Terminating tableau calculi for modal logic K with global counting operators Mohammad Khodadadi, Renate A. Schmidt, Dmitry Tishkovsky School of Computer Science, The University of Manchester, United Kingdom Micha l Zawidzki Department of Logic, University of Lodz, Poland School of Computer Science, The University of Manchester, United Kingdom Abstract This paper presents the first systematic treatment of tableau calculi for modal logic K with global counting operators. Using a recently introduced tableau synthesis framework we establish two terminating tableau calculi for the logic. Whereas the first calculus is a prefix tableau calculus, the second is a refinement that internalises the semantics of the logic without using nominals. We prove the finite model property for the logic and show that adding the unrestricted blocking mechanism does not break soundness and completeness of the calculi and ensures termination in both cases. We have successfully implemented the prefix tableau calculus in the MetTeL 2 tableau prover generation platform. Keywords: modal logic, hybrid logic, tableau, counting operators, finite model property. 1 Introduction Counting modalities were first introduced by Fine in [8] under the name of graded modalities. They allowed expressing a number of successors of a partic- ular world, at which a certain formula holds. In particular, a formula ✸ =n > expresses the fact that the current world has exactly n successors. Some further developments of the theory of graded modalities can be found in [4,5,7]. Van der Hoek and de Rijke [18] established graded modal logics as a modal tool for investigating first-order counting quantifiers and introduced the notion of propositional logic with counting (PLC ) as a name for the logic S5 with graded modalities (see also [3]). With the aim of improving the expressivity of modal logics Areces et al. [2] introduced modal logics with counting operators (MLC ). Global counting operators E >n , E <n and E =n were added to a modal language with the ordinary modalities. Global counting operators increase the expressive power of a logic by allowing nominals, the universal modality, and counting the cardinality of