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 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 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. (shrink)

Sufficient conditions for first-order-based sequent calculi to admit cut elimination by a Schütte–Tait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related to the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal (...) logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus. (shrink)

We present sharpened lower bounds on the size of cut free proofs for first-order logic. Prior lower bounds for eliminating cuts from a proof established superexponential lower bounds as a stack of exponentials, with the height of the stack proportional to the maximum depth d of the formulas in the original proof. Our results remove the constant of proportionality, giving an exponential stack of height equal to d — 0(1). The proof method is based on more efficiently expressing the Gentzen-Solovay (...) cut formulas as low depth formulas. (shrink)

We give examples of calculi that extend Gentzen’s sequent calculusLKby unsound quantifier inferences in such a way that derivations lead only to true sequents, and proofs therein are nonelementarily shorter thanLK-proofs.

We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cut-free proofs we prove corresponding exponential upper and lower bounds.