Dialectica Interpretation with Fine Computational Control

Abstract

Computational proof interpretations enrich the logical meaning of formula connectives and quantifiers with algorithmic relevance and allow to extract the construction contained in a proof. Berger showed that quantifiers can be selectively declared irrelevant for the modified realisability interpretation, thus removing unnecessary computation from the extracted program [1]. Hernest adapted the uniform quantifiers to Gödel’s Dialectica interpretation [2] and later demonstrated together with the author how they can be refined [5]. The present paper gives a further extension, in which the computational meaning can be controlled separately for every component of the Dialectica interpretation. Apart from enriching the possibilities to remove redundancies in extracted programs, this finer approach also allows to independently switch off the postive or negative algorithmic contribution of whole formulas.