Imperative Programs as Proofs via Game Semantics

Abstract

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 thesis 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 a novel 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 expressive imperative total programming language. We can use the first-order structure to express properties on the imperative programs. 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 and faithful 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. The proof system makes novel use of the fact that the sequoid operator allows the exponential modality of linear logic to be expressed as a final coalgebra.

Date of Award	1 Oct 2011
Original language	English
Awarding Institution	University of Bath
Supervisor	Guy McCusker (Supervisor)