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Finitary PCF is not decidable
 Theoretical Computer Science
, 1996
"... The question of the decidability of the observational ordering of finitary PCF was raised [5] to give mathematical content to the full abstraction problem for PCF [9, 14]. We show that the ordering is in fact undecidable. This result places limits on how explicit a representation of the fully abstra ..."
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The question of the decidability of the observational ordering of finitary PCF was raised [5] to give mathematical content to the full abstraction problem for PCF [9, 14]. We show that the ordering is in fact undecidable. This result places limits on how explicit a representation of the fully abstract model can be. It also gives a slight strengthening of the author’s earlier result on typed λdefinability [6].
CallByPushValue: A Subsuming Paradigm
 in Proc. TLCA ’99
, 1999
"... . Callbypushvalue is a new paradigm that subsumes the callbyname and callbyvalue paradigms, in the following sense: both operational and denotational semantics for those paradigms can be seen as arising, via translations that we will provide, from similar semantics for callbypushvalue. To ..."
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. Callbypushvalue is a new paradigm that subsumes the callbyname and callbyvalue paradigms, in the following sense: both operational and denotational semantics for those paradigms can be seen as arising, via translations that we will provide, from similar semantics for callbypushvalue. To explain callbypushvalue, we first discuss general operational ideas, especially the distinction between values and computations, using the principle that "a value is, a computation does". Using an example program, we see that the lambdacalculus primitives can be understood as push/pop commands for an operandstack. We provide operational and denotational semantics for a range of computational effects and show their agreement. We hence obtain semantics for callbyname and callbyvalue, of which some are familiar, some are new and some were known but previously appeared mysterious. 1 Introduction 1.1 Contribution In his invited lecture at POPL '98 [32], Reynolds, surveying over 30 year...
Full Abstraction, Totality and PCF
 Math. Structures Comput. Sci
, 1997
"... ion, Totality and PCF Gordon Plotkin Abstract Inspired by a question of Riecke, we consider the interaction of totality and full abstraction, asking whether full abstraction holds for Scott's model of cpos and continuous functions if one restricts to total programs and total observations. The ..."
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ion, Totality and PCF Gordon Plotkin Abstract Inspired by a question of Riecke, we consider the interaction of totality and full abstraction, asking whether full abstraction holds for Scott's model of cpos and continuous functions if one restricts to total programs and total observations. The answer is negative, as there are distinct operational and denotational notions of totality. However, when two terms are each total in both senses then they are totally equivalent operationally iff they are totally equivalent in the Scott model. Analysing further, we consider sequential and parallel versions of PCF and several models: Scott's model of continuous functions, Milner's fully abstract model of PCF and their effective submodels. We investigate how totality differs between these models. Some apparently rather difficult open problems arise, essentially concerning whether the sequential and parallel versions of PCF have the same expressive power, in the sense of total equivale...
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
Total Functionals and Wellfounded Strategies (Extended Abstract)
, 1999
"... In existing game models, total functionals have no simple characterization neither in term of game strategies, nor in term of the total settheoretical functionals they define. We show that the situation changes if we extend the usual notion of game by allowing infinite plays. Total functionals a ..."
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In existing game models, total functionals have no simple characterization neither in term of game strategies, nor in term of the total settheoretical functionals they define. We show that the situation changes if we extend the usual notion of game by allowing infinite plays. Total functionals are
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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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 ...