TY - GEN

T1 - The true concurrency of Herbrand's theorem

AU - Alcolei, Aurore

AU - Clairambault, Pierre

AU - Hyland, Martin

AU - Winskel, Glynn

PY - 2018/8/29

Y1 - 2018/8/29

N2 - 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.

AB - 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.

KW - game semantics

KW - Herbrand's theorem

KW - true concurrency

U2 - 10.4230/LIPIcs.CSL.2018.5

DO - 10.4230/LIPIcs.CSL.2018.5

M3 - Conference contribution book

AN - SCOPUS:85053038265

SN - 9783959770880

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 5:1-5:22

BT - 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

A2 - Ghica, Dan R.

A2 - Jung, Achim

CY - Dagstuhl, Germany

T2 - 27th Annual EACSL Conference Computer Science Logic, CSL 2018

Y2 - 4 September 2018 through 7 September 2018

ER -