Guy McCusker

A fully abstract game semantics for general references

A games model of a programming language with higher-order store in the style of ML-references is introduced. The category used for the model is obtained by relaxing certain behavioural conditions on a category of games previously used to provide fully abstract models of pure functional languages. The model is shown to be fully abstract by means of factorization arguments which reduce the question of definability for the language with higher-order store to that for its purely functional fragment.

Call-by-Value Games

A general construction of models of call-by-value from models of call-by-name computation is described. The construction makes essential use of the properties of sum types in common denotational models of call-by-name. When applied to categories of games, it yields fully abstract models of the call-by-value functional language PCFv, which can be extended to incorporate recursive types, and of a language with local references as in Standard ML.

Games and Full Abstraction for a Functional Metalanguage with Recursive Types

Introduction.- Full Abstraction.- Game Semantics.- Historical Perspective.- Contribution of This Book.- Prerequisites.- Preliminaries.- Enriched Category Theory.- Intrinsic Preorder.- Games.- Arenas, Views and Legal Positions.- Games and Strategies.- The Category.- Exponential.- A Cartesian Closed Category.- An Alternative Category.- The Extensional Category.- Sums.- Lifting.- Rational Categories and Recursive Types.- Rational Categories.- Recursive Types.- Invariant Relations.- Parameterized Invariant Relations.- IP-Categories.- Axioms for Rationality.- FPC and its Models.- The Language FPC.- Models of FPC.- Semantics of the Recursion Combinator.- Formal Approximation Relations.- Computational Adequacy.- Full Abstraction.- Conclusions.

A fully abstract game semantics for finite nondeterminism

A game semantics of finite nondeterminism is proposed. In this model, a strategy may make a choice between different moves in a given situation; moreover, strategies carry extra information about their possible divergent behaviour. A Cartesian closed category is built and a model of a simple, higher-order nondeterministic imperative language is given. This model is shown to be fully abstract, with respect to an equivalence based on both safety and liveness properties, by means of a factorization theorem which states that every nondeterministic strategy is the composite of a deterministic strategy with a nondeterministic oracle.

Reasoning about Idealized ALGOL Using Regular Languages

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 method of reasoning inherits its desirable theoretical properties. The method is mathematically elementary and formal, which makes it uniquely suitable for automation. We show that reasoning can be carried out using only a meta-language of extended regular expressions, a language for which equivalence is formally decidable.

The regular-language semantics of second-order idealized ALGOL

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.

Full abstraction for idealized Algol with passive expressions

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.

Linearity, Sharing and State: a fully abstract game semantics for Idealized Algol with active expressions: Extended Abstract

Abstract The manipulation of objects with state which changes over time is all-pervasive 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, focuses 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.

Games and full abstraction for the lazy /spl lambda/-calculus

We define a category of games /spl Gscr/, and its extensional quotient /spl Escr/. A model of the lazy X-calculus, a type-free functional language based on evaluation to weak head normal form, is given in /spl Gscr/, yielding an extensional model in /spl Escr/. 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.

Weighted Relational Models of Typed Lambda-Calculi

The category Rel of sets and relations yields one of the simplest denotational semantics of Linear Logic (LL). It is known that Rel is the biproduct completion of the Boolean ring. We consider the generalization of this construction to an arbitrary continuous semiring R, producing a cpo-enriched category which is a semantics of LL, and its (co)Kleisli category is an adequate model of an extension of PCF, parametrized by R. Specific instances of R allow us to compare programs not only with respect to “what they can do”, but also “in how many steps” or “in how many different ways” (for non-deterministic PCF) or even “with what probability” (for probabilistic PCF).