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Geometry of Interaction and Linear Combinatory Algebras
, 2000
"... this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years ea ..."
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Cited by 42 (10 self)
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this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by Stefanescu (Stefanescu 2000).) However, the first author realized, following a stimulating discussion with Gordon Plotkin, that traced monoidal categories provided a common denominator for the axiomatics of both the Girardstyle and AbramskyJagadeesanstyle versions of the Geometry of Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girardstyle GoI was dubbed "particlestyle", since it concerns information particles or tokens flowing around a network, while the AbramskyJagadeesan style GoI was dubbed "wavestyle", since it concerns the evolution of a global information state or "wave". Formally, this distinction is based on whether the tensor product (i.e. the symmetric monoidal structure) in the underlying category is interpreted as a coproduct (particle style) or as a product (wave style). This computational distinction between coproduct and product interpretations of the same underlying network geometry turned out to have been partially anticipated, in a rather di#erent context, in a pioneering paper by E. S. Bainbridge (Bainbridge 1976), as observed by Dusko Pavlovic. These two forms of interpretation, and ways of combining them, have also been studied recently in (Stefanescu 2000). He uses the terminology "additive" for coproductbased (i.e. our "particlestyle") and "multiplicative" for productbased (i.e. our "wavestyle"); this is not suitable for our purposes, because of the clash with Linear Logic term...
A Structural Approach to Reversible Computation
 Theoretical Computer Science
, 2001
"... Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of lowlevel machine models. By contrast, we develop ..."
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Cited by 18 (3 self)
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Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of lowlevel machine models. By contrast, we develop a more structural approach. We show how highlevel functional programs can be mapped compositionally (i.e. in a syntaxdirected fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic—but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis. 1
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding &quot;good &quot; quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the &quot;best &quot; regular category (called its regular completion) that embeds it. The second assigns to
Fully Complete Minimal PER Models for the Simply Typed λcalculus
 CSL'01, LNCS 2142
, 2001
"... We show how to build a fully complete model for the maximal theory of the simply typed λcalculus with k ground constants, k. This is obtained by linear realizability over an affine combinatory algebra of partial involutions from natural numbers into natural numbers. For simplicitly, we give the det ..."
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Cited by 6 (3 self)
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We show how to build a fully complete model for the maximal theory of the simply typed λcalculus with k ground constants, k. This is obtained by linear realizability over an affine combinatory algebra of partial involutions from natural numbers into natural numbers. For simplicitly, we give the details of the construction of a fully complete model for k extended with ground permutations. The fully complete minimal model for k can be obtained by carrying out the previous construction over a suitable subalgebra of partial involutions. The full completeness result is then put to use in order to prove some simple results on the maximal theory.
"Wavestyle" Geometry of Interaction Models in Rel are Graphlike Lambdamodels
"... We study the connections between graph models and \wavestyle " Geometry of Interaction (GoI) models. The latters arise when Abramsky's GoI axiomatization, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical product as tensor, usi ..."
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Cited by 3 (0 self)
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We study the connections between graph models and \wavestyle " Geometry of Interaction (GoI) models. The latters arise when Abramsky's GoI axiomatization, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical product as tensor, using a countable power as the traced strong monoidal functor !.
A Fully Complete PER Model for ML Polymorphic Types
 Proceedings of CSL 2000, Springer LNCS Volume 1862
, 2000
"... . We present a linear realizability technique for building Partial Equivalence Relations (PER) categories over Linear Combinatory Algebras. These PER categories turn out to be linear categories and to form an adjoint model with their coKleisli categories. We show that a special linear combinato ..."
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Cited by 2 (1 self)
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. We present a linear realizability technique for building Partial Equivalence Relations (PER) categories over Linear Combinatory Algebras. These PER categories turn out to be linear categories and to form an adjoint model with their coKleisli categories. We show that a special linear combinatory algebra of partial involutions, arising from Geometry of Interaction constructions, gives rise to a fully and faithfully complete model for ML polymorphic types of system F. Keywords: MLpolymorphic types, linear logic, PER models, Geometry of Interaction, full completeness. Introduction Recently, Game Semantics has been used to define fullycomplete models for various fragments of Linear Logic ([AJ94a,AM99]), and to give fullyabstract models for many programming languages, including PCF [AJM96,HO96,Nic94], richer functional languages [McC96], and languages with nonfunctional features such as reference types and nonlocal control constructs [AM97,Lai97]. All these results are cru...
"Wavestyle" Geometry of Interaction Models are Graphlike λmodels
"... We study the connections between graph models and "wavestyle " Geometry of Interaction (GoI) #models. The latters arise when Abramsky's GoI construction, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical product as tensor, usi ..."
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We study the connections between graph models and "wavestyle " Geometry of Interaction (GoI) #models. The latters arise when Abramsky's GoI construction, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical product as tensor, using the countable power as traced strong monoidal functor !. Abramsky hinted that the category of sets and relations is the basic setting for traditional "static semantics". Here we support this view by showing that a large class of graphlike models can be viewed as arising from a suitable generalization of the GoI construction. Furthermore, we show that the class of untyped #theories induced by wavestyle GoI models is richer than that induced by game models.
Strict Geometry of Interaction Graph Models
, 2003
"... We study a class of \wavestyle" Geometry of Interaction (GoI) models based on the category Rel of sets and relations. Wave GoI models arise when Abramsky's GoI axiomatization, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical ..."
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We study a class of \wavestyle" Geometry of Interaction (GoI) models based on the category Rel of sets and relations. Wave GoI models arise when Abramsky's GoI axiomatization, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical product as tensor, using \countable power" as the traced strong monoidal functor !. Abramsky hinted that the category Rel is the basic setting for traditional denotational \static semantics". However, Rel, together with the cartesian product, apparently escapes Abramsky's original GoI construction. Here we show that Rel can be axiomatized as a strict GoI situation, i.e. a strict variant of Abramsky's GoI situation, which gives rise to a rich class of strict graph models. These are models of restricted calculi in the sense of [HL99], such as Church's Icalculus and the KN calculus.