  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 terminating and non-terminating tableau systems
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 describe these
tableaux with $\mu$-calculus formulas, thus reducing the satisfiability of each
of these logics to the satisfiability of the $\mu$-calculus, resulting in a
general 2EXP upper bound for satisfiability testing.
