Geometry of interaction

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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.
  
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.
   
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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]].
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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> 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 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]]
   
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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.
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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; the execution formula appears as the composition of two automata interacting through a 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 Geometry of Interaction as operators =
   
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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>.
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The original construction of GoI by Girard follows a general pattern already mentionned in the section on [[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>.
   
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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>.
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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>.
   
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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>.
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Such a duality defines an [[orthogonality relation]], with the usual derived definitions and properties.
   
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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>:
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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.
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: <math>X\orth = \{u\in P, \forall v\in X, uv\in\bot\}</math>.
  
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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:
  
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* if <math>X\subset Y</math> then <math>Y\orth\subset X</math>;
  
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* <math>X\subset X\biorth</math>;
  
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* <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>.
  
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>.
   
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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>.
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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>.
   
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== Preliminaries ==
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== [[GoI for MELL: partial isometries|Partial isometries]] ==
   
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We begin by a brief tour of the operations in Hilbert spaces that we use. In this article <math>H</math> will stand for 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> we denote by <math>\langle x,y\rangle</math> their scalar product:
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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>.
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: <math>\langle x, y\rangle = \sum_{n\in\mathbb{N}} x_n\bar y_n</math>.
  
   
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Two vectors of <math>H</math> are ''othogonal'' if their scalar product is nul. We will say that two subspaces are ''disjoint'' when their vectors are pairwise orthogonal; this terminology is slightly misleading as disjoint subspaces always have <math>0</math> in common.
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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.
   
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The ''norm'' of a vector is the square root of the scalar product with itself:
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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.
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: <math>\|x\| = \sqrt{\langle x, x\rangle}</math>.
  
   
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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:
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== [[GoI for MELL: the *-autonomous structure|The *-autonomous structure]] ==
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: <math> x = \sum_{n\in\mathbb{N}} x_ne_n</math>.
  
   
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An ''operator'' on <math>H</math> is a ''continuous'' linear map from <math>H</math> to <math>H</math>. 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:
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The second step is to interpret the linear logic multiplicative operations, most importantly the cut rule.
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: <math>\|u\| = \sup_{\{x\in H,\, \|x\| = 1\}}\|u(x)\|</math>.
  
   
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The set of (bounded) operators is denoted by <math>\mathcal{B}(H)</math>.
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Internalization/externalization is the key for this: typically the type <math>A\tens B</math> is interpreted by a set of <math>p</math>-isometries which are internalizations of operators acting on <math>H\oplus H</math>.
   
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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:
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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.
   
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* <math>\mathrm{Codom}(u) = \{u(x),\, x\in H\}</math>;
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== [[GoI for MELL: exponentials|The exponentials]] ==
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* <math>\mathrm{Ker}(u) = \{x\in H,\, u(x) = 0\}</math>;
  
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* <math>\mathrm{Dom}(u) = \{x\in H,\, \forall y\in\mathrm{Ker}(u), \langle x, y\rangle = 0\}</math>.
  
   
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These three sets are closed subspaces of <math>H</math>.
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Finally we turn to define exponentials, that is connectives managing duplication. To do this we introduce an isomorphism (induced by a <math>p</math>-isometry) between <math>H</math> and <math>H\tens H</math>: the first component of the tensor is intended to hold the address of the the copy whereas the second component contains the content of the copy.
   
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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>.
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We eventually get a quasi-model of full MELL; quasi in the sense that if we can construct <math>p</math>-isometries for usual structural operations in MELL (contraction, dereliction, digging), the interpretation of linear logic proofs is not invariant w.r.t. cut elimination in general. It is however invariant in some good cases, which are enough to get a correction theorem for the interpretation.
   
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A ''projector'' is an idempotent operator of norm <math>0</math> (the projector
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= The Geometry of Interaction as an abstract machine =
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on the null subspace) or <math>1</math>, that is an operator <math>p</math>
  
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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.
  
   
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A ''partial isometry'' is an operator <math>u</math> satisfying <math>uu^* u =
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= Notes and references =
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u</math>; this condition entails that we also have <math>u^*uu^* =
  
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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 codomain are respectively the domain and the codomain of <math>u</math>. The restriction of <math>u</math> to its domain is an isometry. Projectors are particular examples of partial isometries.
  
   
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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>.
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<references/>
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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 and that their direct sum is <math>H</math>.
  
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=== Partial permutations and partial isometries ===
  
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We will now define our proof space which turns out to be the set of partial isometries acting as permutations on a fixed basis of <math>H</math>.
  
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More precisely a ''partial permutation'' <math>\varphi</math> on <math>\mathbb{N}</math> is a function defined on a subset <math>D_\varphi</math> of <math>\mathbb{N}</math> which is one-to-one 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:
  
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* <math>n\in D_{\varphi\circ\psi}</math> iff <math>n\in D_\psi</math> and <math>\psi(n)\in D_\varphi</math>;
  
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* if <math>n\in D_{\varphi\circ\psi}</math> then <math>\varphi\circ\psi(n) = \varphi(\psi(n))</math>;
  
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* the codomain of <math>\varphi\circ\psi</math> is the image of the domain.
  
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Partial permutations are well known to form a structure of ''inverse monoid'' that we detail now.
  
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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.
  
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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.
  
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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:
  
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: <math>\varphi^{-1}\circ\varphi = 1_{D_\varphi}</math>
  
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: <math>\varphi\circ\varphi^{-1} = 1_{C_\varphi}</math>
  
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Given a partial permutation <math>\varphi</math> one defines a partial isometry <math>u_\varphi</math> by:
  
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: <math>u_\varphi(e_n) =
 
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\begin{cases}
  
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e_{\varphi(n)} & \text{ if }n\in D_\varphi,\\
  
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0 & \text{ otherwise.}
  
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\end{cases}
  
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</math>
  
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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:
  
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: <math>y_n = x_{\varphi^{-1}(n)}</math> if <math>n\in C_\varphi</math>, <math>0</math> otherwise.
  
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We will (not so abusively) write <math>e_{\varphi(n)} = 0</math> when <math>\varphi(n)</math> is undefined so that may shorten the definition of <math>u_\varphi</math> into:
  
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: <math>u_\varphi(e_n) = e_{\varphi(n)}</math>.
  
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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>. As a particular case if <math>\varphi</math> is <math>1_D</math> the partial identity on <math>D</math> then <math>u_\varphi</math> is the projector on the subspace spanned by <math>(e_n)_{n\in D}</math>.
  
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{{Proposition|
  
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Let <math>\varphi</math> and <math>\psi</math> be two partial permutations. We have:
  
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: <math>u_\varphi u_\psi = u_{\varphi\circ\psi}</math>.
  
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The adjoint of <math>u_\varphi</math> is:
  
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: <math>u_\varphi^* = u_{\varphi^{-1}}</math>.
  
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In particular the initial projector of <math>u_{\varphi}</math> is given by:
  
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: <math>u_\varphi u^*_\varphi = u_{1_{D_\varphi}}</math>.
  
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and the final projector of <math>u_\varphi</math> is:
  
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: <math>u^*_\varphi u_\varphi = u_{1_{C_\varphi}}</math>.
  
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Projectors generated by partial identities commute; in particular we have:
  
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: <math>u_\varphi u_\varphi^*u_\psi u_\psi^* = u_\psi u_\psi^*u_\varphi u_\varphi^*</math>.
  
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}}
  
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{{Definition|
  
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The ''proof space'' <math>\mathcal{P}</math> is the set of partial isometries of the form <math>u_\varphi</math> for partial permutations <math>\varphi</math> on <math>\mathbb{N}</math>.
  
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}}
  
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In particular note that <math>0\in\mathcal{P}</math>. The set <math>\mathcal{P}</math> is a submonoid of <math>\mathcal{B}(H)</math> but it is not a subalgebra: 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:
  
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{{Proposition|
  
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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:
  
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: <math>u+v\in\mathcal{P}</math> iff <math>uu^*vv^* = u^*uv^*v = 0</math>.
  
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}}
  
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Also note that if <math>u+v=0</math> then <math>u=v=0</math>.
  
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=== From operators to matrices: internalization/externalization ===
  
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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:
  
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: <math>p(e_n) = e_{2n}</math>,
  
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: <math>q(e_n) = e_{2n+1}</math>.
  
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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 orthogonal, thus we have <math>p^* q = q^* p = 0</math>. Note that we also have <math>pp^* + qq^* = 1</math>.
  
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The choice of <math>p</math> and <math>q</math> is actually arbitrary, any two partial isometries with full domain and orthogonal codomains would do the job.
  
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Let <math>U</math> be an operator on <math>H\oplus H</math>. We can write <math>U</math> as a matrix:
  
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: <math>U = \begin{pmatrix}
  
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u_{11} & u_{12}\\
  
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u_{21} & u_{22}
  
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\end{pmatrix}</math>
  
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where each <math>u_{ij}</math> operates on <math>H</math>.
  
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Now through the isomorphism <math>H\oplus H\cong H</math> we may transform <math>U</math> into the operator <math>u</math> on <math>H</math> defined by:
  
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: <math>u = pu_{11}p^* + pu_{12}q^* + qu_{21}p^* + qu_{22}q^*</math>.
  
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We call <math>u</math> the ''internalization'' of <math>U</math>. Internalization is compatible with composition (functorial so to speak): if <math>V</math> is another operator on <math>H\oplus</math> then the internalization of the matrix product <math>UV</math> is the product <math>uv</math>.
  
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Conversely 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>:
  
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: <math>u_{11} = p^*up</math>;
  
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: <math>u_{12} = p^*uq</math>;
  
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: <math>u_{21} = q^*up</math>;
  
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: <math>u_{22} = q^*uq</math>.
  
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The <math>u_{ij}</math>'s are called the ''components'' of <math>u</math>. Note that if <math>u</math> is generated by a partial permutation, that is if <math>u\in\mathcal{P}</math> then so are the <math>u_{ij}</math>'s. Moreover we have:
  
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: <math>u = (pp^*+qq^*)u(pp^*+qq^*) = pu_{11}p^* + pu_{12}q^* + qu_{21}p^* + qu_{22}q^*</math>
  
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which entails that the four terms of the sum have pairwise disjoint domains and pairwise disjoint codomains. This can be verified for example by computing the product of the final projectors of <math>pu_{11}p^*</math> and <math>pu_{12}q^*</math>:
  
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: <math>\begin{align}
  
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(pu_{11}p^*)(pu^*_{11}p^*)(pu_{12}q^*)(qu_{12}^*p^*)
  
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&= (pp^*upp^*)(pp^*u^*pp^*)(pp^*uqq^*)(qq^*u^*pp^*)\\
  
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&= pp^*upp^*u^*pp^*uqq^*u^*pp^*\\
  
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&= pp^*u(pp^*)(u^*pp^*u)qq^*u^*pp^*\\
  
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&= pp^*u(u^*pp^*u)(pp^*)qq^*u^*pp^*\\
  
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&= pp^*uu^*pp^*u(pp^*)(qq^*)u^*pp^*\\
  
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&= 0
  
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\end{align}</math>
  
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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>.
  
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== Interpreting the multiplicative connectives ==
  
−  
−
Recall that when <math>u</math> and <math>v</math> are partial isometries in <math>\mathcal{P}</math> 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 ===
  
−  
−
Given two types <math>A</math> and <math>B</math>, we 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>
  
−  
−
This is an abuse of notations as this operation is more like a direct sum than a tensor. We will stick to this notation 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 see that we have:
  
−
: <math>A\limp B = \{u\in\mathcal{P}\text{ such that } \forall v\in A, \forall w\in B\orth,\, u.(pvp^* + qwq^*) \in\bot\}</math>.
  
−  
−
=== The identity ===
  
−  
−
The interpretation of the identity is an example of the internalization/externalization procedure. 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 ===
  
−  
−
Let <math>A</math> and <math>B</math> be two types and <math>u</math> an operator in <math>A\limp B</math>. By definition this means that given <math>v</math> in <math>A</math> and <math>w</math> in <math>B\orth</math> the operator <math>u.(pvp^* + qwq^*)</math> is nilpotent.
  
−  
−
Let us define <math>u_{11}</math> to <math>u_{22}</math> by externalization as above. If we compute <math>(u.(pvp^* + qwq^*))^n</math> we see that this is a finite sum of operators of the form:
  
−
# <math>q(u_{22}w)^{k_0}u_{21}v(u_{11}v)^{k_1}u_{12}w\dots u_{12}w(u_{22}w)^{k_{p+1}}q^*</math>,
  
−
# <math>p(u_{11}v)^{k_1}u_{12}w\dots u_{12}w(u_{22}w)^{k_{p+1}}q^*</math>,
  
−
# <math>q(u_{22}w)^{k_0}u_{21}v(u_{11}v)^{k_1}u_{12}w\dots (u_{11}v)^{k_p}p^*</math> or
  
−
# <math>p(u_{11}v)^{k_1}u_{12}w\dots (u_{11}v)^{k_p}p^*</math>
  
−
where each of these monimials has exactly <math>n</math> factors of the form <math>u_{i1}v</math> or <math>u_{i2}w</math>.
  
−  
−
From the nilpotency of <math>u.(pvp^* + qwq^*)</math> we deduce that <math>u_{11}v</math> is nilpotent by considering the particular case where <math>w=0</math>. We also have that <math>q^*(u.(pvp^* + qwq^*))^nq</math> is null for <math>n</math> big enough, <strike>which means that monomials of type 1 above are null as soon as their length (the number of factors of the form <math>u_{i1}v</math> or <math>u_{i2}w</math>) is bigger than <math>n</math></strike>.
  
−  
−
This implies that the two following operators are nilpotent:
  
−
: <math>u_{11}v</math> and
  
−
: <math>\bigl(u_{22} + u_{21}v\sum_k(u_{11}v)^k u_{12}\bigr)w</math>.
  
−  
−
Conversely if these two operators are nilpotent then one can show that so is <math>u.(pvp^* + qwq^*)</math>. Moreover we have:
  
−
: <math>q^*\sum_n\bigl(u.(pvp^* + qwq^*)\bigr)^nq = \sum_n\bigl((u_{22} + u_{21}v\sum_k(u_{11}v)^k u_{12})w\bigr)^n</math>.
  
−  
−
We define the ''application of <math>u</math> to <math>v</math>'' as:
  
−
: <math>\mathrm{App}(u, v) = u_{22} + u_{21}v\sum(u_{11}v)^ku_{12}</math>.
  
−
Note that this is well defined as soon as <math>u_{11}v</math> is nilpotent.
  
−  
−
We summarize what has just been shown in the following theorem:
  
−  
−
{{Theorem|
  
−
Let <math>u</math> be an operator, <math>A</math> and <math>B</math> be two types; the following conditions are equivalent:
  
−
* <math>u\in A\limp B</math>;
  
−
* for any <math>v\in A</math>, we both have:
  
−
:: <math>u_{11}v</math> is nilpotent and
  
−
:: <math>\mathrm{App}(u, v)\in B</math>.
  
−
}}
  
−  
−
{{Corollary|
  
−
Under the hypothesis of the theorem 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 =
  

Latest revision as of 15:39, 30 September 2011

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 A\limp B as a morphism from A to B and proof composition (cut rule) as the composition of morphisms. Rather the proof was interpreted as an operator acting on A\limp B, that is a morphism from A\limp B to A\limp B. For proof composition the problem was then, given an operator on A\limp B and another one on B\limp C to construct a new operator on A\limp C. 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 denotational semantics.

The first instance of the GoI was restricted to the MELL 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 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; the execution formula appears as the composition of two automata interacting through a 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 MELL fragment has been reformulated in the framework of traced monoidal categories following an idea originally proposed by Joyal.

Contents

[edit] The Geometry of Interaction as operators

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

Second define a particular subset of P that will be denoted by \bot; then derive a duality on P: for u,v\in P, u and v are dual[1]iff uv\in\bot.

Such a duality defines an orthogonality relation, with the usual derived definitions and properties.

For the GoI, two dualities have proved to work; we will consider the first one: nilpotency, ie, \bot is the set of nilpotent operators in P. Let us explicit this: two operators u and v are dual if there is a nonnegative integer n such that (uv)n = 0. This duality is symmetric: if uv is nilpotent then vu is nilpotent also.

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

The real work[2]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 u is the interpretation of a proof of the formula A then u belongs to the type associated to A.

[edit] Partial isometries

The first step is to build the proof space. This is constructed as a special set of partial isometries on a separable Hilbert space H which turns out to be generated by partial permutations on the canonical basis of H.

These so-called p-isometries enjoy some nice properties, the most important one being that a p-isometry is a sum of p-isometries iff all the terms of the sum have disjoint domains and disjoint codomains. As a consequence we get that a sum of p-isometries is null iff each term of the sum is null.

A second important property is that operators on H can be externalized using p-isometries into operators acting on H\oplus H, and conversely operators on H\oplus H may be internalized into operators on H. This is widely used in the sequel.

[edit] The *-autonomous structure

The second step is to interpret the linear logic multiplicative operations, most importantly the cut rule.

Internalization/externalization is the key for this: typically the type A\tens B is interpreted by a set of p-isometries which are internalizations of operators acting on H\oplus H.

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 p-isometries in respectively A\limp B and A produces an operator in B. From this we deduce the composition and finally obtain a structure of *-autonomous category, that is a model of multiplicative linear logic.

[edit] The exponentials

Finally we turn to define exponentials, that is connectives managing duplication. To do this we introduce an isomorphism (induced by a p-isometry) between H and H\tens H: the first component of the tensor is intended to hold the address of the the copy whereas the second component contains the content of the copy.

We eventually get a quasi-model of full MELL; quasi in the sense that if we can construct p-isometries for usual structural operations in MELL (contraction, dereliction, digging), the interpretation of linear logic proofs is not invariant w.r.t. cut elimination in general. It is however invariant in some good cases, which are enough to get a correction theorem for the interpretation.

[edit] The Geometry of Interaction as an abstract machine

[edit] Notes and references

  1. ↑ In modern terms one says that u and v are polar.
  2. ↑ 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.
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