Geometry of interaction - Revision history

Pierre-Marie Pédrot: ortho

2011-09-30T14:39:39Z

ortho

Pierre-Marie Pédrot: replaced paragraph with a link

2011-09-30T14:38:14Z

replaced paragraph with a link

Laurent Regnier: Exponentials

2010-05-25T07:53:08Z

Exponentials

Laurent Regnier: Summaries of sections

2010-05-15T11:58:39Z

Summaries of sections

Laurent Regnier: Split the page

2010-05-15T11:04:11Z

Split the page

Show changes

Laurent Regnier: /* Execution formula, version 2: composition */ correction

2010-05-15T10:13:27Z

‎Execution formula, version 2: composition: correction

Laurent Regnier: /* Weak distributivity */

2010-05-15T10:07:24Z

‎Weak distributivity

Laurent Regnier: /* The tensor rule */ typo, style

2010-05-15T10:03:29Z

‎The tensor rule: typo, style

Laurent Regnier: /* From operators to matrices: internalization/externalization */ precision

2010-05-14T15:24:29Z

‎From operators to matrices: internalization/externalization: precision

Laurent Regnier: /* Operators, partial isometries */ elementary props of partial isometries

2010-05-14T15:14:25Z

‎Operators, partial isometries: elementary props of partial isometries

@@ Line 19: / Line 19: @@
 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>.
+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 glueing]]''.</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>.
 == [[GoI for MELL: partial isometries|Partial isometries]] ==

@@ Line 13: / Line 13: @@
 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>. This duality is symmetric: if <math>uv</math> is nilpotent then <math>vu</math> is nilpotent also.
+Such a duality defines an [[orthogonality relation]], with the usual derived definitions and properties.
-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>:
+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 nonnegative integer <math>n</math> such that <math>(uv)^n = 0</math>. This duality is symmetric: if <math>uv</math> is nilpotent then <math>vu</math> is nilpotent also.
 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>.
+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>.
 == [[GoI for MELL: partial isometries|Partial isometries]] ==

@@ Line 31: / Line 31: @@
 The first step is to build the proof space. This is constructed as a special set of partial isometries on a separable Hilbert space <math>H</math> which turns out to be generated by partial permutations on the canonical basis of <math>H</math>.
-These so-called <math>p</math>-isometries enjoy some nice properties, the most important one being that a <math>p</math>-isometry is a sum of <math>p</math>-isometries iff all the terms of the sum have disjoint domains and disjoint codomains. As a consequence we get that a sum of <math>p</math>-isometries is null iff each term of the sum is null.
+These so-called ''<math>p</math>-isometries'' enjoy some nice properties, the most important one being that a <math>p</math>-isometry is a sum of <math>p</math>-isometries iff all the terms of the sum have disjoint domains and disjoint codomains. As a consequence we get that a sum of <math>p</math>-isometries is null iff each term of the sum is null.
 A second important property is that operators on <math>H</math> can be ''externalized'' using <math>p</math>-isometries into operators acting on <math>H\oplus H</math>, and conversely operators on <math>H\oplus H</math> may be ''internalized'' into operators on <math>H</math>. This is widely used in the sequel.
-== [[GoI for MELL: the *-autonomous structure| The *-autonomous structure]] ==
+== [[GoI for MELL: the *-autonomous structure|The *-autonomous structure]] ==
 The second step is to interpret the linear logic multiplicative operations, most importantly the cut rule.
@@ Line 42: / Line 42: @@
 The (interpretation of) the cut-rule is defined in two steps: firstly we use nilpotency to define an operation corresponding to lambda-calculus application which given two <math>p</math>-isometries in respectively <math>A\limp B</math> and <math>A</math> produces an operator in <math>B</math>. From this we deduce the composition and finally obtain a structure of *-autonomous category, that is a model of multiplicative linear logic.
 = The Geometry of Interaction as an abstract machine =

@@ Line 28: / Line 28: @@
 == [[GoI for MELL: partial isometries|Partial isometries]] ==
 == [[GoI for MELL: the *-autonomous structure| The *-autonomous structure]] ==
 = The Geometry of Interaction as an abstract machine =

@@ Line 494: / Line 494: @@
 {{Theorem|
 Let GoI(H) be defined by:
-* objects are types, ''ie'' sets <math>A</math> of operators satisfying: <math>A\biorth = A</math>;
+* objects are types, ''ie'' sets <math>A</math> of <math>p</math>-isometries satisfying: <math>A\biorth = A</math>;
-* morphisms from <math>A</math> to <math>B</math> are operators in type <math>A\limp B</math>;
+* morphisms from <math>A</math> to <math>B</math> are <math>p</math>-isometries in type <math>A\limp B</math>;
 * composition is given by the formula above.

@@ Line 452: / Line 452: @@
 ==== Weak distributivity ====
-We can finally get weak distributivity thanks to the operators:
+Similarly we 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>.

@@ Line 355: / Line 355: @@
 === 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:
+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 <math>u\tens u'</math> by:
 : <math>\begin{align}
     u\tens u' &= ppp^*upp^*p^* + qpq^*upp^*p^* + ppp^*uqp^*q^* + qpq^*uqp^*q^*\\
@@ Line 390: / Line 390: @@
   </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>.
+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> 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:
+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>u_{11}v</math> and <math>u'_{11}v'</math> are nilpotent and 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:
+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\\
+      &= \bigl((pu_{11}p^* + 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^*

@@ Line 191: / Line 191: @@
   \end{pmatrix}
 </math>
-then the externalization of <math>uv</math> is <math>UV</math>.
+then the externalization of <math>uv</math> is the matrix product <math>UV</math>.
 As <math>pp^* + qq^* = 1</math> we have:

@@ Line 70: / Line 70: @@
 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>.
+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 then we have:
 === Partial permutations ===