On Herbrand Skeletons

TL;DR

This paper investigates the computational bounds related to Herbrand skeletons and demonstrates that certain expected algorithms for automated theorem proving cannot exist, highlighting fundamental limitations in proof complexity.

Contribution

The paper proves that there are no computable bounds for the size of terms in Herbrand skeletons and that algorithms avoiding existential guesses in theorem proving are impossible.

Findings

No computable bounds for Herbrand skeletons' term sizes.

Existence of algorithms avoiding existential guesses is impossible.

Fundamental limitations in proof complexity and automated theorem proving.

Abstract

Herbrand's theorem plays an important role both in proof theory and in computer science. Given a Herbrand skeleton, which is basically a number specifying the count of disjunctions of the matrix, we would like to get a computable bound on the size of terms which make the disjunction into a quasitautology. This is an important problem in logic, specifically in the complexity of proofs. In computer science, specifically in automated theorem proving, one hopes for an algorithm which avoids the guesses of existential substitution axioms involved in proving a theorem. Herbrand's theorem forms the very basis of automated theorem proving where for a given number $n$ we would like to have an algorithm which finds the terms in the $n$ disjunctions of matrices solely from the shape of the matrix. The main result of this paper is that both problems have negative solutions.