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Polynomial Simulation and Refutation of Complex Formulas of Resolution over Linear Equations in Propositional Proof System

2014
*
American Journal of Modeling and Optimization
*

*In*this paper, we present that the propositional

*proof*system R(lin) (Resolution over Linear Equations) established

*by*Ran Raz and Iddo Tzameret is not a

*super*system, there exists a sequence of

*tautologies*... , which require

*proof*complexity exponential

*in*size of

*tautologies*. ... Since

*in*the late of sixth decade, Tseitin [10] pronounced the first

*super*-

*polynomial*lower bound to the

*lengths*of

*proofs*for resolution; therefore the resolution system is not a

*super*system yet resolution ...

##
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Yet another argument in favour of NP=CoNP
[article]

2020
*
arXiv
*
pre-print

The former

arXiv:2101.00003v1
fatcat:p3na5hk4grcevpoesvjx7stli4
*proof*shows how to obtain*polynomial*and,*polynomial**in*time checkable Dag-like*proofs*for all purely implicational Minimal logic*tautologies*. ... The existence of repeated parts is a consequence of the redundancy theorem for a family of*super*-*polynomial**proofs**in*the purely implicational Minimal logic. ... Thank the*proof*-theory group at Tuebingen-University, led*by*prof. Peter Schroeder-Heister. We want to thank Thomas Piecha and M. Arndt. ...##
###
The Cook-Reckhow definition
[article]

2019
*
arXiv
*
pre-print

*In*this note we shall highlight three particular definitions from the paper: of

*proof*systems, p-simulations and the pigeonhole principle formula, and discuss their role

*in*defining the field. ... to the subject as we understand it today, the most significant being Tseitin's 1968 paper, but none of them introduced general notions that would allow to make an explicit and universal link between

*lengths*-of-

*proofs*... The question about the minimum

*length*of a

*proof*of a statement (measured

*by*either the number of steps or

*by*the size, i.e. the number of symbols) was also studied, primarily

*in*the context of

*speed*-

*up*...

##
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A proper hierarchy of propositional sequent calculi

1996
*
Theoretical Computer Science
*

*In*this paper, we show that LK (

*in*tree form) with a cut rule, where the complexity of cut formulas is bounded

*by*(k + 1) has an exponential

*speed*-

*up*over the one bounded

*by*k. ... We also show that LK (

*in*tree form) with a bounded complexity cut rule does not

*polynomially*simulate cut-free LK

*in*DAG form. ... I would like to thank the anonymous referees for their helpful suggestions which enabled improvement and extention of the results

*in*the first draft of this paper. ...

##
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Refutation Of Hard-Determinable Formulas In The System "Resolution Over Linear Equations" And Its Generalization

2013
*
Pure and Applied Mathematics Journal
*

R (lin) is the generalization of R (Resolution System) and it is known that many

doi:10.11648/j.pamj.20130203.13
fatcat:cn6egqww4zelbjxewxrqu2hxii
*tautologies*, which require the exponential lower bounds of*proof*complexities*in*R, have*polynomially*bounded*proofs**in*... We research the power of the propositional*proof*system R(lin) (Resolution over Linear Equations) described*by*Ran Raz and Iddo Tzameret. ... Acknowledgments This work is supported*by*grant 11-1b023 of SSC of Goverment of Armenia. ...##
###
Proof Complexity of Non-classical Logics
[chapter]

2010
*
Lecture Notes in Computer Science
*

Traditionally, propositional

doi:10.1007/978-3-642-13562-0_3
fatcat:kmxntxzltreofhqdixqkrtsylu
*proofs*have been the main object of investigation*in**proof*complexity. ...*In*this paper we give the first survey of this field concentrating on recent developments*in**proof*complexity of non-classical logics. ★ This paper was written while visiting Sapienza University of Rome ... 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 ...##
###
Lower bounds for propositional proofs and independence results in bounded arithmetic (abstract)
[chapter]

1995
*
Lecture Notes in Computer Science
*

We begin with a highly informal discussion of the role played

doi:10.1007/3-540-60246-1_117
fatcat:4nthzsfwwramnj2bugsxm25rrm
*by*Bounded Arithmetic and propositional*proof*systems*in*the reasoning about the world of feasible computations. ... Then we survey some known lower bounds on the complexity of*proofs**in*various propositional*proof*systems, paying special attention to recent attempts on reducing such bounds to some purely complexity ... Notice that if P 0 is*polynomially*simulated*by*P 1 then it has a*polynomial**speed*-*up*over P 1 , that is s P1 ( ) s P0 ( ) O(1) , 2 TAUT C0 . ...##
###
Propositional proof systems and fast consistency provers
[article]

2020
*
arXiv
*
pre-print

Next it is proved that fast consistency provers do not exist if one considers RE axiomatized theories rather than theories with an axiom set that is recognizable

arXiv:2004.05431v1
fatcat:3bgdita2nfa3bhnvpvra4alaze
*in**polynomial*time. ...*In*this paper we define the notion of an unlikely fast consistency prover and prove that its existence is equivalent to NP= coNP. ...*In*this*proof*, we shall denote*by*T ⊢ ⋆ ϕ the statement that ϕ is provable*in*T*by*a*proof*whose*length*is bounded*by*some*polynomial*on the (*length*of) the parameters of ϕ. ...##
###
Propositional Proof Systems and Fast Consistency Provers

2007
*
Notre Dame Journal of Formal Logic
*

The stay

doi:10.1305/ndjfl/1187031410
fatcat:6lkusc3dujbkvbb2dpi2d4roki
*in*Prague was also financed*by*the Netherlands Organization for Scientific Research (NWO). ... Krajíček and Pudlák proved*in*[5] that the existence of an optimal propositional*proof*system is equivalent to the existence of a fast consistency prover. ...*In*this*proof*, we shall denote*by*T ϕ the statement that ϕ is provable*in*T*by*a*proof*whose*length*is bounded*by*some*polynomial*on the (*length*of) the parameters of ϕ. ...##
###
Towards NP–P via proof complexity and search

2012
*
Annals of Pure and Applied Logic
*

This is a survey of work on

doi:10.1016/j.apal.2011.09.009
fatcat:oud37acsqfgrdio62ir6ci24xi
*proof*complexity and*proof*search from a logicoalgorithmic viewpoint, as motivated*by*the P versus NP problem. ... We discuss propositional*proof*complexity, Cook's program,*proof*automatizability,*proof*search, algorithms for satisfiability, and the state of the art of our (*in*)ability to separate P and NP. ... Supported*in*part*by*NSF grants DMS-0700533 and DMS-1101228. ...##
###
Are there Hard Examples for Frege Systems?
[chapter]

1995
*
Feasible Mathematics II
*

The examples of combinatorial

doi:10.1007/978-1-4612-2566-9_3
fatcat:4e2z5jonanfcnj4rqtcwoddjky
*tautologies*that we consider seem to give at most a quasipolynomial*speed*-*up*of extended Frege*proofs*over Frege*proofs*, with the sole possible exception of*tautologies*based ... Bondy's theorem is shown to have constant-depth,*polynomial*-size*proofs**in*Frege+PHP, and to be equivalent*in*I∆0 to the pigeonhole principle. ...*In*particular, we would like to thank the following people: László Babai, Vašek Chvátal, Stephen Cook, Mauricio Karchmer, Jan Krajíček, Russell Impagliazzo, Steven Rudich, Michael Sipser, Herbert Wilf ...##
###
Relative efficiency of propositional proof systems: resolution vs. cut-free LK

2000
*
Annals of Pure and Applied Logic
*

Symbolic Logic 44 (1979) 36 -50 that tree resolution has

doi:10.1016/s0168-0072(00)00005-1
fatcat:avrku5jnibh5bfvyzeuhddzpiy
*super*-*polynomial**speed*-*up*over (tree) cut-free LK. ...*In*this paper, we introduce a*new*algorithm to eliminate atomic cuts and show that cut-free LK (DAG)*polynomially*simulates resolution when the input formula is expressed as a k-CNF formula. ... It was shown that tree resolution has*super*-*polynomial**speed*-*up*over analytic tableaux*in*[8] ; even truth table is more e cient than analytic tableaux on some class of*tautologies*[10] . ...##
###
Super-Blocked Clauses
[chapter]

2016
*
Lecture Notes in Computer Science
*

Such procedures identify redundant clauses and faithfully remove them, either before solving

doi:10.1007/978-3-319-40229-1_5
fatcat:qgpj7i4thjdw5kwockiij2777i
*in*a preprocessing phase or during solving, resulting*in*a considerable*speed**up*of the SAT solver. ... We introduce more powerful generalizations, called set-blocked clauses and*super*-blocked clauses, respectively. ... The k-*super*-blocking problem is*in*co-NP for all k ∈ N + .*Proof*. ...##
###
Proof Complexity of Non-classical Logics
[chapter]

2012
*
Lecture Notes in Computer Science
*

Traditionally, propositional

doi:10.1007/978-3-642-31485-8_1
fatcat:o2gh5kqzp5fpvb3fataonggkz4
*proofs*have been the main object of investigation*in**proof*complexity. ...*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. ... 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 [105] , the first major achievement*in*this programme ...##
###
Proof Complexity of Propositional Default Logic
[chapter]

2010
*
Lecture Notes in Computer Science
*

*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 . ... Of course,

*in*the presence of

*super*-

*polynomial*lower bounds to the

*proof*size this cannot be done

*in*

*polynomial*time. ...

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