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45
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 46 (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.
Lower Bounds for the Weak Pigeonhole Principle and Random Formulas beyond Resolution
, 2002
"... We work with an extension of Resolution, called Res(2), that allows clauses with conjunctions of two literals. In this system there are rules to introduce and eliminate such conjunctions. We prove that the weak pigeonhole principle PHP cn n and random unsatisfiable CNF formulas require exponentials ..."
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Cited by 36 (12 self)
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We work with an extension of Resolution, called Res(2), that allows clauses with conjunctions of two literals. In this system there are rules to introduce and eliminate such conjunctions. We prove that the weak pigeonhole principle PHP cn n and random unsatisfiable CNF formulas require exponentialsize proofs in this system. This is the strongest system beyond Resolution for which such lower bounds are known. As a consequence to the result about the weak pigeonhole principle, Res(log) is exponentially more powerful than Res(2). Also we prove that Resolution cannot polynomially simulate Res(2), and that Res(2) does not have feasible monotone interpolation solving an open problem posed by Krajcek.
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 34 (6 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
On the Complexity of Resolution with Bounded Conjunctions
 IN 29TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING
, 2004
"... We analyze size and space complexity of Res(k), a family of propositional proof systems introduced by Kraj'icek in [21] which extends Resolution by allowing disjunctions of conjunctions of up to k 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respe ..."
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Cited by 27 (4 self)
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We analyze size and space complexity of Res(k), a family of propositional proof systems introduced by Kraj'icek in [21] which extends Resolution by allowing disjunctions of conjunctions of up to k 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respect to proof size and also with respect to space. Moreover
The complexity of propositional proofs
 Bulletin of Symbolic Logic
"... Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorit ..."
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Cited by 19 (0 self)
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Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents
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 stu ..."
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Cited by 10 (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.
The provable total search problems of bounded arithmetic
, 2007
"... We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument use ..."
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Cited by 8 (4 self)
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We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument uses a translation of first order proofs into large, uniform propositional proofs in a system in which the soundness of the rules can be witnessed by polynomial time reductions between games. We show that ∀ ˆ Σ b 1(α) conservativity of of T i+1 2 (α) over T i 2(α) already implies ∀ ˆ Σ b 1(α) conservativity of T2(α) over T i 2(α). We translate this into propositional form and give a polylogarithmic width CNF GI3 such that if GI3 has small R(log) refutations then so does any polylogarithmic width CNF which has small constant depth refutations. We prove a resolution lower bound for GI3. We use our characterization to give a sufficient condition for the totality of a relativized NP search problem to be unprovable in T i 2(α) in terms of a nonlogical question about multiparty communication protocols.
Improved Bounds on the Weak Pigeonhole Principle and Infinitely Many Primes from Weaker Axioms
, 2001
"... We show that the known boundeddepth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a sizedepth tradeoff upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem ..."
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Cited by 8 (2 self)
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We show that the known boundeddepth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a sizedepth tradeoff upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnjascij's Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in I \Delta 0 with a provably weaker `large number assumption' than previously needed.
Separation Results for the Size of ConstantDepth Propositional Proofs
, 2004
"... This paper proves exponential separations between depth d  LK and depth (d + 2 )  LK for every d 2 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d  LK and depth (d+1)  LK for d N . We investigate the relationship between ..."
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Cited by 5 (3 self)
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This paper proves exponential separations between depth d  LK and depth (d + 2 )  LK for every d 2 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d  LK and depth (d+1)  LK for d N . We investigate the relationship between the sequencesize, treesize and height of depth d  LKderivations for d 2 N , and describe transformations between them.