Optimal Tableau Systems for Propositional Neighborhood Logic over All, Dense, and Discrete Linear Orders

In this paper, we focus our attention on tableau systems for the propositional interval logic of temporal neighborhood (Propositional Neighborhood Logic, PNL for short). PNL is the proper subset of Halpern and Shoham's modal logic of intervals whose modalities correspond to Allen's relations meets and met by. We first prove by a model-theoretic argument that the satisfiability problem for PNL over the class of all (resp., dense, discrete) linear orders is decidable (and NEXPTIME-complete). Then, we develop sound and complete tableau-based decision procedures for all the considered classes of orders, and we prove their optimality. (As a matter of fact, decidability with respect to the class of all linear orders had been already proved via a reduction to the decidable satisfiability problem for the two-variable fragment of first-order logic of binary relational structures over ordered domains).