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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
Disjoint NPPairs
, 2003
"... We study the question of whether the class DisNP of disjoint pairs (A, B) of NPsets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NPsets that is N ..."
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Cited by 20 (6 self)
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We study the question of whether the class DisNP of disjoint pairs (A, B) of NPsets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NPsets that is NPhard. We show under reasonable hypotheses that nonsymmetric disjoint NPpairs exist, which provides additional evidence for the existence of Pinseparable disjoint NPpairs. We construct
Tautologies From PseudoRandom Generators
, 2001
"... We consider tautologies formed from a pseudorandom number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a ..."
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Cited by 16 (0 self)
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We consider tautologies formed from a pseudorandom number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a form of a hardness condition posed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at nonlogicians, of the relation between propositional proof complexity and bounded arithmetic. It is a fundamental problem of mathematical logic to decide if tautologies can be inferred in propositional calculus in substantially fewer steps than it takes to check all possible truth assignments. This is closely related to the famous P/NP problem of Cook [3]. By propositional calculus I mean any textbook system based on a nite number of inference rules and axiom schemes that is sound and complete. The qualication substantially less means that the nu...
How to Lie Without Being (easily) Convicted and the Lengths of Proofs in Propositional Calculus
"... We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the sec ..."
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Cited by 14 (1 self)
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We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie.
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 10 (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.
Complete Problems for Promise Classes by Optimal Proof Systems for Test Sets
 In Proc. 13th Annual IEEE Conference on Computational Complexity, CC 98
, 1998
"... We present a uniform approach to investigate the relationship between the existence of complete sets for promise classes and the existence of (p)optimal proof systems for certain languages. Central to our approach is the notion of a test set which can be used to verify that a given nondeterministic ..."
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Cited by 8 (3 self)
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We present a uniform approach to investigate the relationship between the existence of complete sets for promise classes and the existence of (p)optimal proof systems for certain languages. Central to our approach is the notion of a test set which can be used to verify that a given nondeterministic polynomialtime machine obeys the promise on a given input. Basically, we show that a promise class C has a manyone complete language if and only if there is a test set for C which has a poptimal proof system. As an application we are able to improve earlier results. For example, we show that NP "coN P has a manyone complete language, provided that the set TAUT of all valid boolean formulas as well as the set SAT of all satisfiable boolean formulas have poptimal proof systems. We also apply the result to other classes and show, for example, that the probabilistic complexity classes BPP, RP , and ZPP have manyone complete languages, provided that the set TAUT 2 of all valid \Pi 2 for...
A BottomUp Approach to Foundations of Mathematics
"... this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject ..."
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Cited by 2 (0 self)
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this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject
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.
Scribe Notes
, 2002
"... Contents Lecture #1, Robert Ellis 4 1 Introduction to Propositional Logic 4 2 Propositional Proof Systems 6 Lecture #2, Sashka Davis 10 3 Introduction to Frege Proof Systems 10 4 The Completeness and Implicational Completeness Theorems 12 5 Observations 13 6 Psimulate 13 Lecture #3, Reid Andersen ..."
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Contents Lecture #1, Robert Ellis 4 1 Introduction to Propositional Logic 4 2 Propositional Proof Systems 6 Lecture #2, Sashka Davis 10 3 Introduction to Frege Proof Systems 10 4 The Completeness and Implicational Completeness Theorems 12 5 Observations 13 6 Psimulate 13 Lecture #3, Reid Andersen 15 7 pSimulation 15 8 Extended Frege Sytems 16 Lecture #4, Alan Nash 17 9 The Unification Problem 17 10 Extended Frege Systems (Again) 18 Lecture #5, Tamsen Dunn 20 11 The Pigeon Hole Principle 20 12 TreeLike versus NonTreeLike Proofs 22 Lecture #6, Rosalie Iemho# 24 13 Substitution Frege systems 24 14 The best known lower bounds on proof lengths 26 15 Resolution 27 Lecture #7, Dan Curtis 29 16 Completeness and Soundness of Resolution Proofs 29 Lecture #8, Bryant Forsgren 33 18 Views of Resolution Refutations 33 19 Exponential Lower Bounds on Refutation Proofs of the Pigeon Hole Principle 34 Lecture #9, Nathan Segerlind 37 n 37 22 Lower Bounds for Resolution Proofs of Circuit Lo