Decidable Elementary Modal Logics

In this paper, the modal logic over classes of structures definable by universal first-order Horn formulas is studied. We show that the satisfiability problems for that logics are decidable, confirming the conjecture from [E. Hemaspaandra and H. Schnoor, On the Complexity of Elementary Modal Logics, STACS 08]. We provide a full classification of logics defined by universal first-order Horn formulas, with respect to the complexity of satisfiability of modal logic over the classes of frames they define. It appears, that except for the trivial case of inconsistent formulas for which the problem is in P, local satisfiability is either NP-complete or PSPACE-complete, and global satisfiability is NP-complete, PSPACE-complete, or EXPTIME-complete. While our results holds even if we allow to use equality, we show that inequality leads to undecidability.