Intuitionistic Frege systems are polynomially equivalent

Mint︠s︡, Grigori; Kojevnikov, Arist

doi:10.1007/s10958-006-0116-8

Cited by 7 publications

(13 citation statements)

References 17 publications

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“…Non-classical proof complexity typically deals with Frege systems and equivalent systems like sequent calculi or natural deduction: see for example Buss and Mints [5], Buss and Pudlák [6], Ferrari et al [9], Mints and Kojevnikov [16], Jeřábek [14], Hrubeš [10,11]. (Though sometimes the number of lines is taken as the basic complexity measure instead of size, which E-mail address: jerabek@math.cas.cz.…”

Section: Introductionmentioning

confidence: 99%

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

Jeřábek

2009

Annals of Pure and Applied Logic

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Communicated by U. Kohlenbach MSC: 03F20 03B45 03B55 Keywords: Propositional proof complexity Frege system Modal logic Intermediate logic a b s t r a c tWe investigate the substitution Frege (SF ) proof system and its relationship to extended Frege (EF ) in the context of modal and superintuitionistic (si) propositional logics. We show that EF is p-equivalent to tree-like SF , and we develop a ''normal form'' for SF -proofs. We establish connections between SF for a logic L, and EF for certain bimodal expansions of L.We then turn attention to specific families of modal and si logics. We prove pequivalence of EF and SF for all extensions of KB, all tabular logics, all logics of finite depth and width, and typical examples of logics of finite width and infinite depth. In most cases, we actually show an equivalence with the usual EF system for classical logic with respect to a naturally defined translation.On the other hand, we establish exponential speed-up of SF over EF for all modal and si logics of infinite branching, extending recent lower bounds by P. Hrubeš. We develop a model-theoretical characterization of maximal logics of infinite branching to prove this result.

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Section: Introductionmentioning

confidence: 99%

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

Jeřábek

2009

Annals of Pure and Applied Logic

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“…Here we recall the definitions of Frege systems for Intuitionistic and Johansson's logics, which are given in Mints and Kozhevnikov [10] and Sayadyan and Chubaryan [11] correspondingly.…”

Section: Results For Frege Systems Of Intuitionistic and Johansson's mentioning

confidence: 99%

“…We will use the current concepts of a propositional formula, a classical tautology and non-classical tautologies, sequent, sequent systems for non-classical propositional logics [7][8][9] Frege systems for Intuitionistic and Johansson's logics [10,11] and proof complexity [12]. Let us recall some of them.…”

Section: Preliminariesmentioning

confidence: 99%

Some Properties of Several Proof Systems for Intuitionistic, Johanssons and Monotone Propositional Logics

Chubaryan

Arman

Petrosyan

2018

Journal of Asian Scientific Research

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Article HistoryIn this paper we investigate two properties of some propositional systems of Intuitionistic, Johansson's and Monotone logics: 1) the relations between the proofs complexities of strongly equal tautologies (valid sequents) and 2) the relations between the proofs complexities of minimal tautologies (valid sequents) and of results of substitutions in them. We show that 1) strongly equal tautologies (valid sequents) can have essential different proof complexities in the same system and 2) the result of substitution can be proved easier, than corresponding minimal tautology (valid sequents), therefore the systems, which are considered in this paper, are no monotonous neither by lines nor by size.

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“…Building on a characterization of admissible rules for intuitionistic logic by Ghilardi [Ghi99], Iemhoff [Iem01] constructed an explicit set of rules which forms a basis for all admissible intuitionistic rules. Using this basis, Mints and Kojevnikov [MK06] were able to prove the equivalence of all intuitionistic Frege systems:…”

Section: Simulations Between Non-classical Proof Systemsmentioning

confidence: 99%

“…Theorem 6 (Mints, Kojevnikov [MK06]). All intuitionistic Frege systems in the language →, ∧, ∨, ⊥ are polynomially equivalent.…”

Section: Simulations Between Non-classical Proof Systemsmentioning

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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Intuitionistic Frege systems are polynomially equivalent

Cited by 7 publications

References 17 publications

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

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

Some Properties of Several Proof Systems for Intuitionistic, Johanssons and Monotone Propositional Logics

Theory and Applications of Models of Computation

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