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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 variable-free form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call eectively almost-everywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a -model D...

Abstract. We study orderings ✂S on reductions in the style of Lévy reflecting the growth of information w.r.t. (super)stable sets S of ‘values’ (such as head-normal forms or Böhm-trees). We show that sets of co-initial reductions ordered by ✂S form finitary ω-algebraic complete lattices, and hence form computation and Scott domains. As a consequence, we obtain a relativized version of the computational semantics proposed by Boudol for term rewriting systems. Furthermore, we give a pure domain-theoretic characterization of the orderings ✂S in the spirit of Kahn and Plotkin’s concrete domains. These constructions are carried out in the framework of Stable Deterministic Residual Structures, which are abstract reduction systems with an axiomatized residual relations on redexes, that model all orthogonal (or conflict-free) reduction systems as well as many other interesting computation structures. 1

We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating -terms (Bohm trees, Levy-Longo 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. Keywords Bohm trees, approximants, intersection types. 1 INTRODUCTION A theory of functions like the -calculus, which provides a foundation for the functional programming paradigm in computer science, can be seen essentially as a theory of "programs". This point of view leads naturally to the intuitive idea that the meaning of a -term (program) is represented by the amount of "meaningful information " we can extract from the term by "running it". The formalization of "the information"...

We study normalization by neededness with respect to ‘infinite results’, such as Böhm-trees, in an abstract framework of Stable Deterministic Residual Structures. We formalize the concept of ‘infinite results ’ for finite terms as suitable sets of infinite reductions, and prove an abstract infinitary normalization theorem with respect to such sets. We also give a sufficient and necessary condition for existence of minimal normalizing reductions. 1

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 variable-free form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call eectively almost-everywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a ...

We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating -terms (Bohm trees, L'evy-Longo 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: Bohm trees, approximants, intersection types. 1 Introduction A theory of functions like the -calculus, which provides a foundation for the functional programming paradigm in computer science, can be seen essentially as a theory of "programs". This point of view leads naturally to the intuitive idea that the meaning of a -term (program) is represented by the amount of "meaningful information" we can extract from the term by "running it". The formalization of "the information"...

This thesis is a detailed examination of the application of game semantics to constructing denotational models of the pure untyped λ-calculus. Game semantics is a fairly recent technique, using a formal setting for interaction to model sequential programming languages in an accurate way. syntax-independent 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 λη-algebras 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 λ-models. 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

Abstract. A closed λ-term 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 pre-order 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 co-r.e. (complement of a recursively enumerable) set of easy, but non simple easy, λ-terms. Key words: Lambda calculus, easy terms, simple easy terms, filter models 1