Uniform Interpolation and Admissible Rules

Giessen, Iris van der

doi:10.33540/1486

Cited by 4 publications

(7 citation statements)

References 118 publications

(282 reference statements)

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“…Due to the proof being technically involved, space considerations prevented us from extending the syntactic proof of ULIP to KD5, K45, KD45, KB5, and S5. For S5, layered sequents coincide with hypersequents, and we plan to upgrade the hypersequent-based syntactic proof of UIP from [11] to ULIP (see also [13]). As for KD5, K45, KD45, and KB5, the idea is to modify the method presented here for K5 by using the layered sequent calculus for the respective logic and making other necessary modifications, e.g., to rule dd, to fit the specific structure of the layers.…”

Section: Discussionmentioning

confidence: 99%

“…These are often called pre-interpolants as opposed to their dual post-interpolants that, in classical logic, can be defined as ∃pϕ = ∀pϕ and ∃ℓϕ = ∀ℓϕ (see, e.g., [1,5,11,17] for more explanations).…”

Section: Definition 4 (Uip and Ulip)mentioning

confidence: 99%

“…Let Ξ denote the multiset of these multiformulas ℧ j returned by all these calls. Since Step 2 is the only one used between that call and all the calls comprising (11), it is clear that (6) is their conjunction, i.e.,…”

Section: Whenmentioning

confidence: 99%

“…Towards this goal, we build on the modular treatment of multicomponent calculi to prove the CIP for modal and intermediate logics in [8,18,20,22,23]. First steps have been made by reproving the UIP for modal logics K, D, and T via nested sequents [12] and for S5 via hypersequents [11,13], the first time this is proved proof-theoretically for S5.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, layered sequents are used to provide the first proof-theoretic proof of the ULIP for K5. The method is adapted from [11,13] in which the UIP is proved for S5 based on hypersequents. We provide an algorithm to construct uniform Lyndon interpolants purely by syntactic means using the termination strategy of the proof search.…”

Section: Introductionmentioning

confidence: 99%

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Uniform Interpolation via Nested Sequents

Giessen¹,

Jalali²,

Kuznets³

2021

Logic, Language, Information, and Computation

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We introduce a Gentzen-style framework, called layered sequent calculi, for modal logic K5 and its extensions KD5, K45, KD45, KB5, and S5 with the goal to investigate the uniform Lyndon interpolation property (ULIP), which implies both the uniform interpolation property and the Lyndon interpolation property. We obtain complexityoptimal decision procedures for all logics and present a constructive proof of the ULIP for K5, which to the best of our knowledge, is the first such syntactic proof. To prove that the interpolant is correct, we use modeltheoretic methods, especially bisimulation modulo literals.⋆ Supported by a UKRI Future Leaders Fellowship, 'Structure vs Invariants in Proofs', project reference MR/S035540/1. ⋆⋆ Acknowledges the support of the Netherlands Organization for Scientific Research under grant 639.073.807 and the Czech Science Foundation Grant No. 22-06414L. ⋆ ⋆ ⋆ Supported by the Austrian Science Fund (FWF) ByzDEL project (P33600).

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Section: Discussionmentioning

confidence: 99%

Section: Definition 4 (Uip and Ulip)mentioning

confidence: 99%

Section: Whenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Uniform Interpolation via Nested Sequents

Giessen¹,

Jalali²,

Kuznets³

2021

Logic, Language, Information, and Computation

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Extensions of $$\textsf{K5}$$: Proof Theory and Uniform Lyndon Interpolation

van der Giessen,

Jalali,

Kuznets

2023

Lecture Notes in Computer Science

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We introduce a Gentzen-style framework, called layered sequent calculi, for modal logic $$\textsf{K5}$$ and its extensions $$\textsf{KD5}$$, $$\textsf{K45}$$, $$\textsf{KD45}$$, $$\textsf{KB5}$$, and $$\textsf{S5}$$ with the goal to investigate the uniform Lyndon interpolation property (ULIP), which implies both the uniform interpolation property and the Lyndon interpolation property. We obtain complexity-optimal decision procedures for all logics and present a constructive proof of the ULIP for $$\textsf{K5}$$, which to the best of our knowledge, is the first such syntactic proof. To prove that the interpolant is correct, we use model-theoretic methods, especially bisimulation modulo literals.

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Uniform interpolation via nested sequents and hypersequents

van der Giessen,

Jalali,

Kuznets

2024

Journal of Logic and Computation

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A modular proof-theoretic framework was recently developed to prove Craig interpolation for normal modal logics based on generalizations of sequent calculi (e.g. nested sequents, hypersequents and labelled sequents). In this paper, we turn to uniform interpolation, which is stronger than Craig interpolation. We develop a constructive method for proving uniform interpolation via nested sequents and apply it to reprove the uniform interpolation property for normal modal logics $\textsf{K}$, $\textsf{D}$ and $\textsf{T}$. We then use the know-how developed for nested sequents to apply the same method to hypersequents and obtain the first direct proof of uniform interpolation for $\textsf{S5}$ via a cut-free sequent-like calculus. While our method is proof-theoretic, the definition of uniform interpolation for nested sequents and hypersequents also uses semantic notions, including bisimulation modulo an atomic proposition.

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Uniform Interpolation and Admissible Rules

Cited by 4 publications

References 118 publications

Uniform Interpolation via Nested Sequents

Uniform Interpolation via Nested Sequents

Extensions of $$\textsf{K5}$$: Proof Theory and Uniform Lyndon Interpolation

Uniform interpolation via nested sequents and hypersequents

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