🤖 AI Summary
This paper addresses the decidability of proof search in intuitionistic logic. Method: It extends the coinductive proof search (CoIPS) methodology from minimal implicational logic to polarized intuitionistic logic LJP. Specifically, it establishes, for the first time, a coinductive characterization of focused proofs in LJP alongside an equivalent inductive syntactic representation; introduces a decidability framework based on recursive predicates; and constructs a faithful negative translation from full intuitionistic propositional logic LJT into LJP. Contributions/Results: It achieves full decidability of both existence and finiteness of focused witnesses in LJP. Consequently, it yields novel, unified decidability proofs for inhabitation and finiteness in LJT—enhancing algorithmic reliability through integrated treatment. Moreover, it provides a transferable polarized modeling pathway applicable to variants such as LJQ.
📝 Abstract
The approach to proof search dubbed"coinductive proof search", 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. We prove the decidability of existence of focused inhabitants, and of finiteness of the number of focused inhabitants for polarized intuitionistic logic, by means of such recursive procedures. Moreover, the polarized logic can be used as a platform from which proof search for other logics is understood. We illustrate the technique with LJT, a focused sequent calculus for full intuitionistic propositional logic (including disjunction). For that, we have to work out the"negative translation"of LJT into LJP (that sees all intuitionistic types as negative types), and verify that the translation gives a faithful representation of proof search in LJT as proof search in the polarized logic. We therefore inherit decidability of both problems studied for LJP and thus get new proofs of these results for LJT.