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Decidable modal logic with undecidable admissibility problem. (English. Russian original) Zbl 0782.03005
Algebra Logic 31, No. 1, 53-61 (1992); translation from Algebra Logika 31, No. 1, 83-93 (1992).
The admissibility problem for a given logic $$L$$ is to determine whether an arbitrary given inference rule $$A_ 1(p_ 1,\dots,p_ n),\dots,A_ m(p_ 1,\dots,p_ n)/B(p_ 1,\dots,p_ n)$$ is admissible in $$L$$, i.e., for all formulas $$C_ 1,\dots,C_ n$$, $$B(C_ 1,\dots,C_ n)\in L$$ whenever $$A_ 1(C_ 1,\dots,C_ n)\in L,\dots,A_ m(C_ 1,\dots,C_ n)\in L$$.
As is known, V. Rybakov proved the decidability of the admissibility problem for a number of intermediate and modal logics.
In this paper, the author constructs a decidable normal modal logic for which the admissibility problem is undecidable. The logic is an extension of K4 of width 3 and has infinitely many axioms.

##### MSC:
 03B45 Modal logic (including the logic of norms) 03B25 Decidability of theories and sets of sentences
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##### References:
  V. V. Rybakov, ”Problems of admissibility and substitution, logical equations and restricted theories of free algebras,” in: Logic, Methodology and Philosophy of Science VIII, Elsevier (1989), pp. 121–139. · Zbl 0691.03012  V. V. Rybakov, ”On admissibility of inference rules in modal logicG,” Tr. Inst. Mat. Sib. Otd. Akad. Nauk SSSR,12, 120–138, (1989).  V. V. Rybakov, ”Equations in free closure algebras and the substitution problem,” Dokl. Akad. Nauk SSSR,287, No. 3, 554–557 (1986).  G. D. Birkhoff, Lattice Theory, Amer. Math. Soc. (1979).  K. Fine, ”Logics containing K4, Part I,” J. Symb. Logic,39, No. 1, 31–42 (1974). · Zbl 0287.02010  A. I. Mal’tsev, ”Identical relations on quasigroup varieties,” Mat. Sb.,69, No. 1, 3–12 (1966).
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