Substitution Frege and extended Frege proof systems in non-classical logics

Jeřábek, Emil

doi:10.1016/j.apal.2008.10.005

Cited by 19 publications

(28 citation statements)

References 14 publications

(30 reference statements)

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“…For any modal or superintuitionistic logic, extended Frege and tree-like substitution Frege are polynomially equivalent. 15 This shows that Cook and Reckhow's intuition on extended vs. substitution Frege was indeed correct and is further confirmed by results of Jeřábek [65] who shows that going from extended to substitution Frege corresponds to a conservative strengthening of the underlying logic by a new modal operator. Building on these characterisations, Jeřábek exhibits examples for logics where the EF vs. SF question receives different answers:…”

Section: Simulations Between Non-classical Proof Systemssupporting

confidence: 66%

Lectures on Logic and Computation

Bezhanishvili

Goranko

2012

Lecture Notes in Computer Science

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Abstract. Proof complexity is an interdisciplinary area of research utilising techniques from logic, complexity, and combinatorics towards the main aim of understanding the complexity of theorem proving procedures. Traditionally, propositional proofs have been the main object of investigation in proof complexity. Due their richer expressivity and numerous applications within computer science, also non-classical logics have been intensively studied from a proof complexity perspective in the last decade, and a number of impressive results have been obtained. In these notes we give an introduction to this recent field of proof complexity of non-classical logics. We cover results from proof complexity of modal, intuitionistic, and non-monotonic logics. Some of the results are surveyed, but in addition we provide full details of a recent exponential lower bound for modal logics due to Hrubeš [60] and explain the complexity of several sequent calculi for default logic [16,13]. To make the text self-contained, we also include necessary background information on classical proof systems and non-classical logics.

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Section: Simulations Between Non-classical Proof Systemssupporting

confidence: 66%

Lectures on Logic and Computation

Bezhanishvili

Goranko

2012

Lecture Notes in Computer Science

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“…While there is a rich body of results for propositional proof systems (cf. [18]), proof complexity of non-classical logics has only recently attracted more attention, and a number of exciting results have been obtained for modal and intuitionistic logics [15][16][17]. Starting with Reiter's work [22], several proof-theoretic methods have been developed for default logic (cf.…”

Section: Introductionmentioning

confidence: 99%

Proof Complexity of Propositional Default Logic

Beyersdorff

Meier²,

Müller

et al. 2010

Theory and Applications of Satisfiability Testing – SAT 2010

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Abstract. Default logic is one of the most popular and successful formalisms for non-monotonic reasoning. In 2002, Bonatti and Olivetti introduced several sequent calculi for credulous and skeptical reasoning in propositional default logic. In this paper we examine these calculi from a proof-complexity perspective. In particular, we show that the calculus for credulous reasoning obeys almost the same bounds on the proof size as Gentzen's system LK . Hence proving lower bounds for credulous reasoning will be as hard as proving lower bounds for LK . On the other hand, we show an exponential lower bound to the proof size in Bonatti and Olivetti's enhanced calculus for skeptical default reasoning.

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“…Intuitively therefore, lower bounds to the lengths of proofs in non-classical logic should be easier to obtain, as they "only" target at separating NP and PSPACE. In some sense the results of Hrubeš [Hru09] and Jeřábek [Jeř09] on non-classical Frege systems (see Sect. 4) confirm this intuition: they obtain exponential lower bounds for modal and intuitionistic Frege systems (in fact, even extended Frege) whereas to reach such results in propositional proof complexity we have to overcome a strong current barrier [BBP95].…”

Section: Why Non-classical Logics?mentioning

confidence: 99%

“…As two last examples, 4 is obtained by extending 4 by □ → and 4Grz by extending 4 by □(□( → □ ) → ) → □ . For more information on modal logics we refer to [BdRV01] or the thorough introduction in [Jeř09].…”

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confidence: 99%

Theory and Applications of Models of Computation

2010

Lecture Notes in Computer Science

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Abstract. Proof complexity is an interdisciplinary area of research utilizing techniques from logic, complexity, and combinatorics towards the main aim of understanding the complexity of theorem proving procedures. Traditionally, propositional proofs have been the main object of investigation in proof complexity. Due their richer expressivity and numerous applications within computer science, also non-classical logics have been intensively studied from a proof complexity perspective in the last decade, and a number of impressive results have been obtained. In this paper we give the first survey of this field concentrating on recent developments in proof complexity of non-classical logics. Propositional Proof ComplexityOne of the starting points of propositional proof complexity is the seminal paper of Cook and Reckhow [CR79] where they formalized propositional proof systems as polynomial-time computable functions which have as their range the set of all propositional tautologies. In that paper, Cook and Reckhow also observed a fundamental connection between lengths of proofs and the separation of complexity classes: they showed that there exists a propositional proof system which has polynomial-size proofs for all tautologies (a polynomially bounded proof system) if and only if the class NP is closed under complementation. From this observation the so called Cook-Reckhow programme was derived which serves as one of the major motivations for propositional proof complexity: to separate NP from coNP (and hence P from NP) it suffices to show super-polynomial lower bounds to the size of proofs in all propositional proof systems.Although the first super-polynomial lower bound to the lengths of proofs had already been shown by Tseitin in the late 60's for a sub-system of resolution [Tse68], the first major achievement in this programme was made by Haken in 1985 when he showed an exponential lower bound to the proof size in Resolution for a sequence of propositional formulas describing the pigeonhole principle [Hak85]. In the last two decades these lower bounds were extended to a number of further propositional systems such as the

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Substitution Frege and extended Frege proof systems in non-classical logics

Cited by 19 publications

References 14 publications

Lectures on Logic and Computation

Lectures on Logic and Computation

Proof Complexity of Propositional Default Logic

Theory and Applications of Models of Computation

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