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A Game Semantics For Generic Polymorphism
, 1971
"... Genericity is the idea that the same program can work at many dierent data types. Longo, Milstead and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type 8X ..."
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Genericity is the idea that the same program can work at many dierent data types. Longo, Milstead and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type 8X:A[X ], are equal at any given instance A[T ], then they are equal at all instances. They proved that this rule is admissible in a certain extension of System F, but nding a semantically motivated model satisfying this principle remained an open problem.
Linear realizability and full completeness for typed lambda calculi
 Annals of Pure and Applied Logic
, 2005
"... We present the model construction technique called Linear Realizability. It consists in building a category of Partial Equivalence Relations over a Linear Combinatory Algebra. We illustrate how it can be used to provide models, which are fully complete for various typed λcalculi. In particular, we ..."
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We present the model construction technique called Linear Realizability. It consists in building a category of Partial Equivalence Relations over a Linear Combinatory Algebra. We illustrate how it can be used to provide models, which are fully complete for various typed λcalculi. In particular, we focus on special Linear Combinatory Algebras of partial involutions, and we present PER models over them which are fully complete, inter alia, w.r.t. the following languages and theories: the fragment of System F consisting of MLtypes, the maximal theory on the simply typed λcalculus with finitely many ground constants, and the maximal theory on an infinitary version of this latter calculus. Key words: Typed lambdacalculi, MLpolymorphic types, linear logic, hyperdoctrines, PER models, Geometry of Interaction, (axiomatic) full completeness
"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, using a countable ..."
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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 !.
"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, using the countable pow ..."
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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 product as tens ..."
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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.
Unfixing the Fixpoint: the theories of the λYcalculus
"... Abstract. We investigate the theories of the λYcalculus, i.e. simply typed λcalculus with fixpoint combinators. Nonterminating λYterms exhibit a rich behavior, and one can reflect in λY many results of untyped λcalculus concerning theories. All theories can be characterized as contextual theori ..."
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Abstract. We investigate the theories of the λYcalculus, i.e. simply typed λcalculus with fixpoint combinators. Nonterminating λYterms exhibit a rich behavior, and one can reflect in λY many results of untyped λcalculus concerning theories. All theories can be characterized as contextual theories à la Morris, w.r.t. a suitable set of observables. We focus on theories arising from natural classes of observables, where Y can be approximated, albeit not always initially. In particular, we present the standard theory, induced by terminating terms, which features a canonical interpretation of Y as “minimal fixpoint”, and another theory, induced by pure λterms, which features a noncanonical interpretation of Y. The interest of these two theories is that the term model of the λYcalculus w.r.t. the first theory gives a fully complete model of the maximal theory of the simply typed λcalculus, while the term model of the latter theory provides a fully complete model for the observational equivalence in unary PCF. Throughout the paper we raise open questions and conjectures.