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Undecidable problems for modal definability

The core of our article is the computability of the problem of deciding the modal definability of first-order sentences with respect to classes of frames. It gives a new proof of Chagrova's Theorem telling that, with respect to the class of all frames, the problem of deciding the modal definability of first-order sentences is undecidable. It also gives the proofs of new variants of Chagrova's Theorem