Imperative programs as proofs via game semantics

Martin Churchill, James Laird, Guy McCusker

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Game semantics extends the Curry–Howard isomorphism to a three-way correspondence: proofs, programs, strategies. But the universe of strategies goes beyond intuitionistic logics and lambda calculus, to capture stateful programs. In this paper we describe a logical counterpart to this extension, in which proofs denote such strategies. The system is expressive: it contains all of the connectives of Intuitionistic Linear Logic, and first-order quantification. Use of Lairdʼs sequoid operator allows proofs with imperative behaviour to be expressed. Thus, we can embed first-order Intuitionistic Linear Logic into this system, Polarized Linear Logic, and an imperative total programming language.

The proof system has a tight connection with a simple game model, where games are forests of plays. Formulas are modelled as games, and proofs as history-sensitive winning strategies. We provide a strong full completeness result with respect to this model: each finitary strategy is the denotation of a unique analytic (cut-free) proof. Infinite strategies correspond to analytic proofs that are infinitely deep. Thus, we can normalise proofs, via the semantics.
Original languageEnglish
Pages (from-to)1038-1078
Number of pages41
JournalAnnals of Pure and Applied Logic
Issue number11
Early online date18 Jun 2013
Publication statusPublished - Nov 2013


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