This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the $\mu$-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce a terminating tableau system for the logics we study, based on Kozen's tableau for the $\mu$-calculus, and the one of Fitting and Massacci for modal logic. Finally, we show how to encode the tableaux we introduced into $\mu$-calculus formulas. This encoding provides upper bounds for the satisfiability checking of the few logics we previously did not have algorithms for.

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