Extracting Herbrand systems from refutation schemata

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 cut-elimination by resolution, can be used to analyse 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 introduce a new calculus for the refutation of formula schemata that is simpler and more expressive than previous formalisms. Finally, we show that this new formalism allows the extraction of a structure from the refutation schema, called a Herbrand system, which represents its Herbrand sequent.