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### 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 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.

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 language | English |
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Pages (from-to) | 1038-1078 |

Number of pages | 41 |

Journal | Annals of Pure and Applied Logic |

Volume | 164 |

Issue number | 11 |

Early online date | 18 Jun 2013 |

DOIs | |

Publication status | Published - Nov 2013 |

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## Projects

- 1 Finished

### Semantic Structures for Higher-Order Information Flow

Engineering and Physical Sciences Research Council

20/06/10 → 19/06/12

Project: Research council

## Cite this

Churchill, M., Laird, J., & McCusker, G. (2013). Imperative programs as proofs via game semantics.

*Annals of Pure and Applied Logic*,*164*(11), 1038-1078. https://doi.org/10.1016/j.apal.2013.05.005