On Intuitionistic Proof Transformations,their Complexity,and Application to Constructive Program Synthesis

Abstract

We present a translation of intuitionistic sequent proofs from a multi-succedent calculus ℒ𝒥_mc into a single-succedent calculus ℒ𝒥. The former gives a basis for automated proof search whereas the latter is better suited for proof presentation and program construction from proofs in a system for constructive program synthesis. Well-known translations from the literature have a severe drawback; they use cuts in order to establish the transformation with the undesired consequence that the resulting program term is not intuitive. We establish a transformation based on permutation of inferences and discuss the relevant properties with respect to proof complexity and program terms. As an important result we show that ℒ𝒥 cannot polynomially simulate ℒ𝒥_mc (both without the cut rule), even in the prepositional fragment.