\vdash</math> is not provable.
Subsequently, it is impossible to prove both a formula <math>A</math> and its negation
<math>A\orth</math>; it is impossible to prove <math>\zero</math> or <math>\bot</math>.
}}
{{Proof|
If <math>\vdash\Gamma</math> is a provable sequent, then it is the conclusion of a
cut-free proof.
In each rule except the cut rule, there is at least one formula in
conclusion.
Therefore <math>\vdash</math> cannot be the conclusion of a proof.

The other properties are immediate consequences: if <math>A</math> and <math>A\orth</math> were
provable, then by a cut rule one would get an empty conclusion, which is not
possible.
As particular cases, since <math>\one</math> and <math>\top</math> are provable, their negations
<math>\bot</math> and <math>\zero</math> are not.
}}

=== Expansion of identities ===

Let us write <math>\pi\vdash\Gamma</math> to signify that <math>\pi</math> is a proof with
conclusion <math>\vdash\Gamma</math>.

{{Proposition|title=<math>\eta</math>-expansion|
For every proof <math>\pi</math>, there is a proof <math>\pi'</math> with the same conclusion as
<math>\pi</math> in which the axiom rule is only used with atomic formulas.
If <math>\pi</math> is cut-free, then there is a cut-free <math>\pi'</math>.
}}
{{Proof|
It suffices to prove that for every formula <math>A</math>, the sequent
<math>\vdash A\orth,A</math> has a cut-free proof in which the axiom rule is used only
for atomic formulas.
We prove this by induction on <math>A</math>.
Not that there is a case for each pair of dual connectives.

* If <math>A</math> is atomic, then <math>\vdash A\orth,A</math> is an instance of the atomic
axiom rule.

* If <math>A=A_1\tens A_2</math> then we have

<math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \tens }
\BinRule{ \vdash A_1\orth, A_2\orth, A_1 \tens A_2 }
\LabelRule{ \parr }
\UnaRule{ \vdash A_1\orth \parr A_2\orth, A_1 \tens A_2 }
\DisplayProof
</math>

where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\one</math> or <math>A=\bot</math> then we have

<math>
\LabelRule{ \one }
\NulRule{ \vdash \one }
\LabelRule{ \bot }
\UnaRule{ \vdash \one, \bot }
\DisplayProof
</math>

* If <math>A=A_1\plus A_2</math> then we have

<math>
\AxRule{ \pi_1 \vdash A_1\orth, A_1 }
\LabelRule{ \plus_1 }
\UnaRule{ \vdash A_1\orth, A_1 \plus A_2 }
\AxRule{ \pi_2 \vdash A_2\orth, A_2 }
\LabelRule{ \plus_2 }
\UnaRule{ \vdash A_2\orth, A_1 \plus A_2 }
\LabelRule{ \with }
\BinRule{ \vdash A_1\orth \with A_2\orth, A_1 \plus A_2 }
\DisplayProof
</math>

where <math>\pi_1</math> and <math>\pi_2</math> exist by induction hypothesis.

* If <math>A=\zero</math> or <math>A=\top</math>, we have

<math>
\LabelRule{ \top }
\NulRule{ \vdash \top, \zero }
\DisplayProof
</math>

* If <math>A=\oc B</math> then we have

<math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ d }
\UnaRule{ \pi \vdash \wn B\orth, B }
\LabelRule{ \oc }
\UnaRule{ \pi \vdash \wn B\orth, \oc B }
\DisplayProof
</math>

where <math>\pi</math> exists by induction hypothesis.

* If <math>A=\exists X.B</math> then we have

<math>
\AxRule{ \pi \vdash B\orth, B }
\LabelRule{ \exists }
\UnaRule{ \vdash B\orth, \exists X.B }
\LabelRule{ \forall }
\UnaRule{ \vdash \forall X.B\orth, \exists X.B }
\DisplayProof
</math>

where <math>\pi</math> exists by induction hypothesis.

}}

The interesting thing with <math>\eta</math>-expansion is that, we can always assume that
each connective is explicitly introduced by its associated rule (except in the
case where there is an occurrence of the <math>\top</math> rule).

=== Reversibility ===

{{Definition|title=reversibility|
A connective <math>c</math> is called ''reversible'' if

* for every proof <math>\pi\vdash\Gamma,c(A_1,\ldots,A_n)</math>, there is a proof
<math>\pi'</math> with the same conclusion in which <math>c(A_1,\ldots,A_n)</math> is introduced
by the last rule,

* if <math>\pi</math> is cut-free then there is a cut-free <math>\pi'</math>.

}}

{{Proposition|
The connectives <math>\parr</math>, <math>\bot</math>, <math>\with</math>, <math>\top</math> and <math>\forall</math> are
reversible.
}}
{{Proof|
Using the <math>\eta</math>-expansion property, we assume that the axiom rule is only
applied to atomic formulas.
Then each top-level connective is introduced either by its associated rule
or in an instance of the <math>\top</math> rule.

For <math>\parr</math>, consider a proof <math>\pi\vdash\Gamma,A\parr B</math>.
If <math>A\parr B</math> is introduced by a <math>\parr</math> rule, then if we remove this rule
we get a proof of <math>\vdash\Gamma,A,B</math> (this can be proved by a
straightforward induction).
If it is introduced in the contect of a <math>\top</math> rule, then this rule can be
changed so that <math>A\parr B</math> is replaced by <math>A,B</math>.
In either case, we can apply a final <math>\parr</math> rule to get the expected proof.

For <math>\bot</math>, the same technique applies: if it is introduced by a <math>\bot</math>
rule, then remove this rule to get a proof of <math>\vdash\Gamma</math>, if it is
introduced by a <math>\top</math> rule, remove the <math>\bot</math> from this rule, then apply
the <math>\bot</math> rule at the end of the new proof.

For <math>\with</math>, consider a proof <math>\pi\vdash\Gamma,A\with B</math>.
If the connective is introduced by a <math>\with</math> rule then this rule is applied
in a context like

<math>
\AxRule{ \pi_1 \vdash \Delta, A }
\AxRule{ \pi_2 \vdash \Delta, B }
\LabelRule{ \with }
\BinRule{ \vdash \Delta, A \with B }
\DisplayProof
</math>

Since the formula <math>A\with B</math> is not involved in other rules (except as
context), if we replace this step by <math>\pi_1</math> in <math>\pi</math> we finally get a proof
<math>\pi'_1\vdash\Gamma,A</math>.
If we replace this step by <math>\pi_2</math> we get a proof <math>\pi'_2\vdash\Gamma,B</math>.
Combining <math>\pi_1</math> and <math>\pi_2</math> with a final <math>\with</math> rule we finally get the
expected proof.
The case when the <math>\with</math> was introduced in a <math>\top</math> rule is solved as
before.

For <math>\top</math> the result is trivial: just choose <math>\pi'</math> as an instance of the
<math>\top</math> rule with the appropriate conclusion.

For <math>\forall</math>, consider a proof <math>\pi\vdash\Gamma,\forall X.A</math>.
Up to renaming, we can assume that <math>X</math> occurs free only above the rule that
introduces the quantifier.
If the quantifier is introduced by a <math>\forall</math> rule, then if we remove this
rule, we can check that we get a proof of <math>\vdash\Gamma,A</math> on which we can
finally apply the <math>\forall</math> rule.
The case when the <math>\with</math> was introduced in a <math>\top</math> rule is solved as
before.

Note that, in each case, if the proof we start from is cut-free, our
transformations do not introduce a cut rule.
However, if the original proof has cuts, then the final proof may have more
cuts, since in the case of <math>\with</math> we duplicated a part of the original
proof.
}}

== Variations ==

=== Two-sided sequent calculus ===

The sequent calculus of linear logic can also be presented using two-sided
sequents <math>\Gamma\vdash\Delta</math>, with any number of formulas on the left and
right.
In this case, it is customary to provide rules only for the positive
connectives, then there are left and right introduction rules and a negation
rule that moves formulas between the left and right sides:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\UnaRule{ \Gamma \vdash A\orth, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\UnaRule{ \Gamma, A\orth \vdash \Delta }
\DisplayProof
</math>

Identity group:

<math>
\LabelRule{axiom}
\NulRule{ A \vdash A }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma', A \vdash \Delta' }
\LabelRule{cut}
\BinRule{ \Gamma, \Gamma' \vdash \Delta, \Delta' }
\DisplayProof
</math>

Multiplicative group:

<math>
\AxRule{ \Gamma, A, B \vdash \Delta }
\LabelRule{ \tens_L }
\UnaRule{ \Gamma, A \tens B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\AxRule{ \Gamma' \vdash B, \Delta' }
\LabelRule{ \tens_R }
\BinRule{ \Gamma, \Gamma' \vdash A \tens B, \Delta, \Delta' }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ \one_L }
\UnaRule{ \Gamma, \one \vdash \Delta }
\DisplayProof
\qquad
\LabelRule{ \one_R }
\NulRule{ \vdash \one }
\DisplayProof
</math>

Additive group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\AxRule{ \Gamma, B \vdash \Delta }
\LabelRule{ \plus_L }
\BinRule{ \Gamma, A \plus B \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash A, \Delta }
\LabelRule{ \plus_{R1} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash B, \Delta }
\LabelRule{ \plus_{R2} }
\UnaRule{ \Gamma \vdash A \plus B, \Delta }
\DisplayProof
\qquad
\LabelRule{ \zero_L }
\NulRule{ \Gamma, \zero \vdash \Delta }
\DisplayProof
</math>

Exponential group:

<math>
\AxRule{ \Gamma, A \vdash \Delta }
\LabelRule{ d }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma \vdash \Delta }
\LabelRule{ w }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \Gamma, \oc A, \oc A \vdash \Delta }
\LabelRule{ c }
\UnaRule{ \Gamma, \oc A \vdash \Delta }
\DisplayProof
\qquad
\AxRule{ \oc A_1, \ldots, \oc A_n \vdash B }
\LabelRule{ \oc_R }
\UnaRule{ \oc A_1, \ldots, \oc A_n \vdash \oc B }
\DisplayProof
</math>