Interpolation Properties for Provability Logics GL and GLP

We study interpolation properties of provability logics. We prove the Lyndon interpolation for GL and the uniform interpolation for GLP. DOI: 10.1134/S0081543811060198 The present paper is concerned with provability logics and their interpolation properties. Recall that the Craig interpolation property for a logic L says that if φ implies ψ, then there is an interpolant, that is, a formula θ containing only common variables of φ and ψ such that φ implies θ and θ implies ψ. The Lyndon interpolation property is a strengthening of the Craig one that also takes into consideration negative and positive occurrences of the shared propositional variables; that is, the variables occurring in θ positively (negatively) must also occur both in φ and ψ positively (negatively). The Craig interpolation property for the standard provability logic GL was independently established by C. Smoryński [14] and G. Boolos [5] as early as in 1979. However, the question whether this logic enjoys the Lyndon interpolation seemingly remained open [1]. In the present paper we prove the Lyndon interpolation for the standard provability logic GL and the uniform interpolation for an important provability logic GLP. The polymodal provability logic GLP was introduced by G. Japaridze and shown to be sound and complete with respect to arithmetical semantics [11]. This logic plays a remarkable role in the study of formal arithmetic by means of modal logic. The Craig interpolation property for GLP was first proved by K. Ignatiev in [10], but his proof is not formalizable in the Peano arithmetic. Therefore, L. Beklemishev gave another (finitary) proof of the same fact [3]. In this paper we prove the uniform interpolation for GLP and, in that way, answer the question posed by L. Beklemishev in [3]. A major strengthening of the Craig property is the uniform interpolation. It was first formulated by A. Pitts, who established this property for the intuitionistic propositional logic [12]. The same notion was independently introduced by V. Shavrukov, who proved an analogous result for the provability logic GL [13]. Further research of the uniform interpolation was undertaken by S. Ghilardi and M. Zawadowski [8], and by A. Visser [16]. Since then this property has been studied for many modal logics: for instance, the uniform interpolation holds for K, Grz and T, whereas K4 and S4 do not possess this property. The uniform interpolation is defined as follows. Let υ(φ) be the set of propositional variables of φ. A formula θ is called a p-projection of a formula φ in a logic L, where p is a propositional variable, if υ(θ) ⊂ υ(φ)\{p} and for any ψ with p / ∈ υ(ψ) one has L φ → ψ ⇐⇒ L θ → ψ. Note that the p-projection of φ is unique up to the logical equivalence in L. If there is a p-projection for any formula and for any propositional variable p, then we say that L enjoys the uniform interpolation property. aDepartment of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia.