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The True Concurrency of Herbrand’s Theorem

Abstract : Herbrand's theorem, widely regarded as a cornerstone of proof theory, exposes some of the constructive content of classical logic. In its simplest form, it reduces the validity of a first-order purely existential formula to that of a finite disjunction. In the general case, it reduces first-order validity to propositional validity, by understanding the structure of the assignment of first-order terms to existential quantifiers, and the causal dependency between quantifiers. In this paper, we show that Herbrand's theorem in its general form can be elegantly stated and proved as a theorem in the framework of concurrent games, a denotational semantics designed to faithfully represent causality and independence in concurrent systems, thereby exposing the concurrency underlying the computational content of classical proofs. The causal structure of concurrent strategies, paired with annotations by first-order terms, is used to specify the dependency between quantifiers implicit in proofs. Furthermore concurrent strategies can be composed, yielding a compositional proof of Herbrand's theorem, simply by interpreting classical sequent proofs in a well-chosen denotational model.
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https://hal.archives-ouvertes.fr/hal-01886968
Contributor : Pierre Clairambault Connect in order to contact the contributor
Submitted on : Wednesday, October 3, 2018 - 2:22:21 PM
Last modification on : Thursday, September 29, 2022 - 2:58:07 PM
Long-term archiving on: : Friday, January 4, 2019 - 2:06:23 PM

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Aurore Alcolei, Pierre Clairambault, Martin Hyland, Glynn Winskel. The True Concurrency of Herbrand’s Theorem. 27th EACSL Annual Conference on Computer Science Logic (CSL 2018), Sep 2018, Birmingham, United Kingdom. ⟨10.4230/LIPIcs.CSL.2018.5⟩. ⟨hal-01886968⟩

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