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.

@article{paperbot2735,
  title = {Complexity results for modal logic with recursion via translations and tableaux},
  author = {Luca Aceto and Antonis Achilleos and Elli Anastasiadi and Adrian Francalanza and Anna Ingólfsdóttir},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 20, Issue 3},
  year = {2024},
  doi = {10.46298/lmcs-20(3:14)2024}
}