A sequent calculus and a theorem prover for standard conditional logics

Olivetti, Nicola; Pozzato, Gian Luca; Schwind, Camilla

doi:10.1145/1276920.1276924

Cited by 39 publications

(67 citation statements)

References 41 publications

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“…This reproves known complexity bounds originally shown in [13] (alternative short proofs using coalgebraic semantics are given in [17]). For CKCEM, the bound can be improved to coNP using dynamic programming in the same style as in [20].…”

Section: Complexity Of Proof Searchsupporting

confidence: 75%

“…We note that the complexity of CKMPCEM was explicitly left open in [13]. We also note that regrettably we were not able to reproduce our claim from [18] that CKMPCEM is in coNP, the problem being that the estimate (7.1) breaks down in the presence of (MP g ) and (MPCEM) g ; since no better lower bound than coNP is currently known for CKMPCEM and CKIDMPCEM, this means that the exact complexity of these logics remains open.…”

Section: Complexity Of Proof Searchmentioning

confidence: 72%

“…We note that the latter theorem was left as an open problem for the sequent system presented in [13]. To complete the treatment of conditional logics, we now turn to the system CKCEMMPID that arises by extending CK with the axioms correspoinding to conditional excluded middle, conditional modus ponens and identity.…”

Section: Cut Elimination For Extensions Of Ckcemmentioning

confidence: 99%

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Generic Modal Cut Elimination Applied to Conditional Logics

Pattinson¹,

Schröder²

2011

Logical Methods in Computer Science

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Abstract. We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature.

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Section: Complexity Of Proof Searchsupporting

confidence: 75%

Section: Complexity Of Proof Searchmentioning

confidence: 72%

Section: Cut Elimination For Extensions Of Ckcemmentioning

confidence: 99%

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Generic Modal Cut Elimination Applied to Conditional Logics

Pattinson¹,

Schröder²

2011

Logical Methods in Computer Science

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“…Another topic which deserves investigation is the precise relation of nested sequent calculi with other calculi for conditional logics, in particular with labelled calculi. Some labelled calculi for the conditional logics considered in this work have been presented in [28]. It would be interesting whether we can find a mapping between those calculi and nested sequent ones, as done by Fitting in his compresive work [12], where he establishes a tight correspondence between nested sequent calculi for modal logics and prefixed tableaux.…”

Section: Further Topicsmentioning

confidence: 98%

“…Just to mention some work, to the first stream belongs [2] proposing a calculus for (unnested) cumulative logic C (see below). More recently, [28] presents modular labeled calculi (of optimal complexity) for CK and some of its extensions, basing on the selection function semantics, and [19] presents modular labeled calculi for preferential logic PCL and its extensions. The latter calculi take advantage of a sort of hybrid modal translation.…”

Section: Introductionmentioning

confidence: 99%

Nested sequent calculi for normal conditional logics

Alenda

Olivetti

G.L.

2013

J Logic Computation

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Nested sequent calculi are a useful generalization of ordinary sequent calculi, where sequents are allowed to occur within sequents. Nested sequent calculi have been profitably employed in the area of (multi)-modal logic to obtain analytic and modular proof systems for these logics. In this work, we extend the realm of nested sequents by providing nested sequent calculi for the basic conditional logic CK and some of its significant extensions. We provide also a calculus for Kraus Lehman Magidor cumulative logic C. The calculi are internal (a sequent can be directly translated into a formula), cut-free and analytic. Moreover, they can be used to design (sometimes optimal) decision procedures for the respective logics, and to obtain complexity upper bounds. Our calculi are an argument in favour of nested sequent calculi for modal logics and alike, showing their versatility and power.

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Uniform Lyndon Interpolation for Basic Non-normal Modal Logics

Tabatabai

Iemhoff

Jalali

2021

Lecture Notes in Computer Science

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In this paper we show that the intuitionistic monotone modal logic iM has the uniform Lyndon interpolation property (ULIP). The logic iM is a non-normal modal logic on an intuitionistic basis, and the property ULIP is a strengthening of interpolation in which the interpolant depends only on the premise or the conclusion of an implication, respecting the polarities of the propositional variables. Our method to prove ULIP yields explicit uniform interpolants and makes use of a terminating sequent calculus for iM that we have developed for this purpose. As far as we know, the results that iM has ULIP and a terminating sequent calculus are the first of their kind for an intuitionistic non-normal modal logic. However, rather than proving these particular results, our aim is to show the flexibility of the constructive proof-theoretic method that we use for proving ULIP. It has been developed over the last few years and has been applied to substructural, intermediate, classical (non-)normal modal and intuitionistic normal modal logics. In light of these results, intuitionistic non-normal modal logics seem a natural next class to try to apply the method to, and we take the first step in that direction in this paper.

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A sequent calculus and a theorem prover for standard conditional logics

Cited by 39 publications

References 41 publications

Generic Modal Cut Elimination Applied to Conditional Logics

Generic Modal Cut Elimination Applied to Conditional Logics

Nested sequent calculi for normal conditional logics

Uniform Lyndon Interpolation for Basic Non-normal Modal Logics

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