Least fixpoints as meanings of recursive definitions.

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...

This paper presents two basic results on game-based semantics of FPC, a metalanguage with sums, products, exponentials and recursive types. First we give an axiomatic account of the category of games G introduced in [15], offering a fundamental structural analysis of the category as well as a transparent way to prove computational adequacy. As a consequence we obtain an intensional fullabstraction result through a standard definability argument. Next we extend the category G by introducing a category of games G i with optimised strategies; we show that the denotational semantics in G i gives a compilation of FPC terms into core Pict codes (the asynchronous polyadic -calculus without summation). The process representation follows a pioneering idea of Hyland and Ong [18]. However, we advance their representation by introducing semantically well-founded optimisation techniques; we also exte...

\Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes of domains. In previous work, the present authors introduced the notion of an I-category as an abstraction and enhancement of this idea, with emphasis on the solution of domain equations of the form D = F (D), with F a functor. An important feature of the work is that we are not conned to domains of computation as usually understood; other classes of spaces, more familiar to mathematicians in general, become also accessible. Here we present the idea in terms of what we call information categories, which are concrete I-categories in which the objects are structured sets of \tokens" and morphisms are relations between tokens. This is more in the spirit of information system work, and...