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Admissibility of rules of inference, and logical equations, in modal logics axiomatizing provability
Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 369-390

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This paper examines the modal logics of Gödel-Löb (GL) and Solovay (S) – the smallest and the largest modal representations of arithmetic theories. The problem of recognizing the admissibility of inference rules with parameters (and, in particular, without parameters) in GL and S is shown to be decidable; that is, a positive solution is obtained to analogues of a problem of Friedman. The analogue of a problem of Kuznetsov on finite bases of admissible rules for S and GL is solved in the negative sense. Algorithms are found for recognizing the solvability in GL and S of logical equations and for constructing solutions for them. 
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     author = {V. V. Rybakov},
     title = {Admissibility of rules of inference, and logical equations, in modal logics axiomatizing provability},
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V. V. Rybakov. Admissibility of rules of inference, and logical equations, in modal logics axiomatizing provability. Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 369-390. http://geodesic.mathdoc.fr/item/IM2_1991_36_2_a7/