The true concurrency of Herbrand's theorem

Aurore Alcolei, Pierre Clairambault, Martin Hyland, Glynn Winskel

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

4 Citations (Scopus)
2 Downloads (Pure)


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.

Original languageEnglish
Title of host publication27th EACSL Annual Conference on Computer Science Logic (CSL 2018)
EditorsDan R. Ghica, Achim Jung
Place of PublicationDagstuhl, Germany
Number of pages22
Publication statusPublished - 29 Aug 2018
Event27th Annual EACSL Conference Computer Science Logic, CSL 2018 - Birmingham, United Kingdom
Duration: 4 Sep 20187 Sep 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference27th Annual EACSL Conference Computer Science Logic, CSL 2018
Country/TerritoryUnited Kingdom


  • game semantics
  • Herbrand's theorem
  • true concurrency


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