Decidable and Undecidable Problems for First-Order Definability and Modal Definability

The core of this paper is Chagrova’s Theorems about first-order definability of given modal formulas and modal definability of given elementary conditions. We consider classes of frames for which modal definability is decidable and classes of frames for which first-order definability is trivial. We give a new proof of Chagrova’s Theorem about modal definability and sketches of proofs of new variants of Chagrova’s Theorem about modal definability.