A Coinductive Approach to Proof Search through Typed Lambda-Calculi

Abstract

In reductive proof search, proofs are naturally generalized by solutions, comprising all (possibly infinite) structures generated by locally correct, bottom-up application of inference rules. We propose a rather natural extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by (possibly infinite) terms of the coinductive variant of the typed lambda-calculus that represents proofs. We take this as a starting point for a new, comprehensive approach to proof search; our case study is proof search in the sequent calculus LJT for intuitionistic implication logic. A second, finitary representation is proposed, where the lambda-calculus that represents proofs is extended with a formal greatest fixed point. In the latter system, fixed-point variables enjoy a relaxed form of binding that allows the detection of cycles through the type system. Formal sums are used in both representations to express alternatives in the search process, so that not only individual solutions but actually solution spaces are expressed. Moreover, formal sums are used in the coinductive syntax to define "co-contraction" (contraction bottom-up). A semantics is defined assigning a coinductive λ-term to each finitary term, making use of co-contraction as a semantical match to the relaxed form of binding of fixed-point variables present in the finitary system. The main result is the existence of an equivalent finitary representation for any given solution space expressed coinductively.