We study the complexity of β-reduction for redexes of order 2,3 and 4. The obtained results are: evaluation of boolean expressions can be reduced to β-reduction of order 2 and β-reduction of order 2 is in O(nlog n), β-reduction of order 3 is complete for PTIME, and β-reduction of order 4 is complete for PSPACE. 1

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. 1.

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