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This paper studies the complexity of classical modal logics and of theirextension with fixed-point operators, using translations to transfer resultsacross logics. In particular, we show several complexity results formulti-agent logics via translations to and from the $\mu$-calculus and modallogic, which allow us to transfer known upper and lower bounds. We also usethese translations to introduce a terminating tableau system for the logics westudy, based on Kozen's tableau for the $\mu$-calculus, and the one of Fittingand Massacci for modal logic. Finally, we show how to encode the tableaux weintroduced into $\mu$-calculus formulas. This encoding provides upper boundsfor the satisfiability checking of the few logics we previously did not havealgorithms for.

Complexity results for modal logic with recursion via translations and tableaux