Extracting Herbrand Disjunctions by Functional Interpretation

Abstract

Carrying out a suggestion by Kreisel, we adapt Gödel's functional interpretation to ordinary first-order predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PL-proofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of two main steps: first we extract a functional realizer, next we compute the β-normal-form of the realizer from which the Herbrand terms can be read off. Even though the extraction is carried out in the extended language, the terms are ordinary PL-terms. In contrast to approaches to Herbrand's theorem based on cut elimination or ε-elimination this extraction technique is, except for the normalization step, of low polynomial complexity, fully modular and furthermore allows an analysis of the structure of the Herbrand terms, in the spirit of Kreisel ([13]), already prior to the normalization step. It is expected that the implementation of functional interpretation in Schwichtenberg's MINLOG system can be adapted to yield an efficient Herbrand-term extraction tool.

Gödel's Functional Interpretation and its Use in Current Mathematics*

“…As the latter can be viewed as a generalization of Herbrand's theorem it is natural to look also for an extraction algorithm based on functional interpretation of valid Herbrand disjunctions from proofs in classical logic. This was suggested already by G. Kreisel in Kreisel 1967 and carried out finally in Gerhardy and Kohlenbach 2005.…”

A herbrandized functional interpretation of classical first-order logic

“…, t k } and the above formula is which is obviously a tautology. The way our interpretation works for this example is somewhat different from the Gerhardy-Kohlenbach analysis in [9]. Compare also with Example 3 below.…”

“…The original plan of the paper was to state soundness with a semantical verification of the interpreting formulas. The verification would say that certain formulas of L mix are true in the finite-order set-theoretical structure naturally arising from a given first-order structure for L (see definition 6 of [9]). Actually, in [9], the verifications are done within a finite-order logical theory (of which the set-theoretical finiteorder stuctures are models).…”

“…The verification would say that certain formulas of L mix are true in the finite-order set-theoretical structure naturally arising from a given first-order structure for L (see definition 6 of [9]). Actually, in [9], the verifications are done within a finite-order logical theory (of which the set-theoretical finiteorder stuctures are models). When writing this paper, it dawned on the first author that the verification can be done in a particularly nice way, via the notion of tautology in character of Section 3.…”

“…In this paper, we consider classical first-order predicate logic without equality. This is also the calculus analyzed by Philipp Gerhardy and Ulrich Kohlenbach in [9]. Their paper presents an interpretation of classical logic based on the original (functional) dialectica interpretation of Kurt Gödel (cf.…”