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Fully Complete Models for ML Polymorphic Types
 In Proc. of MFCS'2000
, 1999
"... We present an axiomatic characterization of models fullycomplete for MLpolymorphic types of system F. This axiomatization is given for hyperdoctrine models, which arise as adjoint models, i.e. coKleisli categories of suitable linear categories. Examples of adjoint models can be obtained from cate ..."
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We present an axiomatic characterization of models fullycomplete for MLpolymorphic types of system F. This axiomatization is given for hyperdoctrine models, which arise as adjoint models, i.e. coKleisli categories of suitable linear categories. Examples of adjoint models can be obtained from categories of Partial Equivalence Relations over Linear Combinatory Algebras. We show that a special linear combinatory algebra of partial involutions induces an hyperdoctrine which satisfies our axiomatization, and hence it provides a fullycomplete model for MLtypes. Introduction In this paper we address the problem of full completeness for system F. A categorical model of a type theory (or logic) is said to be fullycomplete ([AJ94a]) if, for all types (formulae) A; B, all morphisms f : [[A]] ! [[B]], from the interpretation of A into the interpretation of B, are denotations of a proofterm of the intailment A ` B. The notion of fullcompleteness is the counterpart of the notion of full...
Innocent Game Models of Untyped λCalculus
 Theoretical Computer Science
, 2000
"... We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [9] and Nickau [17], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and ..."
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We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [9] and Nickau [17], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and DREC as global sections of a reexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a strong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call eectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a model D...
Innocent Game Models of Untyped λCalculus
, 2000
"... We present a new denotation model for the untyped λcalculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between "question" and "answer" moves is removed. We first construct mod ..."
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We present a new denotation model for the untyped λcalculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between "question" and "answer" moves is removed. We first construct models D and DREC as global sections of a reflexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a stong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call effectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a ...
A universal innocent game model for the Bohm tree lambda theory
 In Computer Science Logic: Proceedings of the 8th Annual Conference on the EACSL
, 1999
"... Abstract. We present a game model of the untyped λcalculus, with equational theory equal to the Böhm tree λtheory B, which is universal (i.e. every element of the model is definable by some term). This answers a question of Di Gianantonio, Franco and Honsell. We build on our earlier work, which us ..."
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Abstract. We present a game model of the untyped λcalculus, with equational theory equal to the Böhm tree λtheory B, which is universal (i.e. every element of the model is definable by some term). This answers a question of Di Gianantonio, Franco and Honsell. We build on our earlier work, which uses the methods of innocent game semantics to develop a universal model inducing the maximal consistent sensible theory H ∗. To our knowledge these are the first syntaxindependent universal models of the untyped λcalculus. 1
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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. 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...
More Universal Game Models of Untyped λCalculus: The Böhm Tree Strikes Back
 STRIKES BACK, CSL'99 CONF. PROC., LNCS
, 1999
"... We present a game model of the untyped λcalculus, with equational theory equal to the Bohm tree λtheory B, which is universal (i.e. every element of the model is definable by some term). This answers a question of Di Gianantonio, Franco and Honsell. We build on our earlier work, which uses the met ..."
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We present a game model of the untyped λcalculus, with equational theory equal to the Bohm tree λtheory B, which is universal (i.e. every element of the model is definable by some term). This answers a question of Di Gianantonio, Franco and Honsell. We build on our earlier work, which uses the methods of innocent game semantics to develop a universal model inducing the maximal consistent sensible theory H*. To our knowledge these are the first syntaxindependent universal models of the untyped λcalculus.
Innocent Game Models of Untyped
, 2000
"... We present a new denotation model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models ..."
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Cited by 1 (0 self)
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We present a new denotation model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and DREC as global sections of a reexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a stong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call eectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a ...