Herbrand's Theorem in Refutation Schemata

TL;DR

This paper develops a new formal framework for schematic substitutions and unification algorithms to analyze inductive proofs via schematic cut-elimination, enabling extraction of Herbrand schemata.

Contribution

It introduces a novel formalism for schematic substitutions and unification, facilitating Herbrand schema extraction from proof schemata with induction.

Findings

Developed a unification algorithm for resolution schemata.

Constructed a framework for schematic substitutions.

Enabled extraction of Herbrand schemata from refutation schemas.

Abstract

An inductive proof can be represented as a proof schema, i.e. as a parameterized sequence of proofs defined in a primitive recursive way. A corresponding cut-elimination method, called schematic CERES, can be used to analyze these proofs, and to extract their (schematic) Herbrand sequents, even though Herbrand's theorem in general does not hold for proofs with induction inferences. This work focuses on the most crucial part of the schematic cut-elimination method, which is to construct a refutation of a schematic formula that represents the cut-structure of the original proof schema. We develop a new framework for schematic substitutions and define a unification algorithm for resolution schemata. Moreover, we show that this new formalism allows the extraction of a structure from the refutation schema, called a Herbrand schema, which represents its Herbrand sequent.