Uniform Lyndon interpolation property in propositional modal logics

Abstract

We introduce and investigate the notion of uniform Lyndon interpolation property (ULIP) which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including K, KB, GL and Grz enjoy ULIP. Our proofs are modifications of Visser's proofs of uniform interpolation property using bounded bisimulations [33]. Also we give a new upper bound on the complexity of uniform interpolants for GL and Grz. * kurahashi@n.kisarazu.ac.jp recently settled affirmatively for GL by Shamkanov [29] and for Grz by Maksimova [24]. Recently, Kuznets [18] proved LIP for a wider class of propositional modal logics including the logics in the so-called modal cube of [12]. Maksimova [21] showed that there exist normal extensions of S5 having CIP but do not have LIP (see also [11]).Pitts [26] proved that intuitionistic propositional logic has the uniform interpolation property. A logic L is said to have the uniform interpolation property (UIP) if for any formula ϕ and any finite set P of propositional variables, there exists a formula θ such that θ does not contain propositional variables in P and it uniformly interpolates all L-provable implications ϕ → ψ in L where ψ does not contain propositional variables in P . Shavrukov [30] proved that the propositional modal logic GL has UIP. UIP for K, Grz, and KT were proved by Ghilardi [13] and Visser [33], Visser [33], and Bílková [2], respectively. See also [3,17]. However, it was proved by Ghilardi and Zawadowski [14] that the modal logic S4 does not enjoy UIP, and Bílková [2] also showed the same result for K4.So far, it has been studied separately that each logic has UIP and that logic has LIP. In this paper, we give a framework which can simultaneously derive that a logic enjoys both UIP and LIP. Namely, we introduce the notion of uniform Lyndon interpolation property (ULIP), and investigate this newly introduced notion.In Section 2, we show that ULIP is actually stronger than both UIP and LIP. Also we prove several basic behaviors of ULIP. Then we show that ULIP for the propositional modal logics K5, KD5, K45, KD45, KB5 and S5 easily follows from LIP for each of them. In Section 3, we introduce the notion of bounded (P, Q)-bisimulation between Kripke models which is a main tool of our proofs. ULIP for the propositional modal logics K, KD, KT, KB, KDB and KTB is proved in Section 4. Consequently, we obtain both UIP and LIP for these logics. UIP for KB, KDB and KTB are probably new. At last, we prove ULIP for GL and Grz in Section 5. Our proofs of ULIP are modifications of Visser's proofs [33] of UIP using bounded bisimulations. Especially for GL and Grz, we give a new upper bound on the complexity of uniform interpolants. Interpolation properties in propositional modal logicsIn this section, we introduce some variations of interpolation property. In particular, we newly introduce the notion of uniform Lyndon interpolation property, and we investigate several basic behaviors of uniform Lyndon interpolation propert...

Uniform Interpolation via Nested Sequents

“…For each y / ∈ Range(I ′ ) such that ρR ′ y, there is x such that (y, x) ∈ P and •d : ξ ∈ ℧ y,x because Θ y,x ⊇ Θ ∋ ξ. Hence, M ′ , y ξ by IH (16). Finally, for each x / ∈ Range(I ′ ) such that not ρR ′ x, there is y such that (y, x) ∈ P and d : ξ ∈ ℧ y,x because Φ y,x ⊇ Φ ∋ ξ.…”

“…It strengthens the Craig interpolation property (CIP) by making interpolants depend on only one formula of an implication, either the premise or conclusion. A lot of work has gone into proving the UIP, and it is shown to be useful in various areas of computer science, including knowledge representation [16] and description logics [24]. Early results on the UIP in modal logic include positive results proved semantically for logics GL and K (independently in [9,31,34]) and negative results for logics S4 [10] and K4 [5].…”

Uniform Lyndon Interpolation for Basic Non-normal Modal Logics

“…Uniform Lyndon interpolation (ULIP) is a strengthening of UIP in which the interpolant respects the polarity of propositional variables (a definition follows in the next section). It first occurred in [10], where it was shown that several normal modal logics, including K and KD, have that property. In this paper we show that the non-normal modal logics E, M, EN, MN, MC, K, their conditional versions, CE, CM, CEN, CMN, CMC, CK in addition to CKID have uniform Lyndon interpolation and the interpolant can be constructed explicitly from the proof.…”

Uniform interpolation via nested sequents and hypersequents

“…Visser provided purely semantic proofs for K, GL, and IPC based on bounded bisimulation up to atomic propositions [31]. This method was later applied to prove the stronger Lyndon UIP for a wide range of modal logics [17]. The semantic interpretation of uniform interpolation is called bisimulation quantifiers, see [7] for an extended explanation.…”