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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 15 (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.
The Complexity of the Hajós Calculus
 SIAM J. Disc. Math
, 1992
"... The Haj'os Calculus is a simple, nondeterministic procedure which generates the class of non3colorable graphs. Mansfield and Welsch [MW] posed the question of whether there exist graphs which require exponentialsized Haj'os constructions. Unless NP 6= coNP , there must exist graphs which require ..."
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Cited by 11 (1 self)
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The Haj'os Calculus is a simple, nondeterministic procedure which generates the class of non3colorable graphs. Mansfield and Welsch [MW] posed the question of whether there exist graphs which require exponentialsized Haj'os constructions. Unless NP 6= coNP , there must exist graphs which require exponentialsized constructions, but to date, little progress has been made on this question, despite considerable effort. In this paper, we prove that the Haj'os Calculus generates polynomialsized constructions for all non3colorable graphs if and only if Extended Frege systems are polynomially bounded. Extended Frege systems are a very powerful family of proof systems for proving tautologies, and proving superpolynomial lower bounds for these systems is a longstanding, important problem in logic and complexity theory. We also establish a relationship between a complete subsystem of the Haj'os Calculus, and boundeddepth Frege systems; this enables us to prove exponential lower bounds on ...
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.
Exponential lower bounds for the Treelike Hajós Calculus
 Inf. Process. Lett
, 1997
"... The Haj'os Calculus is a simple, nondeterministic procedure which generates precisely the class of non3colorable graphs. In this note, we prove exponential lower bounds on the size of treelike Haj'os constructions. 1 Introduction The Haj'os calculus is a simple, nondeterministic procedure for g ..."
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Cited by 5 (0 self)
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The Haj'os Calculus is a simple, nondeterministic procedure which generates precisely the class of non3colorable graphs. In this note, we prove exponential lower bounds on the size of treelike Haj'os constructions. 1 Introduction The Haj'os calculus is a simple, nondeterministic procedure for generating the class of graphs that are not kcolorable [Haj]. Mansfield and Welsh [MW] have posed the problem of determining the complexity of this procedure; in particular, it is an open problem whether or not there exists a polynomialsize Haj'os construction for every non3 colorable graph. Because graph 3colorability is NPcomplete, if there were polynomialsize Haj'os constructions of all non3colorable graphs, then NP = coNP , so we expect that the Haj'os calculus is not polynomially bounded. However, there has been very little progress toward a proof of this conjecture, despite considerable effort. Pitassi and Urquhart [PU] have recently shown that the Haj'os calculus is polynomiall...
Separating daglike and treelike proof systems
 Accepted in LICS
, 2007
"... We show that treelike (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cutfree PK. Generally, we exhibit a family of sequents that have polynomial size cutfree proofs but requires superpolynomial treelike proofs even when the cut rule is allowed on ..."
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We show that treelike (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cutfree PK. Generally, we exhibit a family of sequents that have polynomial size cutfree proofs but requires superpolynomial treelike proofs even when the cut rule is allowed on a class of cutformulas that satisfies some plausible hardness assumption. This gives (in some cases, conditional) negative answers to several questions from a recent work of Maciel and Pitassi (LICS 2006). Our technique is inspired by the technique from Maciel and Pitassi. While the sequents used in earlier work are derived from the Pigeonhole principle, here we generalize Statman’s sequents. This gives the desired separation, and at the same time provides stronger results in some cases. 1
A Propositional Proof System for. . .
"... . In this paper we introduce Gentzenstyle quantied propositional proof systems L i for the theories R i 2 . We formalize the systems L i within the bounded arithmetic theory R 1 2 and we show that for i 1, R i 2 can prove the validity of a sequent derived by an L i proof. This stateme ..."
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. In this paper we introduce Gentzenstyle quantied propositional proof systems L i for the theories R i 2 . We formalize the systems L i within the bounded arithmetic theory R 1 2 and we show that for i 1, R i 2 can prove the validity of a sequent derived by an L i proof. This statement is formally called iRFN(L i ). We show if R i 2 ` 8xA(x) where A 2 b i , then for each integer n there is a translation of the formula A into quantied propositional logic such that R i 2 proves there is an L i proof of this translated formula. Using the proofs of these two facts we show that L i is in some sense the strongest system for which R i 2 can prove iRFN and we show for i j 2 that the 8 b j consequences of R i 2 are nitely axiomatized. 1. Introduction Propositional proof systems and bounded arithmetic are closely connected. Cook [10] introduced the equational arithmetic theory PV of polynomial time computable functions and showed PV co...
Lifting Lower Bounds for TreeLike Proofs
, 2011
"... It is known that constantdepth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider treelike Sequent Calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to b ..."
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It is known that constantdepth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider treelike Sequent Calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning smalldepth Boolean circuits, we prove exponential lower bounds for such proofs. We prove these lower bounds directly from the computational hardness assumption. We start with a lower bound for cutfree proofs and “lift ” it so it applies to proofs with constantdepth cuts. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known unconditional lower bound in the case where modular connectives are not used. We establish a conditional exponential separation between the power of constantdepth proofs that use different modular connectives. We show that these treelike proofs with constantdepth cuts cannot polynomially simulate similar daglike proofs, even when the daglike proofs are cutfree. We present a new proof of the nonfinite axiomatizability of the theory of bounded arithmetic I∆0(R). Finally, under a plausible hardness assumption concerning the polynomialtime hierarchy, we show that the hierarchy G ∗ i of quantified propositional proof systems does not collapse.