Semantics of interaction

Abstract

The “classical” paradigm for denotational semantics models data types as domains, i.e. 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, computational complexity, and optimality of reduction strategies, have either not been captured at all denotationally, or not in a fully satisfactory fashion. Moreover, once languages with features beyond the purely functional are considered, the appropriateness of modelling programs by functions is increasingly open to question. Neither concurrency nor “advanced” imperative features such as local references have been captured denotationally in a fully convincing fashion. This analysis suggests a desideratum of Intensional Semantics, interpolating between denotational and operational semantics as traditionally conceived. This should combine the good mathematical structural properties of denotational semantics with the ability to capture dynamical aspects and to embody computational intuitions of operational semantics. Thus we may think of Intensional semantics as “Denotational semantics + time (dynamics)”, or as “Syntax-free operational semantics”. A number of recent developments (and, with hindsight, some older ones) can be seen as contributing to this goal of Intensional Semantics. We will focus on the recent work on Game semantics, which has led to some striking advances in the Full Abstraction problem for PCF and other programming languages (Abramsky et al. 1995) (Abramsky and McCusker 1995) (Hyland and Ong 1995) (McCusker 1996a) (Ong 1996). Our aim is to give a genuinely elementary first introduction; we therefore present a simplified version of game semantics, which nonetheless