In the chapter, we provide tableau-based decision procedures for the logic M(En). In the existing literature, several approaches for deciding modal/description logics with counting operators can be found. [6] describes a decision procedure for modal logics with counting operators that exploits the translation function from the modal counting language to the hybrid language with the universal modality H(A). Tableau-based decision procedures for analogs of modal logics with counting operators were established in the field of description logics, where counting operators are known under the guise of cardinality constraints. Sound, complete and terminating tableau-calculi for these logics can be found in [65, 60, 31]. These calculi, in general, do not differ in the rules for cardinalities, however, they utilise different blocking mechanisms for ensuring termination, such as pairwise blocking or pattern-based blocking.

We exploit the framework from [84] to synthesise a sound, complete and terminating prefixed tableau calculus for M(En). We also provide a refinement of this calculus consisting in internalizing the semantics of the logic within the language of the logic. We only exploit global counting operators to dispense with meta-linguistic expressions like M, x ⊧ ϕ or R(x, y) to achieve a full internalization.

We also describe a successful implementation of the calculus using the MetTeL2 tableau prover generator [90, 1].

The chapter is structured as follows. In Section 8.1, we introduce a prefixed tableau calculus TpM(En) for the logic MEn. We also prove soundness and completeness of the calculus. Termination of TpM(En) is proven in Section 8.2. Section 8.3 is devoted to possible refinements of particular rules of TpM(En). The internalized version of the calculus named TiM(En) is presented in Section 8.4. We briefly comment on a successful implementation of TiM(En) on MetTeL2 prover. Section 8.6 is an extensive overview of other decision procedures for logics with global counting operators.