Game Semantics for the Pure Lazy lambda-calculus

1995

Abstract We define a category of games 𝒢, and its extensional quotient ℰ. A model of the lazy X-calculus, a type-free functional language based on evaluation to weak head normal form, is given in 𝒢, yielding an extensional model in ℰ. This model is shown to be fully abstract with respect to applicative simulation. This is, so fear as we known, the first purely semantic construction of a fully abstract model for a reflexively-typed sequential language

Lecture Notes in Computer Science, 1999

We study extensional models of the untyped lambda calculus in the setting of the game semantics introduced by Abramsky, Hyland et alii. In particular we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in a standard category of games, induce the same λ-theory. This is H * , the maximal theory induced already by the classical C.P.O. model D∞, introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction.

1999

Abstract. We study extensional models of the untyped lambda calculus in the setting of the game semantics introduced by Abramsky, Hyland et alii. In particular we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in a standard category of games, induce the same λ-theory. This is H∗, the maximal theory induced already by the classical C.P.O. model D∞, introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction.

. We study extensional models of the untyped lambda calculus in the setting of game semantics. In particular, we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in the category of games G, introduced by Abramsky, Jagadeesan and Malacaria, induce the same -theory. This is H , the maximal theory induced already by the classical CPO model D1 , introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction. Introduction -theories are congruences over -terms, which extend pure fi-conversion. Their interest lies in the fact that they correspond to the possible operational (obser- vational) semantics of -calculus. Although researchers have mainly focused on only three such operational semantics, namely those given by head reduction, head lazy reduction or call-by-value reduction, the class of -theories is, in effect, unfathomly rich, see e.g. [6...

2015

Into A-calculus we introduce lazy lsts $\tilde{a} $ whose naive meaning is an infinite tit consisting of variables, ($a_{0}, $ $a_{1} $ , a2, $\ldots$). It is shown that there exist maps which form aGalois connection ffom Parigot’s $\mathrm{A}/\mathrm{i}$-calculus to the A-calculus with lazy list. The translations form not only an equational correspondence but also areduction corre-spondence between the two calculi. 1Introduction We introduce lazy lists into A-calculus. The introduction of infinite lists is motived by a study on denotational semantics of type-free Ap-calculus [Pari92, Pari97, BHF99, BHFOI]. Given domains $U\mathrm{x}U\cong U\cong[Uarrow U] $ such as in Lambek-Scott [LS86], we have established acontinuation denotational semantics of type-free $\lambda\mu$-calculus[Fuji02], which formally coincides with the CPS-translation [HS97, SR98, FujiOl] followed by the direct denotational semantics of the A-calculus [ScOt72, StOy77]. See also the literature [HS97, SR98, SeliOl]...

1994

Abstract: In this paper we define the Lazy Lambda Calculus with constants, which extends Abramsky's pure lazy Lambda Calculus. This calculus forms a model for modern lazy functional programming languages. Such languages usually provide a call-by-value facility which is able to distinguish between the values _| _ and\ x. _| _. We study the operational and denotational semantics of this calculus both with and without a superimposed type inference system.

1993

this paper an alternative description of the game semantics for the untyped lambda calculus is given. More precisely, we introduce a finitary description of lambda terms. This description turns out to be equivalent to a particular game denotational semantics of the lambda calculus. Introduction

Lecture Notes in Computer Science, 2009

We recently introduced an extensional model of the pure λ-calculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the non-deterministic features of this model. Unlike most traditional approaches, our way of interpreting non-determinism does not require any additional powerdomain construction: we show that our model provides a straightforward semantics of non-determinism (may convergence) by means of unions of interpretations as well as of parallelism (must convergence) by means of a binary, non-idempotent, operation available on the model, which is related to the mix rule of Linear Logic. More precisely, we introduce a λ-calculus extended with non-deterministic choice and parallel composition, and we define its operational semantics (based on the may and must intuitions underlying our two additional operations). We describe the interpretation of this calculus in our model and show that this interpretation is sensible with respect to our operational semantics: a term converges if, and only if, it has a non-empty interpretation.

We recently introduced an extensional model of the pure λ-calculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the non-deterministic features of this model. Unlike most traditional approaches, our way of interpreting non-determinism does not require any additional powerdomain construction: we show that our model provides a straightforward semantics of non-determinism (may convergence) by means of unions of interpretations as well as of parallelism (must convergence) by means of a binary, non-idempotent, operation available on the model, which is related to the mix rule of Linear Logic. More precisely, we introduce a λ-calculus extended with non-deterministic choice and parallel composition, and we define its operational semantics (based on the may and must intuitions underlying our two additional operations). We describe the interpretation of this calculus in our model and show that this interpretation is sensible with respect to our operational semantics: a term converges if, and only if, it has a non-empty interpretation.

Biosystems, 2005

This paper presents a new game system formalism. The system describes both strategies and a game master (who computes scores in a given game system) in terms of λ-calculus. This formalism revisits the prisoner's dilemma game, to discuss how meta-strategies emerge in this classical game, even without repetition. We have also examined the evolution of meta-strategies in λ formalism.