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A compositional account of Herbrand's theorem via concurrent games

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Herbrand’s theorem, often 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 ∃x.φ(x) (with φ quantifier-free) to that of a finite disjunction φ(t₁) ∨ … ∨ φ(tₙ). More generally, it gives a reduction of 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 talk, I will show that Herbrand’s theorem in its general form can be elegantly stated as a theorem in the framework of concurrent games. The causal structure of concurrent strategies, paired with annotations by first-order terms, is used to specify the dependency between quantifiers. 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. I assume no prior knowledge of Herbrand’s theorem or concurrent games.

This is joint work with Aurore Alcolei, Martin Hyland and Glynn Winskel.

This talk is part of the Logic and Semantics Seminar (Computer Laboratory) series.

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