Complexity of finite-variable fragments of propositional modal logics of symmetric frames

While finite-variable fragments of the propositional modal logic S5—complete with respect to reflexive, symmetric and transitive frames—are polynomial-time decidable, the restriction to finite-variable formulas for logics of reflexive and transitive frames yields fragments that remain ‘intractable.’ The role of the symmetry condition in this context has not been investigated. We show that symmetry either by itself or in combination with reflexivity produces logics that behave just like logics of reflexive and transitive frames, i.e. their finite-variable fragments remain intractable, namely PSPACE-hard. This raises the question of where exactly the borderline lies between modal logics whose finite-variable fragments are tractable and the rest.