Geodesic
Decidability of admissibility in the modal system~$\mathrm{Grz}$ and in intuitionistic logic
Izvestiya. Mathematics , Tome 28 (1987) no. 3, pp. 589-608.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IM2_1987_28_3_a6,
     author = {V. V. Rybakov},
     title = {Decidability of admissibility in the modal system~$\mathrm{Grz}$ and in intuitionistic logic},
     journal = {Izvestiya. Mathematics },
     pages = {589--608},
     publisher = {mathdoc},
     volume = {28},
     number = {3},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1987_28_3_a6/}
}
TY  - JOUR
AU  - V. V. Rybakov
TI  - Decidability of admissibility in the modal system~$\mathrm{Grz}$ and in intuitionistic logic
JO  - Izvestiya. Mathematics 
PY  - 1987
SP  - 589
EP  - 608
VL  - 28
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1987_28_3_a6/
LA  - en
ID  - IM2_1987_28_3_a6
ER  - 
%0 Journal Article
%A V. V. Rybakov
%T Decidability of admissibility in the modal system~$\mathrm{Grz}$ and in intuitionistic logic
%J Izvestiya. Mathematics 
%D 1987
%P 589-608
%V 28
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1987_28_3_a6/
%G en
%F IM2_1987_28_3_a6
V. V. Rybakov. Decidability of admissibility in the modal system~$\mathrm{Grz}$ and in intuitionistic logic. Izvestiya. Mathematics , Tome 28 (1987) no. 3, pp. 589-608. http://geodesic.mathdoc.fr/item/IM2_1987_28_3_a6/

[1] Maksimova L. L., Rybakov V. V., “O reshetke normalnykh modalnykh logik”, Algebra i logika, 13 (1974), 188–216 | MR | Zbl

[2] Mints G. E., “Dopustimye i proizvodnye pravila”, Zapiski nauchnykh seminarov LOMI AN SSSR, 8, 1968, 189–191 | Zbl

[3] Mints G. E., “Proizvodnost dopustimykh pravil”, Zapiski nauchnykh seminarov LOMI AN SSSR, 32, 1972, 85–89 | MR

[4] Raseva E., Sikorskii R., Matematika metamatematiki, Nauka, M., 1972 | MR

[5] Rybakov V. V., “Dopustimye pravila predtablichnykh modalnykh logik”, Algebra i logika, 20 (1981), 440–464 | MR | Zbl

[6] Rybakov V. V., “Dopustimye pravila dlya logik, vklyuchayuschikh $S4.3$”, Sib matem. zh., 25 (1984), 141–145 | MR | Zbl

[7] Rybakov V. V., “Razreshimost problemy dopustimosti v konechnosloinykh modalnykh logikakh”, Algebra i logika, 23 (1984), 100–116 | MR | Zbl

[8] Rybakov V. V., “Elementarnye teorii svobodnykh algebr zamykanii”, 16 Vsesoyuznaya algebraicheskaya konferentsiya. Tezisy dokladov, Leningrad, 1981, 116

[9] Tsitkin A. I., “O dopustimykh pravilakh intuitsionistskoi logiki vyskazyvanii”, Matem. sbornik, 102(144) (1977), 314–323 | Zbl

[10] Tsitkin A. I., “O strukturalno polnykh superintuitsionistskikh logikakh”, Dokl. AN SSSR, 241 (1978), 40–43 | MR | Zbl

[11] Shekhtman V. B., “Lestnitsy Rigera–Nishimury”, Dokl. AN SSSR, 241 (1978), 1288–1291 | MR | Zbl

[12] Freedman H., “One hundred and two problem in mathematical logic”, J. of Symbolic Logic, 40 (1975), 113–130 | DOI | MR

[13] Harrop R., “Concerning formulas of the types $A\rightarrow B\bigvee C,\,A\rightarrow (Ex)B(x)$ in intuitionistic formal systems”, J. Symbolic Logic, 25 (1960), 27–32 | DOI | MR | Zbl

[14] Port J., “The deducibilities of $S5$”, J. of Philosophical logic, 10 (1981), 409–422 | DOI | MR | Zbl

[15] Segerberg K., An ascay in classical modal logic, University of Uppsala, North Holland, 1971 | MR | Zbl