Single Step Tableaux for Modal Logics Computational Properties, Complexity and Methodology

Single Step Tableaux (SST) are the basis of a calculus for modal logics that combines different features of sequent and prefixed tableaux into a simple, modular, strongly analytic, and effective calculus for a wide range of modal logics. The paper presents a number of the computational results about SST (confluence, decidability, space complexity, modularity, etc.) and compares SST with other formalisms such as translation methods, modal resolution, and Gentzen-type tableaux. For instance, it discusses the feasibility and infeasibility of deriving decision procedures for SST and translation-based methods by replacing loop checking techniques with simpler termination checks. The complexity of searching for validity and logical consequence with SST and other methods is discussed. Minimal conditions on SST search strategies are proven to yield PSPACE (and NPTIME for S5 and KD45) decision procedures. The paper also presents the methodology underlying the construction of the correctness and completeness proofs.