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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 33 (4 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.
On the ConstantDepth Complexity of kClique
"... We prove a lower bound of ω(n k/4) on the size of constantdepth circuits solving the kclique problem on nvertex graphs (for every constant k). This improves a lower bound of ω(n k/89d2) due to Beame where d is the circuit depth. Our lower bound has the advantage that it does not depend on the cons ..."
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Cited by 10 (2 self)
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We prove a lower bound of ω(n k/4) on the size of constantdepth circuits solving the kclique problem on nvertex graphs (for every constant k). This improves a lower bound of ω(n k/89d2) due to Beame where d is the circuit depth. Our lower bound has the advantage that it does not depend on the constant d in the exponent of n, thus breaking the mold of the traditional sizedepth tradeoff. Our kclique lower bound derives from a stronger result of independent interest. Suppose fn: {0, 1} (n2) − → {0, 1} is a sequence of functions computed by constantdepth circuits of size O(n t). Let G be an ErdősRényi random graph with vertex set {1,..., n} and independent edge probabilities n −α where α ≤ 1 2t−1. Let A be a uniform random kelement subset of {1,..., n} (where k is any constant independent of n) and let KA denote the clique supported on A. We prove that fn(G) = fn(G ∪ KA) asymptotically almost surely. These results resolve a longstanding open question in finite model theory (going back at least to Immerman in 1982). The mvariable fragment of firstorder logic, denoted by FO m, consists of the firstorder sentences which involve at most m variables. Our results imply that the bounded variable hierarchy FO 1 ⊂ FO 2 ⊂ · · · ⊂ FO m ⊂ · · · is strict in terms of expressive power on finite ordered graphs. It was previously unknown that FO 3 is less expressive than full firstorder logic on finite ordered graphs.
Some Aspects of Model Theory and Finite Structures
, 2002
"... this paper is to highlight some of these aspects of the model theory of nite structures, where the nite and in nite interact fruitfully, in order to dispel the perhaps too common perception that ( rstorder) model theory has little to say about nite structures ..."
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Cited by 9 (0 self)
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this paper is to highlight some of these aspects of the model theory of nite structures, where the nite and in nite interact fruitfully, in order to dispel the perhaps too common perception that ( rstorder) model theory has little to say about nite structures
Averagecase complexity of detecting cliques
, 2010
"... The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for kClique, as this problem is known. Our results show that, in certain mod ..."
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Cited by 3 (0 self)
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The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for kClique, as this problem is known. Our results show that, in certain models of computation, solving kClique in the average case requires Ω(n k/4) resources (moreover, k/4 is tight). Here the models of computation are boundeddepth Boolean circuits and unboundeddepth monotone circuits, the complexity measure is the number of gates, and the input distributions are random graphs with an appropriate density of edges. Such random graphs (the wellstudied ErdősRényi graphs) are widely believed to be a source of computationally hard instances for clique problems, a hypothesis first articulated by Karp in 1976. This thesis gives the first unconditional lower bounds supporting this hypothesis. Significantly, our result for boundeddepth Boolean circuits breaks out of the traditional
The expressive power of twovariable least fixedpoint logics
"... Abstract. The present paper gives a classification of the expressive power of twovariable least fixedpoint logics. The main results are: 1. The twovariable fragment of monadic least fixedpoint logic with parameters is as expressive as full monadic least fixedpoint logic (on binary structures). ..."
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Abstract. The present paper gives a classification of the expressive power of twovariable least fixedpoint logics. The main results are: 1. The twovariable fragment of monadic least fixedpoint logic with parameters is as expressive as full monadic least fixedpoint logic (on binary structures). 2. The twovariable fragment of monadic least fixedpoint logic without parameters is as expressive as the twovariable fragment of binary least fixedpoint logic without parameters. 3. The twovariable fragment of binary least fixedpoint logic with parameters is strictly more expressive than the twovariable fragment of monadic least fixedpoint logic with parameters (even on finite strings). 1.
The Computational Complexity Column
, 1998
"... Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful too ..."
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Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], AmbosSpies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resourcebounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resourcebounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their
Logical complexity of graphs: a survey
 CONTEMPORARY MATHEMATICS
, 2004
"... We discuss the definability of finite graphs in firstorder logic with two relation symbols for adjacency and equality of vertices. The logical depth D(G) of a graph G is equal to the minimum quantifier depth of a sentence defining G up to isomorphism. The logical width W(G) is the minimum number of ..."
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We discuss the definability of finite graphs in firstorder logic with two relation symbols for adjacency and equality of vertices. The logical depth D(G) of a graph G is equal to the minimum quantifier depth of a sentence defining G up to isomorphism. The logical width W(G) is the minimum number of variables occurring in such a sentence. The logical length L(G) is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the WeisfeilerLehman algorithm in isomorphism testing, the evolution of a random graph, quantitative characteristics of the zeroone law, or the contribution of Frank Ramsey to the research on Hilbert’s Entscheidungsproblem). Also, we trace the behavior of the descriptive complexity of a graph as the logic becomes more restrictive (for example, only definitions with a bounded number of variables or quantifier alternations are allowed) or more expressible