Abstract
Game semantics extends the CurryHoward isomorphism to a threeway 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 firstorder quantification. Use of a novel sequoid operator allows proofs with imperative behaviour to be expressed. Thus, we can embed firstorder Intuitionistic Linear Logic into this system, Polarized Linear Logic, and an expressive imperative total programming language. We can use the firstorder 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 historysensitive 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 (cutfree) 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.
Language  English 

Qualification  Ph.D. 
Awarding Institution 

Supervisors/Advisors 

Award date  1 Oct 2011 
Status  Unpublished  15 Feb 2012 
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Imperative Programs as Proofs via Game Semantics. / Churchill, Martin.
2012. 251 p.Research output: Thesis › Doctoral Thesis
}
TY  THES
T1  Imperative Programs as Proofs via Game Semantics
AU  Churchill, Martin
PY  2012/2/15
Y1  2012/2/15
N2  Game semantics extends the CurryHoward isomorphism to a threeway 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 firstorder quantification. Use of a novel sequoid operator allows proofs with imperative behaviour to be expressed. Thus, we can embed firstorder Intuitionistic Linear Logic into this system, Polarized Linear Logic, and an expressive imperative total programming language. We can use the firstorder 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 historysensitive 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 (cutfree) 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.
AB  Game semantics extends the CurryHoward isomorphism to a threeway 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 firstorder quantification. Use of a novel sequoid operator allows proofs with imperative behaviour to be expressed. Thus, we can embed firstorder Intuitionistic Linear Logic into this system, Polarized Linear Logic, and an expressive imperative total programming language. We can use the firstorder 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 historysensitive 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 (cutfree) 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.
M3  Doctoral Thesis
ER 