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Decidability of the admissibility problem in layer-finite logics. (English. Russian original) Zbl 0576.03012
Algebra Logic 23, 75-87 (1984); translation from Algebra Logika 23, No. 1, 100-116 (1984).
The aim of the paper is to study admissible rules of inference in layer- finite propositional modal logics. Let $$\lambda$$ be a modal propositional logic containing S4 and eq($$\lambda)$$ be the variety of closure algebras (or, dually, interior algebras) corresponding to $$\lambda$$. Among the logics of fixed layer n, the smallest (weakest) logic is $$S4+\delta_ n$$. The following statements are proved: (1) The universal theory of free algebras of the variety $$eq(S4+\delta_ n)$$ is decidable for any $$n<\omega$$. (As corollary it follows that the admissibility problem in each logic $$S4+\delta_ n$$ has an affirmative answer: there exists an algorithm which for each rule decides whether it is admissible or not.) (2) The quasivariety generated by the free algebras of infinite rank in the variety $$eq(S4+\delta_ n)$$ has no basis of quasiidentities in a finite number of variables for $$n\geq 2$$. (It follows that all logics $$S4+\delta_ n$$ for $$n\geq 2$$ have no finite basis of admissible rules.) (3) The elementary theory of the free algebras of the variety $$eq(S4+\delta_ n)$$ is completely undecidable for any $$2\leq n<\omega.$$
Remark: author proved analogous results for the modal system S4 and the intuitionistic propositional logic H. These results are published in Algebra Logika 23, No.5, 546-572 (1984) and ibid. 24, No.1, 87-107 (1985). In particular, the admissibility problem for H is decidable - positive solution of problem 40 of H. Friedman [One hundred and two problems in mathematical logic, J. Symb. Logic 40, 113-129 (1975; Zbl 0318.02002)].

##### MSC:
 03B45 Modal logic (including the logic of norms) 03B25 Decidability of theories and sets of sentences
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##### References:
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