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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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Cited by 3 (3 self)
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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
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 ..."
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Cited by 3 (1 self)
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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 ...
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 mod ..."
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Cited by 3 (1 self)
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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...
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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Cited by 2 (0 self)
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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.
Infinite λcalculus and Types
, 1998
"... Recent work on infinitary versions of the lambda calculus has shown that the infinite lambda calculus can be a useful tool to study the unsolvable terms of the classical lambda calculus. Working in the framework of the intersection type disciplines, we devise a type assignment system such that two t ..."
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Cited by 1 (0 self)
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Recent work on infinitary versions of the lambda calculus has shown that the infinite lambda calculus can be a useful tool to study the unsolvable terms of the classical lambda calculus. Working in the framework of the intersection type disciplines, we devise a type assignment system such that two terms are equal in the infinite lambda calculus iff they can be assigned the same types in any basis. A novel feature of the system is the presence of a type constant to denote the set of all terms of order zero, and the possibility of applying a type to another type. We prove a completeness and an approximation theorem for our system. Our results can be considered as a first step towards the goal of giving a denotational semantics for the lambda calculus which is suited for the study of the unsolvable terms. However some noncontinuity phenomena of the infinite lambda calculus make a full realization of this idea (namely the construction of a filter model) a quite difficult task.
Intersection Types for λTrees
"... We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating λterms (Böhm trees, LévyLongo trees,...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by th ..."
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We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating λterms (Böhm trees, LévyLongo trees,...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by the corresponding tree representation of terms. More precisely, for each family of trees, two terms have the same tree if and only if they get assigned the same types in the corresponding type assignment system.
A lambda calculus for D∞
, 2002
"... We define an extension of lambda calculus which is fully abstract for Scott's D_infinitymodels. We do so by constructing an infinitary lambda calculus which not only has the confluence property, but also is normalising: every term has its infinity etaBöhm tree as unique normal form. The exten ..."
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We define an extension of lambda calculus which is fully abstract for Scott's D_infinitymodels. We do so by constructing an infinitary lambda calculus which not only has the confluence property, but also is normalising: every term has its infinity etaBöhm tree as unique normal form. The extension incorporates...
Department of Computing, Imperial College,
"... We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluatingterms (Böhm trees, LévyLongo trees,...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by the ..."
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We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluatingterms (Böhm trees, LévyLongo trees,...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by the corresponding tree representation of terms. More precisely, for each family of trees, two terms have the same tree if and only if they get assigned the same types in the corresponding type assignment system. Key words: Böhm trees, approximants, intersection types. 1
Types for Trees
, 1999
"... We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating λterms (Böhm trees, LévyLongo trees, ...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by t ..."
Abstract
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We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating λterms (Böhm trees, LévyLongo trees, ...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by the corresponding tree representation of terms. More precisely, for each family of trees two terms have the same tree if and only if they get assigned the same types in the corresponding type assignment system.