Modal Definability over a Class of Structures with Two Equivalence Relations

More than 40 years the correspondence between modal logic and first-order logic, when they are interpreted in relational structures, is on the main stream of the investigations of many modal logicians. The most interesting in this direction is a series of results on modal and first-order definability proved by Chagrova in the 1990s. In particular, from them it follows the undecidability of both types definability. These results stimulate creating of different algorithms which in many cases succeed to find first-order equivalent for any formula from a reach set of modal formulas. On the other hand, Balbiani and Tinchev have proved for several modal languages that both definability problems, restricted to the class of all partitions, are decidable and have found their complexity. A straightforward transfer of these results to the case of modal language with two modalities interpreted by equivalence relations is impossible. Moreover, the correspondence problem is undecidable in the mentioned case. We consider the class K of all structures .W;R1;R2/, where W ¤ ;, R1 and R2 are equivalence relations over W and R1 R2. Let L be the first-order language with formal equality and exactly two binary predicate symbols. Let M be the propositional modal language with two or three unary modalities. Any structure from K can be considered as a Kripke structure for M, in the presence of the third modality, it is interpreted as the universal modality by the Cartesian square of the universe. We prove that the modal definability for the sentence from L by a modal formula from M over the class K is a decidable problem. We demonstrate the complexity of the above definability problem.