Deciding modal logics using tableaux and set theory

We propose a tableau-like decision procedure for deciding the satisfiability of set-theoretical formulae with restricted universal quantifiers and the powerset operator. Our result apply to a rather large class of set theories. The procedure we define can be used as a subroutine to decide the same class of formulae both in Set Theory and in non well-founded set theories, since we assume neither Regularity nor any form of anti-foundation axiom. Moreover, the decidability result presented allow to characterize a class of decidable modal logics. Thanks to the 2-as-P (box-as-powerset) translation our procedure can be used to uniformly study a large class of modal logics which includes K, T , S4, S5, S4.3.