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41
Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations
, 1997
"... We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, i ..."
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Cited by 135 (5 self)
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We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.
Lower Bounds for Cutting Planes Proofs with Small Coefficients
, 1995
"... We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the cl ..."
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Cited by 77 (19 self)
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We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of smallweight CP , our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems : (1) Treelike CP proofs cannot polynomially simulate nontreelike CP proofs. (2) Treelike CP proofs and BoundeddepthFrege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP proof system. In particular, they work for CP with a deduction rule, and also for proof systems that allow any formula with small communication complexity, and any set of sound rules of inference. 1 Introduction One of the most fundamental questions in pro...
On the Weak Pigeonhole Principle
, 2001
"... We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of Ramsey theorem. In particular, we link the proof complexity of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of ..."
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Cited by 69 (3 self)
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We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of Ramsey theorem. In particular, we link the proof complexity of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of (extensions of) treelike resolution proofs of Ramsey theorem. We establish a connection between provability of WPHP in fragments of bounded arithmetic and cryptographic assumptions (the existence of oneway functions). In particular, we show that functions violating WPHP 2n n are oneway and, on the other hand, that oneway permutations give rise to functions violating PHP n+1 n , and that strongly collisionfree families of hash functions give rise to functions violating WPHP 2n n (all in suitable models of bounded arithmetic). Further we formulate few problems and conjectures; in particular, on the structured PHP (introduced here) and on the unrelativised WPHP. The symbol WPHP m n...
On Interpolation and Automatization for Frege Systems
, 2000
"... The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the ..."
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Cited by 51 (7 self)
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The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexitytheoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC 0 Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC 0 Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial size TC 0 Frege. As a corollary, we obtain that TC 0 Frege as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integ...
Resolution is Not Automatizable Unless W[P] is Tractable
 In 42nd Annual IEEE Symposium on Foundations of Computer Science
, 2001
"... We show that neither Resolution nor treelike Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is xedparameter tractable by randomized algorithms with onesided error. ..."
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Cited by 48 (0 self)
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We show that neither Resolution nor treelike Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is xedparameter tractable by randomized algorithms with onesided error.
A Study of Proof Search Algorithms for Resolution and Polynomial Calculus
, 1999
"... This paper is concerned with the complexity of proofs and of searching for proofs in two propositional proof system: Resolution and Polynomial Calculus (PC). For the former system we show that the recently proposed algorithm of [BW99] for searching for proofs cannot give better than weakly exponenti ..."
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Cited by 46 (5 self)
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This paper is concerned with the complexity of proofs and of searching for proofs in two propositional proof system: Resolution and Polynomial Calculus (PC). For the former system we show that the recently proposed algorithm of [BW99] for searching for proofs cannot give better than weakly exponential performance. This is a consequence of showing optimality of their general relationship reffered to in [BW99] as sizewidth tradeoff. We moreover obtain the optimality of the sizewidth tradeoff for the widely used restrictions of resolution: Regular, DavisPutnam, Negative, Positive and Linear. As for the second system, we show that the translation to polynomials of a CNF formula having short resolution proofs, cannot be refuted in PC with degree less than \Omega\Gammaan/ n). A consequence of this is that the simulation of resolution by PC of [CEI97] cannot be improved to better than quasipolynomial in the case we start with small resolution proofs. We conjecture that the simu...
Pseudorandom Generators Hard for kDNF Resolution and Polynomial Calculus Resolution
, 2003
"... A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the ..."
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Cited by 41 (4 self)
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A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the propositional proof system that extends Resolution by allowing kDNFs instead of clauses.
On the Automatizability of Resolution and Related Propositional Proof Systems
, 2002
"... We analyse the possibility that a system that simulates Resolution is automatizable. We call this notion "weak automatizability". We prove ..."
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Cited by 37 (5 self)
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We analyse the possibility that a system that simulates Resolution is automatizable. We call this notion "weak automatizability". We prove
Nonautomatizability of boundeddepth Frege proofs
, 1999
"... In this paper, we show how to extend the argument due to Bonet, Pitassi and Raz to show that boundeddepth Frege proofs do not have feasible interpolation, assuming that factoring of Blum integers or computing the DiffieHellman function is sufficiently hard. It follows as a corollary that boundedde ..."
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Cited by 29 (10 self)
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In this paper, we show how to extend the argument due to Bonet, Pitassi and Raz to show that boundeddepth Frege proofs do not have feasible interpolation, assuming that factoring of Blum integers or computing the DiffieHellman function is sufficiently hard. It follows as a corollary that boundeddepth Frege is not automatizable; in other words, there is no deterministic polynomialtime algorithm that will output a short proof if one exists. A notable feature of our argument is its simplicity.