ion for the Lazy -calculus Samson Abramsky Guy McCusker Department of Computing Imperial College of Science, Technology and Medicine 180 Queen's Gate London SW7 2BZ United Kingdom Abstract We define a category of games G, and its extensional quotient E . A model of the lazy -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 far as we know, the first purely semantic construction of a fully abstract model for a reflexively-typed sequential language. 1 Introduction Full Abstraction is a key concept in programming language semantics [9, 12, 23, 26]. The ingredients are as follows. We are given a language L, with an `observational preorder' - on terms in L such that P - Q means that every observable property of P is also satisfied by Q; and a denotational model MJ\DeltaK. The model M is then said to be f...

The “classical ” paradigm for denotational semantics models data types as domains, ��� � structured sets of some kind, and programs as (suitable) functions between domains. The semantic universe in which the denotational modelling is carried out is thus a category with domains as objects, functions as morphisms, and composition of morphisms given by function composition. A sharp distinction is then drawn between denotational and operational semantics. Denotational semantics is often referred to as “mathematical semantics ” because it exhibits a high degree of mathematical structure; this is in part achieved by the fact that denotational semantics abstracts away from the dynamics of computation—from time. By contrast, operational semantics is formulated in terms of the syntax of the language being modelled; it is highly intensional in character; and it is capable of expressing the dynamical aspects of computation. The classical denotational paradigm has been very successful, but has some definite limitations. Firstly, fine-structural features of computation, such as sequentiality,

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This paper extends Curry-Howard interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec -calculus and the linear rec -calculus respectively, are given sound categorical interpretations. The embedding of proofs of IL into proofs of ILL given by the Girard Translation is extended with the rules for recursion, such that an embedding of terms of the rec -calculus into terms of the linear rec -calculus is induced via the extended Curry-Howard isomorphisms. This embedding is shown to be sound with respect to the categorical interpretations. Full version of paper to appear in Proceedings of CSL '94, LNCS 933, 1995. y Basic Research in Computer Science, Centre of the Danish National Research Foundation. Contents 1 Introduction 4 2 The Categorical Picture 6 2.1 Previous Work and Related Results : : : : : : : : : : : : : : : : : : : : : : 6 2.2 How to deal with parameters : : : : : : : ...

While Game Semantics has been remarkably successful at modelling, often in a fully abstract manner, a wide range of features of programming languages, there has to date been no attempt at applying it to subtyping. We show how the simple device of explicitly introducing error values in the syntax of the calculus leads to a notion of subtyping for game semantics. We construct an interpretation of a simple -calculus with subtyping and show how the range of the interpretation of types is a complete lattice thus yielding an interpretation of bounded quantification.

Genericity is the idea that the same program can work at many dierent data types. Longo, Milstead and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type 8X:A[X ], are equal at any given instance A[T ], then they are equal at all instances. They proved that this rule is admissible in a certain extension of System F, but nding a semantically motivated model satisfying this principle remained an open problem.

This paper shows how a fully abstract model for a rich metalanguage like FPC can be used to prove theorems about other languages. In particular, we use results obtained from a game semantics of FPC to show that the natural translation of the lazy -calculus into the metalanguage is fully abstract, thus obtaining a new full abstraction result from an old one. The proofs involved are very easy---all the hard work was done in giving the original games model. So far we have been unable to prove the completeness of our translation without recourse to the denotational model; we therefore have an indication of the worth of such fully abstract models. 1 Introduction Plotkin, in his CSLI notes [18], showed how denotational semantics can be viewed as a two-stage process. First one defines a metalanguage which describes elements of the intended semantic model, usually some category of domains. Then to give semantics to a language L it suffices to translate it into the metalanguage. While this is ...

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

In this paper we present a fully abstract game model for the pure lazy -calculus, i.e. the lazy -calculus without constants. In order to obtain this result we introduce a new category of games, the monotonic games, whose main characteristic consists in having an order relation on moves.

After informally reviewing the main concepts from game semantics and placing the development of the field in a historical context we examine its main applications. We focus in particular on finite state model checking, higher order model checking and more recent developments in hardware design. 1. Chronology, methodology, ideology Game Semantics is a denotational semantics in the conventional sense: for any term, it assigns a certain mathematical object as its meaning, which is constructed compositionally from the meanings of its sub-terms in a way that is independent of the operational semantics of the object language. What makes Game Semantics particular, peculiar maybe, is that the mathematical objects it operates with