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23
A Switching Lemma for Small Restrictions and Lower Bounds for kDNF Resolution (Extended Abstract)
 SIAM J. Comput
, 2002
"... We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with kDNFs instead of cla ..."
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Cited by 45 (7 self)
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We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with kDNFs instead of clauses. We also obtain an exponential separation between depth d circuits of k + 1.
Resolution lower bounds for perfect matching principles
 Journal of Computer and System Sciences
"... For an arbitrary hypergraph H, letPM(H) be the propositional formula asserting that H contains a perfect matching. We show that every resolution refutation of PM(H) musthavesize exp Ω δ(H) λ(H)r(H)(log n(H))(r(H)+logn(H)) where n(H) is the number of vertices, δ(H) is the minimal degree of a vertex, ..."
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Cited by 42 (4 self)
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For an arbitrary hypergraph H, letPM(H) be the propositional formula asserting that H contains a perfect matching. We show that every resolution refutation of PM(H) musthavesize exp Ω δ(H) λ(H)r(H)(log n(H))(r(H)+logn(H)) where n(H) is the number of vertices, δ(H) is the minimal degree of a vertex, r(H) is the maximal size of an edge, and λ(H) is the maximal number of edges incident to two different vertices. For ordinary graphs G our general bound considerably simplifies to exp Ω (implying an exp(Ω(δ(G) 1/3)) lower bound that depends on the minimal degree only). As a direct corollary, every resolution proof of the functional ( ( onto)) version of must have size exp Ω (which the pigeonhole principle onto − FPHP m n n (log m) 2 δ(G) (log n(G)) 2 becomes exp ( Ω(n 1/3) ) when the number of pigeons m is unbounded). This in turn immediately implies an exp(Ω(t/n 3)) lower bound on the size of resolution proofs of the principle asserting that the circuit size of the Boolean function fn in n variables is greater than t. Inparticular,Resolution does not possess efficient proofs of NP ⊆ P/poly. These results relativize, in a natural way, to a more general principle M(UH) asserting that H contains a matching covering all vertices in U ⊆ V (H).
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.
Space Complexity In Propositional Calculus
 SIAM JOURNAL OF COMPUTING
, 2002
"... We study space complexity in the framework of propositional proofs. We consider a natural model analogous to Turing machines with a readonly input tape and such popular propositional proof systems as resolution, polynomial calculus, and Frege systems. We propose two di#erent space measures, corresp ..."
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Cited by 40 (8 self)
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We study space complexity in the framework of propositional proofs. We consider a natural model analogous to Turing machines with a readonly input tape and such popular propositional proof systems as resolution, polynomial calculus, and Frege systems. We propose two di#erent space measures, corresponding to the maximal number of bits, and clauses/monomials that need to be kept in the memory simultaneously. We prove a number of lower and upper bounds in these models, as well as some structural results concerning the clause space for resolution and Frege systems.
An exponential separation between regular and general resolution
 Theory of Computing
"... Dedicated to the memory of Misha Alekhnovich Abstract: This paper gives two distinct proofs of an exponential separation between regular resolution and unrestricted resolution. The previous best known separation between these systems was quasipolynomial. ACM Classification: F.2.2, F.2.3 AMS Classif ..."
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Cited by 34 (5 self)
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Dedicated to the memory of Misha Alekhnovich Abstract: This paper gives two distinct proofs of an exponential separation between regular resolution and unrestricted resolution. The previous best known separation between these systems was quasipolynomial. ACM Classification: F.2.2, F.2.3 AMS Classification: 03F20, 68Q17 Key words and phrases: resolution, proof complexity, lower bounds
Theories for Complexity Classes and their Propositional Translations
 Complexity of computations and proofs
, 2004
"... We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus. ..."
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Cited by 30 (7 self)
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We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.
The Proof Complexity of Linear Algebra
 IN SEVENTEENTH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS
, 2002
"... We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the rst feasible proofs of the CayleyHamilton theorem and other properties of the determina ..."
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Cited by 17 (6 self)
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We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the rst feasible proofs of the CayleyHamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities such as AB = I ! BA = I .
Monotone Simulations of Nonmonotone Proofs
, 2001
"... We show that an LK proof of size m of a monotone sequent (a sequent that contains only formulas in the basis ; ) can be turned into a proof containing only monotone formulas of size O(log m) and with the number of proof lines polynomial in m. Also we show that some interesting special cases, n ..."
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Cited by 17 (2 self)
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We show that an LK proof of size m of a monotone sequent (a sequent that contains only formulas in the basis ; ) can be turned into a proof containing only monotone formulas of size O(log m) and with the number of proof lines polynomial in m. Also we show that some interesting special cases, namely the functional and the onto versions of PHP and a version of the Matching Principle, have polynomial size monotone proofs. We prove that LK is polynomially bounded if and only if its monotone fragment is.
Optimal Proof Systems Imply Complete Sets For Promise Classes
 INFORMATION AND COMPUTATION
, 2001
"... A polynomial time computable function h : whose range is a set L is called a proof system for L. In this setting, an hproof for x 2 L is just a string w with h(w) = x. Cook and Reckhow de ned this concept in [11] and in order to compare the relative strength of dierent proof systems for ..."
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Cited by 14 (1 self)
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A polynomial time computable function h : whose range is a set L is called a proof system for L. In this setting, an hproof for x 2 L is just a string w with h(w) = x. Cook and Reckhow de ned this concept in [11] and in order to compare the relative strength of dierent proof systems for the set TAUT of tautologies in propositional logic, they considered the notion of psimulation. Intuitively, a proof system h psimulates h if any hproof w can be translated in polynomial time into an h for h(w). Krajcek and Pudlak [18] considered the related notion of simulation between proof systems where it is only required that for any hproof w there exists an h whose size is polynomially bounded in the size of w.
Proof complexity of pigeonhole principles
 IN PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON DEVELOPMENTS IN LANGUAGE THEORY
, 2002
"... The pigeonhole principle asserts that there is no injective mapping from m pigeons to n holes as long as m> n. It is amazingly simple, expresses one of the most basic primitives in mathematics and Theoretical Computer Science (counting) and, for these reasons, is probably the most extensively studie ..."
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Cited by 12 (1 self)
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The pigeonhole principle asserts that there is no injective mapping from m pigeons to n holes as long as m> n. It is amazingly simple, expresses one of the most basic primitives in mathematics and Theoretical Computer Science (counting) and, for these reasons, is probably the most extensively studied combinatorial principle. In this survey we try to summarize what is known about its proof complexity, and what we would still like to prove. We also mention some applications of the pigeonhole principle to the study of efficient provability of major open problems in computational complexity, as well as some of its generalizations in the form of general matching principles.