Pasquale Malacaria and Chris Hankin Dept. of Computing Imperial College LONDON SW7 2BZ pm5,clh@doc.ic.ac.uk Abstract We present a unifying framework for using game semantics as a basis for program analysis. Also, we present a case study of the techniques. The unifying framework presents games-based program analysis as an abstract interpretation of an appropriate games category in the category of non-deterministic games. The case study concerns an application to security.

The denotational semantics approach to the semantics of programming languages interprets the language constructions by assigning elements of mathematical structures to them. The structures form so-called categories of domains and the study of their closure properties is the subject of domain theory [Sco70, Sco82, Plo83, GS90, AJ94]. Typically, categories of domains consist of suitably complete partially ordered sets together with continuous maps. But, what is a category of domains? The main aim of axiomatic domain theory is to answer this question by axiomatising the structure needed on a mathematical universe so that it can be considered a category of domains. Criteria required from categories of domains can be of the most varied sort. For example, we could ask them to * have a rich collection of type constructors: sums, products, exponentials, powerdomains, dependent types, polymorphic types, etc; * have fixed-point operators for programs and type constructors; * have only computable maps [Sco76, Smy77, Mul81, McC84, Ros86, Pho90, Lon95]; * have a Stone dual providing a logic of observable properties [Abr87, Vic89, Zha91]. An additional aim of the axiomatic approach is to relate these mathematical criteria with computational criteria. As we indicate below an axiomatic treatment of various of the above aspects is now available but much research remains to be done.

We present results concerning the solution of recursive domain equations in the category G of games, which is a modified version of the category presented in [AJM94]. New constructions corresponding to lifting and separated sum for games are presented, and are used to generate games for two simple recursive types: the vertical and lazy natural numbers. Recently, the "game semantics" paradigm has been used to model the multiplicative fragment of linear logic [AJ94], and to provide a solution to the full abstraction problem for PCF [AJM94, HO94], where the intensional structure of the games model captures both the sequential and functional nature of the language. In the light of these results, it is natural to ask whether recursive types can be handled in this setting. Here we show that they can: for a wide class of functors \Phi, including all of the usual type constructors, the equation D ' \Phi(D) has a (canonical) solution. In fact we solve this equation up to identity, and the solut...

We develop a control flow analysis algorithm for PCF based on game semantics. The analysis is closely related to Shivers' 0-CFA analysis and the algorithm is shown to be cubic. The game semantics basis for the algorithm means that it can be naturally extended to handle strict languages and languages with imperative features. These extensions are discussed in the paper. We sketch the correctness proof for the algorithm. We also illustrate an algorithm for computing k-limited CFA.

We introduce a stratified version of the coherent spaces model where an object is given by a sequence of coherent spaces. The intuition behind it is that each level gives a di#erent degree of precision on the computation, an appearance. A morphism is required to satisfy a coherence condition at each level and this setting gives a model of Elementary Linear Logic. We then introduce a measure function on the web meant to describe the di#erence between the number of output and input requests in the computation. The locally bounded morphisms defined thanks to this measure give a subcategory which is a model of Light Linear Logic. 1

We construct a model for FPC, a purely functional, sequential, call-by-value language. The model is built from partial continuous functions, in the style of Plotkin, further constrained to be uniform with respect to a class of logical relations. We prove that the model is fully abstract.

ion and PCF John Longley Gordon Plotkin March 15, 1996 Abstract We introduce the concept of logical full abstraction, generalising the usual equational notion. We consider the language PCF and two extensions with "parallel" operations. The main result is that, for standard interpretations, logical full abstraction is equivalent to equational full abstraction together with universality; the proof involves constructing enumeration operators. We also consider restrictions on logical complexity and on the level of types. 1 Introduction The study of denotational semantics seeks to provide mathematical descriptions of programming languages by giving denotations of programs in terms of previously understood mathematical structures. For example, if P is a program that takes an input and produces an output, we might take its denotation to be a function from a set of input-values to a set of output-values. The most widely-known approach to denotational semantics is that of traditiona...

Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of low-level machine models. By contrast, we develop a more structural approach. We show how high-level functional programs can be mapped compositionally (i.e. in a syntax-directed fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic—but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis. 1

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.

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