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77
Exponential Lower Bounds for the Pigeonhole Principle
, 1992
"... In this paper we prove an exponential lower bound on the size of boundeddepth Frege proofs for the pigeonhole principle (PHP). We also obtain an ~(log log rz)depth lower bound for any polynomialsized Frege proof of the pigeonhole principle. Our theorem nearly completes the search for the exact ..."
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Cited by 122 (28 self)
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In this paper we prove an exponential lower bound on the size of boundeddepth Frege proofs for the pigeonhole principle (PHP). We also obtain an ~(log log rz)depth lower bound for any polynomialsized Frege proof of the pigeonhole principle. Our theorem nearly completes the search for the exact complexity of the PHP, as Sam Buss has constructed polynomialsize, log ndepth Frege proofs for the PHP. The main lemma in our proof can be viewed as a general H&.stadstyle Switching Lemma for restrictions that are partial matchings. Our lower bounds for the pigeonhole principle improve on previous superpolynomial lower bounds.
Simplified and Improved Resolution Lower Bounds
 IN PROCEEDINGS OF THE 37TH IEEE FOCS
, 1996
"... We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probabili ..."
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Cited by 103 (8 self)
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We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probability approaching 1, any Resolution refutation of a randomly chosen 3CNF formula with at most n 6=5\Gammaffl clauses requires exponential size. Previous bounds applied only when the number of clauses was at most linear in the number of variables. For the pigeonhole principle our bound is a small improvement over previous bounds. Our proofs are more elementary than previous arguments, and establish a connection between Resolution proof size and maximum clause size.
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 76 (16 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...
An Exponential Lower Bound to the Size of Bounded Depth Frege . . .
, 1994
"... We prove lower bounds of the form exp (n ffl d ) ; ffl d ? 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for whic ..."
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Cited by 68 (10 self)
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We prove lower bounds of the form exp (n ffl d ) ; ffl d ? 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.
Lower Bounds For The Polynomial Calculus
, 1998
"... We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumpt ..."
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Cited by 56 (6 self)
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We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumptions. We also show that for some modifications of PHP n , expressible by polynomials of at most logarithmic degree, our bound can be improved to linear in the number of variables. Finally, we show that for any Boolean function f n in n variables, every polynomial calculus proof of the statement "f n cannot be computed by any circuit of size t," must have degree t=n). Loosely speaking, this means that low degree polynomial calculus proofs do not prove NP 6 P=poly.
Resolution Proofs of Generalized Pigeonhole Principles
, 1988
"... We extend results of A. Haken to give an exponential lower bound on the size of resolution proofs for propositional formulas encoding a generalized pigeonhole principle. These propositional formulas express the fact that there is no oneone mapping from c n objects to n objects when c > 1 ..."
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Cited by 47 (4 self)
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We extend results of A. Haken to give an exponential lower bound on the size of resolution proofs for propositional formulas encoding a generalized pigeonhole principle. These propositional formulas express the fact that there is no oneone mapping from c n objects to n objects when c > 1 . As a corollary, resolution proof systems do not psimulate constant formula depth Frege proof systems.
A Theoretical Analysis of Reasoning By Symmetry in FirstOrder Logic (Extended Abstract)
 In AAAI Workshop on Tractable Reasoning
, 1992
"... ) James M. Crawford AT&T Bell Laboratories 600 Mountain Ave. Murray Hill, NJ 079740636 jc@research.att.com Abstract Many classes of reasoning problems display a large amount of symmetry. In mathematical and commonsense reasoning, such symmetries are often used to reduce the difficulty of r ..."
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Cited by 42 (1 self)
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) James M. Crawford AT&T Bell Laboratories 600 Mountain Ave. Murray Hill, NJ 079740636 jc@research.att.com Abstract Many classes of reasoning problems display a large amount of symmetry. In mathematical and commonsense reasoning, such symmetries are often used to reduce the difficulty of reasoning. In this paper we show how symmetries can be used in automated reasoning both to reduce or avoid case analysis, and to reduce the scope of existential quantification. We show further that the computational complexity of the symmetry detection problem is equivalent (within a polynomial factor) to that of the graph isomorphism problem. This result is significant because the complexity of the graph isomorphism problem is a hard open problem (it is believed to lie between P and NP). Further, for graphs with bounded degree, graph isomorphism is known to be polynomial decidable. We then show how P'olya's theorem can be used to count the number of interpretations of a logical theory which are ...
On the Proof Complexity of Deep Inference
, 2000
"... We obtain two results about the proof complexity of deep inference: 1) deepinference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deepinference proof systems that exhibit an exponential ..."
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Cited by 41 (14 self)
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We obtain two results about the proof complexity of deep inference: 1) deepinference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deepinference proof systems that exhibit an exponential speedup over analytic Gentzen proof systems that they polynomially simulate.
Upper and Lower Bounds for Treelike Cutting Planes Proofs
 In 9th IEEE Symposium on Logic in Computer Science
, 1994
"... In this paper we study the complexity of Cutting Planes (CP) refutations, and treelike CP refutations. Treelike CP proofs are natural and still quite powerful. In particular, the propositional pigeonhole principle (PHP) has been shown to have polynomialsized treelike CP proofs. Our main result s ..."
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Cited by 40 (8 self)
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In this paper we study the complexity of Cutting Planes (CP) refutations, and treelike CP refutations. Treelike CP proofs are natural and still quite powerful. In particular, the propositional pigeonhole principle (PHP) has been shown to have polynomialsized treelike CP proofs. Our main result shows that a family of tautologies, introduced in this paper requires exponentialsized treelike CP proofs. We obtain this result by introducing a new method which relates the size of a CP refutation to the communication complexity of a related search problem. Because these tautologies have polynomialsized Frege proofs, it follows that treelike CP cannot polynomially simulate Frege systems. 1 Introduction An important open problem is to determine whether there exists a propositional proof system that admits short (polynomial size) proofs for all tautologies, or equivalently, whether or not NP equals coNP. In order to attack Research supported by NSF NYI grant CCR92570979 y Research su...
Resolution and the weak pigeonhole principle
 IN CSL
, 1997
"... We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give lower bounds for treelike resolution proofs. We present a normal form for resolution proofs of pigeonhole principles based on a new monotone resolution rule. ..."
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Cited by 36 (3 self)
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We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give lower bounds for treelike resolution proofs. We present a normal form for resolution proofs of pigeonhole principles based on a new monotone resolution rule.