### Abstract

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.

Language | English |
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Pages | 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 | |

Status | Published - Nov 2013 |

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### Cite this

**Imperative programs as proofs via game semantics.** / Churchill, Martin; Laird, James; McCusker, Guy.

Research output: Contribution to journal › Article

*Annals of Pure and Applied Logic*, vol. 164, no. 11, pp. 1038-1078. DOI: 10.1016/j.apal.2013.05.005

}

TY - JOUR

T1 - Imperative programs as proofs via game semantics

AU - Churchill,Martin

AU - Laird,James

AU - McCusker,Guy

N1 - Special issue on Seventh Workshop on Games for Logic and Programming Languages (GaLoP VII)

PY - 2013/11

Y1 - 2013/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84880699170&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1016/j.apal.2013.05.005

U2 - 10.1016/j.apal.2013.05.005

DO - 10.1016/j.apal.2013.05.005

M3 - Article

VL - 164

SP - 1038

EP - 1078

JO - Annals of Pure and Applied Logic

T2 - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

SN - 0168-0072

IS - 11

ER -