A Hypersequent Calculus with Clusters for Linear Frames

Overview & Synopsis

The logic Kt4.3 is the basic modal logic of linear frames. Along with its extensions, it is found at the core of linear-time temporal logics and logics on words. In this paper, we consider the problem of designing proof systems for these logics, in such a way that proof search yields decision procedures for validity with an optimal complexity—coNP in this case. In earlier work, Indrzejczak has proposed an ordered hypersequent calculus that is sound and complete for Kt4.3 but does not yield any decision procedure. We refine his approach, using a hypersequent structure that corresponds to weak rather than strict total orders, and using annotations that reflect the model-theoretic insights given by small models for Kt4.3. We obtain a sound and complete calculus with an associated coNP proof search algorithm. These results extend naturally to the cases of unbounded and dense frames, and to the complexity of the two-variable fragment of first-order logic over total orders.