Japaridze's polymodal logic , is a system of provability logic with infinitely many modal operators. This system has played an important role in some applications of provability algebras in proof theory, and has been extensively studied since the late 1980s. It is named after Giorgi Japaridze.
Language and axiomatization
The language of GLP extends that of the language of classical propositional logic by including the infinite series ,,,... of “necessity” operators. Their dual “possibility” operators <0>,<1>,<2>,... are defined by <n>p = ¬¬p. The axioms of GLP are all classical tautologies and all formulas of one of the following forms:
Consider a “sufficiently strong” first-order theory T such as Peano ArithmeticPA. Define the series T0,T1,T2,... of theories as follows:
T0 is T
Tn+1 is the extension of Tn through the additional axioms ∀xF for each formula F such that Tn proves all of the formulas F,F,F,...
For each n, let Prn be a natural arithmetization of the predicate “x is the Gödel number of a sentence provable in Tn. A realization is a function * which sends each nonlogical atom a of the language of GLP to a sentence a * of the language of T. It extends to all formulas of the language of GLP by stipulating that * commutes with the Boolean connectives, and that * is Pr_n, where ‘F *’ stands for the the Gödel number of F *. An arithmetical completeness theorem for GLP states that a formula F is provable in GLP if and only if, for every interpretation *, the sentence F * is provable in T. The above understanding of the series T0,T1,T2,... of theories is not the only natural understanding yielding the soundness and completeness of GLP. For instance, each theory Tn can be understood as T augmented with all true ∏n sentences as additional axioms. George Boolos showed that GLP remains sound and complete with analysis in the role of the base theoryT.
Other semantics
GLP has been shown to be incomplete with respect to any class of Kripke frames. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. GLP is complete w.r.t. the class of all GLP-spaces.
Computational complexity
The problem of being a theorem of GLP is PSPACE-complete. So is the same problem restricted to only variable-free formulas of GLP.
History
GLP, under the name GP, was introduced by Giorgi Japaridze in his PhD thesis "Modal Logical Means of Investigating Provability" and published two years later along with the completeness theorem for GLP with respect to its provability interpretation and a proof that Kripke frames for GLP do not exist. Beklemishev also conducted a more extensive study of Kripke models for GLP. Topological models for GLP were studied by Beklemishev, Bezhanishvili, Icard and Gabelaia. The decidability of GLP in polynomial space was proven by I. Shapirovsky, and the PSPACE-hardness of its variable-free fragment was proven by F.Pakhomov. Among the most notable applications of GLP has been its use in proof-theoretically analyzing Peano arithmetic, elaborating a canonical way for recovering ordinal notation system up to ɛ0 from the corresponding algebra, and constructing simple combinatorial independent statements . An extensive survey of GLP in the context of provability logics in general was given by George Boolos in his book “The Logic of Provability”.
Literature
L. Beklemishev, . Annals of Pure and Applied Logic 128, pp. 103–123.
L. Beklemishev, . Annals of Pure and Applied Logic 161, 756–774.
L. Beklemishev, G. Bezhanishvili and T. Icar, “On topological models of GLP”. Ways of proof theory, Ontos Mathematical Logic, 2, eds. R. Schindler, Ontos Verlag, Frankfurt, 2010, pp. 133–153.
L. Beklemishev, “On the Craig interpolation and the fixed point properties of GLP”. Proofs, Categories and Computations. S. Feferman et al., eds., College Publications 2010. pp. 49–60.
