Coinductive proof search for polarized logic with applications to full intuitionistic propositional logic

Abstract

The approach to proof search dubbed "coinductive proof search" (CoIPS), and previously developed by the authors for implicational intuitionistic logic, is in this paper extended to LJP, a focused sequent-calculus presentation of polarized intuitionistic logic, including an array of positive and negative connectives. As before, this includes developing a coinductive description of the search space generated by a sequent, an equivalent inductive syntax describing the same space, and decision procedures for inhabitation problems in the form of predicates defined by recursion on the inductive syntax. Inhabitation is taken in a generalized sense, because we also consider when a sequent has a solution, that is a (possibly infinite) run of bottom-up proof search which never fails to apply another inference rule. We provide a very general scheme whose instances are decision problems concerning solutions in LJP having algorithms through the inductive syntax. Polarized logic and LJP can be used as a platform from which proof search for other logics is understood. We illustrate this for the proof systems LJT and LJQ for intuitionistic logic, both equipped with all the connectives. For that we work out respectively a negative and a positive interpretation into LJP, which map formulas of the source logic into formulas in LJP of the said polarity; and this even at the level of the coinductive versions of the three involved proof systems. The interpretations are proved to be, not only faithful, but actually full embeddings, establishing a bijection between the solutions (resp. proofs) of a sequent and the solutions (resp. proofs) of its polarized interpretation. This allows the inheritance to the source systems of the decidability and other results previously obtained for LJP, thereby vastly generalizing the previous results of CoIPS, which were confined to LJT and implicational intuitionistic logic.