Pi-calculus, dialogue games and PCF

J. Martin E. Hyland
C.-H. Luke Ong

In Proceedings of 7th Annual ACM Conference on Functional Programming Languages and Computer Architecture (La Jolla, California, June 26--28, 1995). ACM - Association for Computing Machinery, 1995

Available as PostScript.


Game semantics is an unusual denotational semantics in that it captures the intensional (or algorithmic) and dynamical aspects of the computation. This makes it an ideal semantical framework in which to seek to unify analyses of both the qualitative (correctness) as well as the quantitative (efficiency) properties of programming languages. This paper reports work arising from a recent construction of an order (or inequationally) fully abstract model for Scott's functional programming language PCF based on a kind of two-person (Player and Opponent) dialogue game of questions and answers [HO94]. In this model types are interpreted as games and terms as innocent strategies. The fully abstract game model may be said to be canonical for the semantical analysis of sequential functional languages. Unfortunately even for relatively simple PCF-terms, precise description of their denotations as strategies in [HO94] very rapidly becomes unwieldy and opaque. What is needed to remedy the situation is an expressive formal language which lends itself to a succinct and economical representation of innocent strategies. In this paper we give just such a representation in terms of an appropriately sorted polyadic -calculus, reading input -actions as Opponent's moves, and output -actions as Player's moves. This correspondence captures every essential aspect of the dialogue game paradigm so precisely that the -representation may as well be taken to be the basis for its formal definition. Although the -representation of strategies already gives an encoding of PCF in the -calculus --- indirectly via the fully abstract denotation of PCF-terms as innocent strategies, we define by recursion a new encoding of PCF in the -calculus directly. The two -encodings of PCF are weakly bisimilar; if the latter is regarded as a compilation, then the former amounts to a more efficient version with compile-time optimization.
J. M. E. Hyland and C.-H. L. Ong. On full abstraction for PCF: I, II and III. preliminary version, 110 pages, ftp-able at theory.doc.ic.ac.uk in directory papers/Ong, 1994. Alternatively get it, [FA-HO94], here.

Authors addresses: DPMMS, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB ENGLAND
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD ENGLAND and DISCS, National University of Singapore.

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