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118
Short Proofs are Narrow  Resolution made Simple
 Journal of the ACM
, 2000
"... The width of a Resolution proof is de ned to be the maximal number of literals in any clause of the proof. In this paper we relate proof width to proof length (=size), in both general Resolution, and its treelike variant. The following consequences of these relations reveal width as a crucial " ..."
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Cited by 205 (14 self)
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The width of a Resolution proof is de ned to be the maximal number of literals in any clause of the proof. In this paper we relate proof width to proof length (=size), in both general Resolution, and its treelike variant. The following consequences of these relations reveal width as a crucial "resource" of Resolution proofs. In one direction, the relations allow us to give simple, unified proofs for almost all known exponential lower bounds on size of resolution proofs, as well as several interesting new ones. They all follow from width lower bounds, and we show how these follow from natural expansion property of clauses of the input tautology. In the other direction, the widthsize relations naturally suggest a simple dynamic programming procedure for automated theorem proving  one which simply searches for small width proofs. This relation guarantees that the running time (and thus the size of the produced proof) is at most quasipolynomial in the smallest treelike proof. This algorithm is never much worse than any of the recursive automated provers (such as DLL) used in practice. In contrast, we present a family of tautologies on which it is exponentially faster.
Some Consequences of Cryptographical Conjectures for . . .
, 1995
"... We show that there is a pair of disjoint NPsets, whose disjointness is provable in S 1 2 and which cannot be separated by a set in P=poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S 1 2 admittin ..."
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Cited by 70 (14 self)
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We show that there is a pair of disjoint NPsets, whose disjointness is provable in S 1 2 and which cannot be separated by a set in P=poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S 1 2 admitting an NP definition of primes about which it can prove that no number satisfying the definition is composite. As a corollary we obtain that the Extended Frege (EF) proof system does not admit feasible interpolation theorem unless the RSA cryptosystem is not secure, and that an extension of EF by tautologies p (p primes), formalizing that p is not composite, as additional axioms does not admit feasible interpolation theorem unless factoring and the discrete logarithm are in P=poly . The NP 6= coNP conjecture is equivalent to the statement that no propositional proof system (as defined in [6]) admits polynomial size proofs of all tautologies. However, only for few proof systems occur...
Space Bounds for Resolution
, 2000
"... We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition [11] the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer ..."
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Cited by 65 (3 self)
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We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition [11] the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer analysis of the space in the refutation, ranging from constant to linear space. Moreover, the new definition allows to relate the space needed in a resolution proof of a formula to other well studied complexity measures. It coincides with the complexity of a pebble game in the resolution graphs of a formula, and as we show, has relationships to the size of the refutation. We also give upper and lower bounds on the space needed for the resolution of unsatisfiable formulas. We show that Tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n \Gamma c for some constant c. Measured on the number of clauses, this result is the best possible. We also show that the formulas expressing the general Pigeonhole Principle with n holes and more than n pigeons, need space n + 1 independently of the number of pigeons. Since a matching space upper bound of n + 1 for these formulas exist, the obtained bound is exact. We also point to a possible connection between resolution space and resolution width, another measure for the complexity of resolution refutations.
Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs
, 1999
"... We develop data structures for dynamic closest pair problems with arbitrary distance functions, that do not necessarily come from any geometric structure on the objects. Based on a technique previously used by the author for Euclidean closest pairs, we show how to insert and delete objects from an n ..."
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Cited by 65 (1 self)
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We develop data structures for dynamic closest pair problems with arbitrary distance functions, that do not necessarily come from any geometric structure on the objects. Based on a technique previously used by the author for Euclidean closest pairs, we show how to insert and delete objects from an nobject set, maintaining the closest pair, in O(nlog² n) time per update and O(n) space. With quadratic space, we can instead use a quadtreelike structure to achieve an optimal time bound, O(n) per update. We apply these data structures to hierarchical clustering, greedy matching, and TSP heuristics, and discuss other potential applications in machine learning, Gröbner bases, and local improvement algorithms for partition and placement problems. Experiments show our new methods to be faster in practice than previously used heuristics.
A Combinatorial Characterization of Resolution Width
 In 18th IEEE Conference on Computational Complexity
, 2002
"... We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result i ..."
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Cited by 52 (6 self)
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We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a wellknown open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the Dense Linear Order Principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the sizewidth relationship is tight.
Pseudorandom Generators Hard for kDNF Resolution and Polynomial Calculus. Unpublished
, 2003
"... Abstract A pseudorandom generator Gn : {0, 1} n → {0, 1} m is hard for a propositional proof system P if (roughly speaking) P cannot efficiently prove the statement Gn(x1, . . . , xn) = b for any string b ∈ {0, 1} m . We present a func ) generator which is hard for Res(ε log n); here Res(k) is the ..."
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Cited by 50 (4 self)
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Abstract A pseudorandom generator Gn : {0, 1} n → {0, 1} m is hard for a propositional proof system P if (roughly speaking) P cannot efficiently prove the statement Gn(x1, . . . , xn) = b for any string b ∈ {0, 1} m . We present a func ) generator which is hard for Res(ε log n); here Res(k) is the propositional proof system that extends Resolution by allowing kDNFs instead of clauses. As a direct consequence of this result, we show that whenever t ≥ n 2 , every Res(ε log t) proof of the principle ¬Circuitt(fn) (asserting that the circuit size of a Boolean function fn in n variables is greater than t) must have size exp(t Ω(1) ). In particular, Res(log log N ) (N ∼ 2 n is the overall number of propositional variables) does not possess efficient proofs of NP ⊆ P/poly. Similar results hold also for the system PCR (the natural common extension of Polynomial Calculus and Resolution) when the characteristic of the ground field is different from 2. As a byproduct, we also improve on the small restriction switching lemma due to Segerlind, Buss and Impagliazzo by removing a square root from the final bound. This in particular implies that the (moderately) weak pigeonhole principle PHP 2n n is hard for Res(ε log n/ log log n).
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 (7 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...
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 45 (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.
Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1999
"... This paper gives nearly optimal lower bounds on the minimum degree of polynomial calculus refutations of Tseitin's graph tautologies and the mod p counting principles, p 2. The lower bounds apply to the polynomial calculus over fields or rings. These are the first linear lower bounds for ..."
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Cited by 36 (9 self)
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This paper gives nearly optimal lower bounds on the minimum degree of polynomial calculus refutations of Tseitin's graph tautologies and the mod p counting principles, p 2. The lower bounds apply to the polynomial calculus over fields or rings. These are the first linear lower bounds for the polynomial calculus for kCNF formulas. As a