The True Concurrency of Herbrand's Theorem

Authors Aurore Alcolei, Pierre Clairambault, Martin Hyland, Glynn Winskel

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Author Details

Aurore Alcolei
  • Univ Lyon, ENS de Lyon, CNRS, UCB Lyon 1, LIP, Lyon, France
Pierre Clairambault
  • Univ Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP, Lyon, France
Martin Hyland
  • DPMMS, University of Cambridge , Cambridge, United Kingdom
Glynn Winskel
  • Computer Laboratory, University of Cambridge, Cambridge, United Kingdom

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Aurore Alcolei, Pierre Clairambault, Martin Hyland, and Glynn Winskel. The True Concurrency of Herbrand's Theorem. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • Theory of computation → Denotational semantics
  • Herbrand's theorem
  • Game semantics
  • True concurrency


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