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27
NearOptimal Separation of Treelike and General Resolution
 Electronic Colloquium in Computation Complexity
, 2000
"... We present the best known separation between treelike and general resolution, improving on the recent exp(n ) separation of [BEGJ98]. ..."
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Cited by 47 (3 self)
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We present the best known separation between treelike and general resolution, improving on the recent exp(n ) separation of [BEGJ98].
On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems
, 2000
"... An exponential lower bound for the size of treelike Cutting Planes refutations of a certain family of CNF formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the treelike versions and the daglike versions of resolution and Cutting Planes. ..."
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Cited by 39 (9 self)
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An exponential lower bound for the size of treelike Cutting Planes refutations of a certain family of CNF formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the treelike versions and the daglike versions of resolution and Cutting Planes. In both cases only superpolynomial separations were known [29, 18, 8]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [25] are extended to monotone real circuits. An exponential separation is also proved between treelike resolution and several refinements of resolution: negative resolution and regular resolution. Actually this last separation also provides a separation between treelike resolution and ordered resolution, thus the corresponding superpolynomial separation of [29] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [13].
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
Exponential Separations between Restricted Resolution and Cutting Planes Proof Systems
, 1998
"... We prove an exponential lower bound for treelike Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between treelike and daglike proofs for both CuttingPlanes and resolution; in both cases only superpolynomial se ..."
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Cited by 28 (5 self)
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We prove an exponential lower bound for treelike Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between treelike and daglike proofs for both CuttingPlanes and resolution; in both cases only superpolynomial separations were known before [30, 20, 10]. In order to prove this, we extend the lower bounds on the depth of monotone circuits of Raz and McKenzie [26] to monotone real circuits. In the case of resolution, we further improve this result by giving an exponential separation of treelike resolution from (daglike) regular resolution proofs. In fact, the refutation provided to give the upper bound respects the stronger restriction of being a DavisPutnam resolution proof. This extends the corresponding superpolynomial separation of [30]. Finally, we prove an exponential separation between DavisPutnam resolution and unrestricted resolution proofs; only a superpolynomial separation was previously...
Size Space tradeoffs for Resolution
, 2002
"... We investigate tradeoffs of various important complexity measures such as size, space and width. We show examples of CNF formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results, the first of their ..."
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Cited by 22 (3 self)
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We investigate tradeoffs of various important complexity measures such as size, space and width. We show examples of CNF formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results, the first of their kind, have implications on the efficiency (or rather, inefficiency) of some commonly used SAT solving heuristics. Our proof
Communication Complexity: Towards Lower Bounds on Circuit Depth
, 2000
"... Karchmer, Raz, and Wigderson, 1991, discuss the circuit depth complexity of n bit Boolean functions constructed by composing up to d = log n= log log n levels of k = log n bit boolean functions. Any such function is in AC . They conjecture that circuit depth is additive under composition, whic ..."
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Cited by 17 (0 self)
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Karchmer, Raz, and Wigderson, 1991, discuss the circuit depth complexity of n bit Boolean functions constructed by composing up to d = log n= log log n levels of k = log n bit boolean functions. Any such function is in AC . They conjecture that circuit depth is additive under composition, which would imply that any (bounded fanin) circuit for this problem requires dk 2 \Omega\Gamma119 n= log log n) depth. This would separate AC from NC . They recommend using the communication game characterization of circuit depth. In order to develop techniques for using communication complexity to prove circuit lower bounds, they suggest an intermediate communication complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dk \Gamma O(d ) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits k) circuit depth. Although this fact can be easily established using a counting argument, we hope that the ideas in our proof will be incorporated more easily into subsequent arguments which use communication complexity to prove circuit depth bounds.
JAMRESISTANT COMMUNICATION WITHOUT SHARED SECRETS THROUGH THE USE OF CONCURRENT CODES
, 2007
"... We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop an ..."
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Cited by 16 (9 self)
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We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop and analyze this algorithm within the framework of a new type of code, concurrent codes, which are those superimposed codes that allow efficient decoding. Finally, we propose the Universal Concurrent Code algorithm, and prove that it covers all possible concurrent codes, and give connections between its theory and that of monotone Boolean functions.
Short proofs may be spacious: An optimal separation of space and length in resolution
, 2008
"... A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negat ..."
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Cited by 11 (8 self)
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A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Ω(n / log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n / log n). Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the blackwhite pebbling price of the graph. The proof is somewhat simpler than previous results (in particular, those reported in [Nordström 2006, Nordström and H˚astad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard blackwhite pebbling price.
Homogenization and the Polynomial Calculus
 Computational Complexity
, 1999
"... In standard implementations of the Grobner basis algorithm, the original polynomials are homogenized so that each term in a given polynomial has the same degree. In this paper, we study the effect of homogenization on the runtime of the Grobner basis algorithm for instances of satisfiability, and ..."
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Cited by 10 (4 self)
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In standard implementations of the Grobner basis algorithm, the original polynomials are homogenized so that each term in a given polynomial has the same degree. In this paper, we study the effect of homogenization on the runtime of the Grobner basis algorithm for instances of satisfiability, and also on the degree of Polynomial Calculus (PC) and Nullstellensatz refutations of unsatisfiable formulas. We show that the Nullstellensatz degree is equivalent to the homogenized PC degree. Using this relationship, we prove nearly linear separations between Nullstellensatz and PC degree, for 3CNF formulas. Research partially supported by NSF grant CCR9457782 and a scholarship from the Arizona Chapter of the ARCS Foundation. + Research supported by NSF CCR9734911, Sloan Research Fellowship BR3311, and by a cooperative research grant INT9600919 /ME103 from NSF and the M SMT (Czech Republic), and USAIsraelBSF Grant 9700188 # Research supported by NSF grant CCR9457782 and USI...
Conjunctive query answering with OWL 2 QL
 IN: PROC. OF KR. AAAI PRESS
, 2012
"... We present a novel rewriting technique for conjunctive query answering over OWL 2 QL ontologies. In general, the obtained rewritings are not necessarily correct and can be of exponential size in the length of the query. We argue, however, that in most, if not all, practical cases the rewritings are ..."
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Cited by 10 (7 self)
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We present a novel rewriting technique for conjunctive query answering over OWL 2 QL ontologies. In general, the obtained rewritings are not necessarily correct and can be of exponential size in the length of the query. We argue, however, that in most, if not all, practical cases the rewritings are correct and of polynomial size. Moreover, we prove some sufficient conditions, imposed on queries and ontologies, that guarantee correctness and succinctness. We also support our claim by experimental results.