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Effectivizing Inseparability
, 1991
"... Smullyan's notion of effectively inseparable pairs of sets is not the best effective /constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our notion, and ..."
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Smullyan's notion of effectively inseparable pairs of sets is not the best effective /constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our notion, and show that the pairs of index sets effectively inseparable in Smullyan's sense are the same as those effectively inseparable in ours. In fact we characterize the pairs of index sets effectively inseparable in either sense thereby generalizing Rice's Theorem. For subrecursive index sets we have sufficient conditions for various inseparabilities to hold. For inseparability by sets in the same subrecursive class we have a characterization. The latter essentially generalizes Kozen's (and Royer's later) Subrecursive Rice Theorem, and the proof of each result about subrecursive index sets is presented "Rogers style" with care to observe subrecursive restrictions. There are pairs of sets effectively inseparab...
Modal Definability over a Class of Structures with Two Equivalence Relations
"... More than 40 years the correspondence between modal logic and firstorder 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 firstorder definabilit ..."
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More than 40 years the correspondence between modal logic and firstorder 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 firstorder 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 firstorder 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 ยค;, R1 and R2 are equivalence relations over W and R1 R2. Let L be the firstorder 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.