Uniform interpolation and sequent calculi in modal logic

Iemhoff, Rosalie

doi:10.1007/s00153-018-0629-0

Cited by 20 publications

(19 citation statements)

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“…We do not know whether there are examples of interesting logics that have calculi that consist solely of focused rules. In the classical setting [17] we use a notion of focused 4 that extends the one in this paper and is such that all rules of G3p are focused. The same does not hold for G3ip, but in the next section we show that IPC has a calculus consisting of focused and nonfocused rules, for which there exists a sound interpolant assignment.…”

Section: Corollary 16 Any Intermediate Logic That Does Not Have Unifo...mentioning

confidence: 99%

“…In [17], a method has been developed to prove uniform interpolation for any modal logic with a sequent calculus consisting of so-called focused and focused modal rules. This method provides a single framework via which to prove in a uniform way existing and new results on uniform interpolation, such as the result from [6] that K has uniform interpolation and the new result that KD has uniform interpolation.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we extend the method of [17] to intermediate and intuitionistic modal logics, where the latter are modal logics that contain IPC. This is not a straightforward extension, since in contrast to CPC, already for IPC itself the proof of uniform interpolation is highly nontrivial.…”

Section: Introductionmentioning

confidence: 99%

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Uniform interpolation and the existence of sequent calculi

Iemhoff

2019

Annals of Pure and Applied Logic

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This paper presents a uniform and modular method to prove uniform interpolation for several intermediate and intuitionistic modal logics. The proof-theoretic method uses sequent calculi that are extensions of the terminating sequent calculus G4ip for intuitionistic propositional logic. It is shown that whenever the rules in a calculus satisfy certain structural properties, the corresponding logic has uniform interpolation. It follows that the intuitionistic versions of K and KD (without the diamond operator) have uniform interpolation. It also follows that no intermediate or intuitionistic modal logic without uniform interpolation has a sequent calculus satisfying those structural properties, thereby establishing that except for the seven intermediate logics that have uniform interpolation, no intermediate logic has such a sequent calculus.

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Section: Corollary 16 Any Intermediate Logic That Does Not Have Unifo...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Uniform interpolation and the existence of sequent calculi

Iemhoff

2019

Annals of Pure and Applied Logic

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confidence: 99%

Universal Proof Theory: Semi-analytic Rules and Uniform Interpolation

Tabatabai,

Jalali

2018

Preprint

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In [7] and [8], Iemhoff introduced a connection between the existence of a terminating sequent calculi of a certain kind and the uniform interpolation property of the super-intuitionistic logic that the calculus captures. In this paper, we will generalize this relationship to also cover the substructural setting on the one hand and a much more powerful class of rules on the other. The resulted relationship then provides a uniform method to establish uniform interpolation property for the logics FL e , FL ew , CFL e , CFL ew , IPC, CPC and their K and KD-type modal extensions. More interestingly though, on the negative side, we will show that no extension of FL e can enjoy a certain natural type of terminating sequent calculus unless it has the uniform interpolation property. It excludes almost all super-intutionistic logics and the logics K4 and S4 from having such a reasonable calculus.

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“…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.…”

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confidence: 99%

Uniform Lyndon interpolation property in propositional modal logics

Kurahashi

2020

Arch. Math. Logic

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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...

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Uniform interpolation and sequent calculi in modal logic

Cited by 20 publications

References 9 publications

Uniform interpolation and the existence of sequent calculi

Uniform interpolation and the existence of sequent calculi

Universal Proof Theory: Semi-analytic Rules and Uniform Interpolation

Uniform Lyndon interpolation property in propositional modal logics

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