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59
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.
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.
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 39 (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).
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 39 (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.
Lower Bounds for Polynomial Calculus: NonBinomial Case
, 2001
"... We generalize recent linear lower bounds for Polynomial Calculus based on binomial ideals. We produce a general hardness criterion (that we call immunity) which is satisfied by a random function and prove linear lower bounds on the degree of PC refutations for a wide class of tautologies based on im ..."
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Cited by 37 (9 self)
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We generalize recent linear lower bounds for Polynomial Calculus based on binomial ideals. We produce a general hardness criterion (that we call immunity) which is satisfied by a random function and prove linear lower bounds on the degree of PC refutations for a wide class of tautologies based on immune functions. As some applications of our techniques, we introduce mod p Tseitin tautologies in the Boolean case (e.g. in the presence of axioms x 2 i = x i ), prove that they are hard for PC over fields with characteristic different from p, and generalize them to Flow tautologies which are based on the MAJORITY function and are proved to be hard over any field. We also show the Ω(n) lower bound for random kCNF's over fields of characteristic 2.
Setting 2 variables at a time yields a new lower bound for random 3SAT (Extended Abstract)
 STOC
, 2000
"... Let X be a set of n Boolean variables and denote by C(X) the set of all 3clauses over X, i.e. the set of all 8(3) possible disjunctions of three distinct, noncomplementary literais from variables in X. Let F(n, m) be a random 3SAT formula formed by selecting, with replacement, m clauses uniformly ..."
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Cited by 35 (4 self)
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Let X be a set of n Boolean variables and denote by C(X) the set of all 3clauses over X, i.e. the set of all 8(3) possible disjunctions of three distinct, noncomplementary literais from variables in X. Let F(n, m) be a random 3SAT formula formed by selecting, with replacement, m clauses uniformly at random from C(X) and taking their conjunction. The satisfiability threshold conjecture asserts that there exists a constant ra such that as n+ c¢, F(n, rn) is satisfiable with probability that tends to 1 if r < ra, but unsatisfiable with probability that tends to 1 if r:> r3. Experimental evidence suggests rz ~ 4.2. We prove rz> 3.145 improving over the previous best lower bound r3> 3.003 due to Frieze and Suen. For this, we introduce a satisfiability heuristic that works iteratively, permanently setting the value of a pair of variables in each round. The framework we develop for the analysis of our heuristic allows us to also derive most previous lower bounds for random 3SAT in a uniform manner and with little effort.
Computational properties of argument systems satisfying graphtheoretic constraints
 Artificial Intelligence
, 2007
"... One difficulty that arises in abstract argument systems is that many natural questions regarding argument acceptability are, in general, computationally intractable having been classified as complete for classes such as NP, coNP, and ¢¡ £. In consequence, a number of researchers have considered me ..."
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Cited by 31 (8 self)
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One difficulty that arises in abstract argument systems is that many natural questions regarding argument acceptability are, in general, computationally intractable having been classified as complete for classes such as NP, coNP, and ¢¡ £. In consequence, a number of researchers have considered methods for specialising the structure of such systems so as to identify classes for which efficient decision processes exist. In this paper the effect of a number of graphtheoretic restrictions is considered: ¤partite systems (¤¦¥¨ § ) in which the set of arguments may be partitioned into ¤ sets each of which is conflictfree; systems in which the numbers of attacks originating from and made upon any argument are bounded; planar systems; and, finally, those of ¤bounded treewidth. For the class of bipartite graphs, it is shown that determining the acceptability status of a specific argument can be accomplished in polynomialtime under both credulous and sceptical semantics. In addition we establish the existence of polynomial time methods for systems having bounded treewidth when deciding the following: whether a given (set of) arguments is credulously accepted; if the system has a nonempty preferred extension; has a stable extension; is coherent;
Conflictdriven disjunctive answer set solving
 IN KR’08, AAAI PRESS
, 2008
"... We elaborate a uniform approach to computing answer sets of disjunctive logic programs based on stateoftheart Boolean constraint solving techniques. Starting from a constraintbased characterization of answer sets, we develop advanced solving algorithms, featuring backjumping and conflictdriven l ..."
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Cited by 26 (10 self)
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We elaborate a uniform approach to computing answer sets of disjunctive logic programs based on stateoftheart Boolean constraint solving techniques. Starting from a constraintbased characterization of answer sets, we develop advanced solving algorithms, featuring backjumping and conflictdriven learning using the FirstUIP scheme as well as sophisticated unfounded set checking. As a final result, we obtain a competitive solver for Σ P 2complete problems, taking advantage of Boolean constraint solving technology without using any legacy solvers as black boxes.
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 21 (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
Generalizing Boolean satisfiability I: Background and survey of existing work
 Journal of Artificial Intelligence Research
, 2004
"... This is the first of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern highperformance solvers. The fundamental idea underlying zap is that many problems passed to such engines contain ..."
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Cited by 19 (3 self)
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This is the first of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern highperformance solvers. The fundamental idea underlying zap is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper is a survey of the work underlying zap, and discusses previous attempts to improve the performance of the DavisPutnamLogemannLoveland algorithm by exploiting the structure of the problem being solved. We examine existing ideas including extensions of the Boolean language to allow cardinality constraints, pseudoBoolean representations, symmetry, and a limited form of quantification. While this paper is intended as a survey, our research results are contained in the two subsequent articles, with the theoretical structure of zap described in the second paper in this series, and zap’s implementation described in the third. 1.