The manipulation of objects with state which changes over time is allpervasive in computing. Perhaps the simplest example of such objects are the program variables of classical imperative languages. An important strand of work within the study of such languages, pioneered by John Reynolds, focusses on "Idealized Algol", an elegant synthesis of imperative and functional features. We present a novel semantics for Idealized Algol using games, which is quite unlike traditional denotational models of state. The model takes into account the irreversibility of changes in state, and makes explicit the difference between copying and sharing of entities. As a formal measure of the accuracy of our model, we obtain a full abstraction theorem for Idealized Algol with active expressions.

This paper considers the consequences of relaxing the bracketing condition on `dialogue games', showing that this leads to a category of games which can be `factorized' into a well-bracketed substructure, and a set of classically typed morphisms. These are shown to be sound denotations for control operators, allowing the factorization to be used to extend the definability result for PCF to one for PCF with control operators at atomic types. Thus we define a fully abstract and effectively presentable model of a functional language with non-local control as part of a modular approach to modelling non-functional features using games. 1.

Introduction SAMSON ABRAMSKY (samson@comlab.ox.ac.uk) Oxford University Computing Laboratory 1. Introduction Game Semantics has emerged as a powerful paradigm for giving semantics to a variety of programming languages and logical systems. It has been used to construct the first syntax-independent fully abstract models for a spectrum of programming languages ranging from purely functional languages to languages with non-functional features such as control operators and locally-scoped references [4, 21, 5, 19, 2, 22, 17, 11]. A substantial survey of the state of the art of Game Semantics circa 1997 was given in a previous Marktoberdorf volume [6]. Our aim in this tutorial presentation is to give a first indication of how Game Semantics can be developed in a new, algorithmic direction, with a view to applications in computer-assisted verification and program analysis. Some promising steps have already been taken in this

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,

ion for Idealized Algol with Passive Expressions Samson Abramsky University of Edinburgh Department of Computer Science James Clerk Maxwell Building Edinburgh EH9 3JZ Scotland samson@dcs.ed.ac.uk Guy McCusker St John's College Oxford OX1 3JP, England mccusker@comlab.ox.ac.uk Abstract A fully abstract games model of Reynolds' Idealized Algol is described. The model gives a semantic account of the distinction between active types, such as commands, which admit side-effecting behaviour, and passive types, such as expressions, which do not. Keywords: Algol-like languages, game semantics, full abstraction. 1 Introduction Our aim in this paper is to give the first syntax-independent construction of a fully abstract model for Idealized Algol. John Reynolds proposed Idealized Algol as capturing the essence of Algol 60 [32]; it is an elegant synthesis of the features of a simple block-structured imperative programming language with those of higher-order functional programming. As such it...

This thesis examines the use of denotational semantics to reason about control flow in sequential, basically functional languages. It extends recent work in game semantics, in which programs are interpreted as strategies for computation by interaction with an environment. Abramsky has suggested that an intensional hierarchy of computational features such as state, and their fully abstract models, can be captured as violations of the constraints on strategies in the basic functional model. Non-local control flow is shown to fit into this framework as the violation of strong and weak ‘bracketing ’ conditions, related to linear behaviour. The language µPCF (Parigot’s λµ with constants and recursion) is adopted as a simple basis for higher-type, sequential computation with access to the flow of control. A simple operational semantics for both call-by-name and call-by-value evaluation is described. It is shown that dropping the bracketing condition on games models of PCF yields fully abstract models of µPCF.

We give an axiomatic treatment of fixed-point operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixed-point operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !-continuous functions between !-complete pointed partial orders. This possesses a least-fixed-point oper...

We explain how recent developments in game semantics can be applied to reasoning about equivalence of terms in a non-trivial fragment of Idealized Algol (IA) by expressing sets of complete plays as regular languages. Being derived directly from the fully abstract game semantics for IA, our model inherits its good theoretical properties; in fact, for second-order IA taken as a stand-alone language the regular language model is fully abstract. The method is algorithmic and formal, which makes it suitable for automation. We show how reasoning is carried out using a meta-language of extended regular expressions, a language for which equivalence is decidable.

We explain how game semantics can be used to reason about term equivalence in a finitary imperative first order language with arrays. For this language, the game-semantic interpretation of types and terms is fully characterized by their sets of complete plays. Because these sets are regular over the alphabet of moves, they are representable by (extended) regular expressions. The formal apparatus of game semantics is greatly simplified but the good theoretical properties of the model are preserved. The principal advantage of this approach is that it is mathematically elementary, while fully formalized. Since language equivalence for regular languages is decidable, this method of proving term equivalence is suitable for automation.

Important progresses in logic are leading to interactive and dynamical models. Geometry of Interaction and Games Semantics are two major examples. Ludics, initiated by Girard, is a further step in this direction. The objects