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Stable computational semantics of conflictfree rewrite systems (Draft). Available at http://www.sys.uea.ac.uk/~zurab
, 2000
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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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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...
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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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.
Khasidashvili Z. An abstract Böhmnormalization. in
 Proc. WRS’02, Electronic Notes in Computer Science, Elsevier Science B.V
, 2002
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Types for Trees
 IFIP 1996
, 1996
"... We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating terms (Bohm trees, LevyLongo 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 terms (Bohm trees, LevyLongo 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.
Problem 19
"... Abstract. A closed λterm M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent, while it is simple easy if, given an arbitrary intersection type τ, one can find a suitable preorder on types which allows to derive τ for M. Simple easiness implies easiness. Th ..."
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Abstract. A closed λterm M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent, while it is simple easy if, given an arbitrary intersection type τ, one can find a suitable preorder on types which allows to derive τ for M. Simple easiness implies easiness. The question whether easiness implies simple easiness constitutes Problem 19 in the TLCA list of open problems. In this paper we negatively answer the question providing a nonempty cor.e. (complement of a recursively enumerable) set of easy, but non simple easy, λterms. Key words: Lambda calculus, easy terms, simple easy terms, filter models 1
On the equational consistency of ordertheoretic models of the λcalculus
"... Answering a question by Honsell and Plotkin, we show that there are two equations between λterms, the socalled subtractive equations, consistent with λcalculus but not satisfied in any partially ordered model with bottom element. We also relate the subtractive equations to the open problem of the ..."
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Answering a question by Honsell and Plotkin, we show that there are two equations between λterms, the socalled subtractive equations, consistent with λcalculus but not satisfied in any partially ordered model with bottom element. We also relate the subtractive equations to the open problem of the orderincompleteness of λcalculus.
Programming Research Group
"... This thesis is a detailed examination of the application of game semantics to constructing denotational models of the pure untyped λcalculus. Game semantics is a fairly recent technique, using a formal setting for interaction to model sequential programming languages in an accurate way. syntaxinde ..."
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This thesis is a detailed examination of the application of game semantics to constructing denotational models of the pure untyped λcalculus. Game semantics is a fairly recent technique, using a formal setting for interaction to model sequential programming languages in an accurate way. syntaxindependent model of PCF; the only difference is that in our setting the distinction between “question ” and “answer ” moves is removed. Many of the standard results for PCF games carry through into this setting. Cartesian closed categories of arenas and innocent strategies are constructed, leading to ληalgebras D and DREC. By a method of approximation, these are shown to be sensible models (i.e. all unsolvable terms are equated) but they contain many undefinable elements and are not λmodels. By introducing a new “economical ” representation of innocent strategies we are able to prove a precise syntactic connexion between a term and its denotation. This