Describing proofs by short tautologies

Abstract

Herbrand’s theorem is one of the most fundamental results about first-order logic. In the context of proof analysis, Herbrand-disjunctions are used for describing the constructive content of cut-free proofs. However, given a proof with cuts, the computation of a Herbrand-disjunction is of significant computational complexity, as the cuts in the proof have to be eliminated first. In this paper we prove a generalization of Herbrand’s theorem: From a proof with cuts, one can read off a small (linear in the size of the proof) tautology composed of instances of the end-sequent and the cut formulas. This tautology describes the proof in the following way: Each cut induces a (propositional) formula stating that a disjunction of instances of the cut formula implies a conjunction of instances of the cut formula. All these cut-implications together then imply the already existing instances of the end-sequent. The proof that this formula is a tautology is carried out by transforming the instances in the proof to normal forms and using characteristic clause sets to relate them. These clause sets have first been studied in the context of cut-elimination. This extended Herbrand theorem is then applied to cut-elimination sequences in order to show that, for the computation of an Herbrand-disjunction, the knowledge of only the term substitutions performed during cut-elimination is already sufficient.