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Izv. Akad. Nauk SSSR Ser. Mat., 1986, Volume 50, Issue 3, Pages 598–616 (Mi izv1521)  

This article is cited in 5 scientific papers (total in 5 papers)

Decidability of admissibility in the modal system $\mathrm{Grz}$ and in intuitionistic logic

V. V. Rybakov


Abstract: A criterion for admissibility of rules in the modal system $\mathrm{Grz}\rightleftharpoons\mathrm S4+\Box(\Box(p\supset\Box p)\supset p)\supset p$ is found. On the basis of it an algorithm is constructed that recognizes the admissibility of rules in $\mathrm{Grz}$. The decidability of admissibility in $\mathrm{Grz}$, proved in the paper, yields as a corollary a positive solution of the Kuznetsov–Friedman problem of algorithmic decidability of the admissibility problem in intuitionistic propositional logic. Algebraic analogues of the results obtained here are the decidability of the universal theories of a free pseudo-Boolean algebra and a free topo-Boolean algebra in the variety of algebras corresponding to the system $\mathrm{Grz}$. The elementary theories of these free algebras are hereditarily undecidable.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1987, 28:3, 589–608

Bibliographic databases:

UDC: 517.11+519.48
MSC: Primary 03B45; Secondary 03F55
Received: 16.04.1984

Citation: V. V. Rybakov, “Decidability of admissibility in the modal system $\mathrm{Grz}$ and in intuitionistic logic”, Izv. Akad. Nauk SSSR Ser. Mat., 50:3 (1986), 598–616; Math. USSR-Izv., 28:3 (1987), 589–608

Citation in format AMSBIB
\Bibitem{Ryb86}
\by V.~V.~Rybakov
\paper Decidability of admissibility in the modal system~$\mathrm{Grz}$ and in intuitionistic logic
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1986
\vol 50
\issue 3
\pages 598--616
\mathnet{http://mi.mathnet.ru/izv1521}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=854597}
\zmath{https://zbmath.org/?q=an:0624.03009}
\transl
\jour Math. USSR-Izv.
\yr 1987
\vol 28
\issue 3
\pages 589--608
\crossref{https://doi.org/10.1070/IM1987v028n03ABEH000902}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Rybakov, “Criteria for admissibility of rules of inference with parameters in the intuituonistc propositional calculus”, Math. USSR-Izv., 37:3 (1991), 693–703  mathnet  crossref  mathscinet  zmath  adsnasa
    2. V. V. Rybakov, “Admissibility of rules of inference, and logical equations, in modal logics axiomatizing provability”, Math. USSR-Izv., 36:2 (1991), 369–390  mathnet  crossref  mathscinet  zmath  adsnasa
    3. V.V. Rybakov, “Problems of substitution and admissibility in the modal system Grz and in intuitionistic propositional calculus”, Annals of Pure and Applied Logic, 50:1 (1990), 71  crossref
    4. E.-J. Thiele, “European Summer Meeting of the Association for Symbolic Logic”, J. symb. log, 57:01 (1992), 282  crossref
    5. Ronald Fagin, J.Y.. Halpern, M.Y.. Vardi, “What is an inference rule?”, J. symb. log, 57:03 (1992), 1018  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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