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The relative efficiency of propositional proof systems
 Journal of Symbolic Logic
, 1979
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Cited by 331 (5 self)
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http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
Are there Hard Examples for Frege Systems?
"... It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. S ..."
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Cited by 20 (2 self)
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It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speedup of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl. It is
On Godel's theorems on lengths of proofs II: Lower bounds for recognizing k symbol provability
 in Feasible Mathematics II, P. Clote and
, 1995
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Bounded Arithmetic and Propositional Proof Complexity
 in Logic of Computation
, 1995
"... This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of t ..."
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Cited by 11 (0 self)
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This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cutfree proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.
Some Remarks on Lengths of Propositional Proofs
, 2002
"... We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum lengt ..."
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Cited by 10 (1 self)
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We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depth d Frege proofs of m lines can be transformed into depth d proofs of O(m^(d+1)) symbols. We show that renaming Frege proof systems are pequivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed.
The Deduction Rule and Linear and Nearlinear Proof Simulations
"... ... that a Frege proof of n lines can be transformed into a treelike Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus treelike systems simulate Frege systems with proof lengths bounded by O(n log n). ..."
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Cited by 10 (5 self)
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... that a Frege proof of n lines can be transformed into a treelike Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus treelike systems simulate Frege systems with proof lengths bounded by O(n log n).
Number of symbols in Frege proofs with and without the deduction rule
 in Arithmetic, Proof Theory and Computational Complexity, P. Clote and
, 1993
"... Abstract Frege systems with the deduction rule produce at most quadratic speedup over Frege systems using as a measure of length the number of symbols in the proof. We study whether that speedup is in reality smaller. We show that the speedup is linear when the Frege proofs are treelike. Also, two ..."
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Cited by 6 (0 self)
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Abstract Frege systems with the deduction rule produce at most quadratic speedup over Frege systems using as a measure of length the number of symbols in the proof. We study whether that speedup is in reality smaller. We show that the speedup is linear when the Frege proofs are treelike. Also, two groups of formulas, permutation formulas and transitive closure formulas, that seemed most likely to produce an almost quadratic speedup when using the deduction rule, are shown to produce only log n and log 2 n factors respectively. 1 Introduction A Frege proof system is an inference system for propositional logic in which the only rule of inference is Modus Ponens.
Propositional proof complexity — an introduction
 In Ulrich Berger and Helmut Schwichtenberg, editors, Computational Proof Theory
, 1997
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Are There Hard Examples for Frege Proof Systems?
, 1995
"... It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatoriM tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surpri ..."
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Cited by 1 (0 self)
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It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatoriM tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speedup of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl.
Descriptive Complexity and Finite Models
"... This paper introduces algebraic proof systems for the propositional calculus. We present new results concerning the relative efficiency of these systems, and also survey what is currently known. Many open problems are presented. 1 Introduction A fundamental problem in logic and computer science is ..."
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This paper introduces algebraic proof systems for the propositional calculus. We present new results concerning the relative efficiency of these systems, and also survey what is currently known. Many open problems are presented. 1 Introduction A fundamental problem in logic and computer science is understanding the efficiency of propositional proof systems. It has been known for a long time that NP = coNP if and only if there exists an efficient propositional proof system, but despite 25 years of research, this problem is still not resolved. (See [46] for an excellent survey of this area.) The intention of the present article is to introduce a new algebraic approach to this problem. Our proof systems are simpler than classical proof systems, and purely algebraic. It is our hope that by studying proof complexity in this light, that new upper and lower bound techniques may emerge. The use of the Nullstellensatz for propositional refutations may have been first suggested in a paper by Lo...