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84
Singular Combinatorics
 ICM 2002 VOL. III 13
, 2002
"... Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit probability distributions present in large random structures. "Sing ..."
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Cited by 387 (11 self)
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Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit probability distributions present in large random structures. "Singularity analysis" reviewed here provides constructive estimates that are applicable in several areas of combinatorics. It constitutes a complexanalytic Tauberian procedure by which combinatorial constructions and asymptoticprobabilistic laws can be systematically related.
Computing With FirstOrder Logic
, 1995
"... We study two important extensions of firstorder logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtaine ..."
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Cited by 53 (13 self)
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We study two important extensions of firstorder logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtained using a powerful normal form for while which shows that each while computation over an unordered domain can be reduced to a while computation over an ordered domain via a fixpoint query. The fixpoint query computes an equivalence relation on tuples which is a congruence with respect to the rest of the computation. The same technique is used to show that equivalence of tuples and structures with respect to FO formulas with bounded number of variables is definable in fixpoint. Generalizing fixpoint and while, we consider more powerful languages which model arbitrary computation interacting with a database using a finite set of FO queries. Such computation is modeled by a relational machine...
Groups and Constraints: Symmetry Breaking during Search
 In Proceedings of CP02, LNCS 2470
, 2002
"... We present an interface between the ECL constraint logic programming system and the GAPcompu tational abstract algebra system. The interface provides a method for e#ciently dealing with large nu mbers of symmetries of constraint satisfaction problems for minimal programming e#ort. We als ..."
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Cited by 53 (13 self)
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We present an interface between the ECL constraint logic programming system and the GAPcompu tational abstract algebra system. The interface provides a method for e#ciently dealing with large nu mbers of symmetries of constraint satisfaction problems for minimal programming e#ort. We also report an implementation of SBDSu sing the GAPECL interface which is capable of handling many more symmetries than previou s implementations and provides improved search performance for symmetric constraint satisfaction problems.
On the computational complexity of some classical equivalence relations on boolean functions
 Forschungsberichte Mathematische Logik, Universitat Heidelberg, Bericht Nr. 18, Dezember
, 1998
"... Abstract. The paper analyzes in terms of polynomial time manyone reductions the computational complexity of several natural equivalence relations on Boolean functions which derive from replacing variables by expressions, one of them is the Boolean isomorphism relation. Most of these computational p ..."
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Cited by 20 (4 self)
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Abstract. The paper analyzes in terms of polynomial time manyone reductions the computational complexity of several natural equivalence relations on Boolean functions which derive from replacing variables by expressions, one of them is the Boolean isomorphism relation. Most of these computational problems turn out to be between coNP and � p 2. 1.
The distribution of nodes of given degree in random trees
 J. Graph Theory
, 1999
"... Abstract. Let Tn denote the set of unrooted unlabeled trees of size n and let k ≥ 1 be given. By assuming that every tree of Tn is equally likely it is shown that the limiting distribution of the number of nodes of degree k is normal with mean value ∼ µkn and variance ∼ σ2 kn with positive constants ..."
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Cited by 15 (6 self)
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Abstract. Let Tn denote the set of unrooted unlabeled trees of size n and let k ≥ 1 be given. By assuming that every tree of Tn is equally likely it is shown that the limiting distribution of the number of nodes of degree k is normal with mean value ∼ µkn and variance ∼ σ2 kn with positive constants µk and σk. Besides, the asymptotic behavior of µk and σk for k → ∞ as well as the corresponding multivariate distributions are derived. Furthermore, similar results can be proved for plane trees, for labeled trees, and for forests. 1.
Recursion and growth Estimates in renormalizable quantum field theory
"... Abstract. In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent oneparticle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. The radius of convergence turns ou ..."
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Cited by 14 (11 self)
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Abstract. In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent oneparticle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. The radius of convergence turns out to be min{ρ, 1/b1}, where ρ is the bound on the convergent ones, the instanton radius, and b1 the first coefficient of the βfunction. 1. The setup 1.1. Introduction. In this paper we explore the recursive structure of the shortdistance sector of a renormalizable quantum field theory. Such theories have a finite number of distinct amplitudes r ∈ R ⊂ A which need renormalization. We decompose each Green function in accordance with structure functions which need renormalization, the remaining structure functions and superficially convergent Green
Enumerating Markov Equivalence Classes of Acyclic Digraph Models
 PROC. OF THE CONF. ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE
, 2001
"... Graphical Markov models determined by acyclic digraphs (ADGs), also called directed acyclic graphs (DAGs), are widely studied in statistics, computer science (as Bayesian networks), operations research (as influence diagrams), and many related fields. Because different ADGs may determine the s ..."
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Cited by 14 (0 self)
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Graphical Markov models determined by acyclic digraphs (ADGs), also called directed acyclic graphs (DAGs), are widely studied in statistics, computer science (as Bayesian networks), operations research (as influence diagrams), and many related fields. Because different ADGs may determine the same Markov equivalence class, it long has been of interest to determine the efficiency gained in model specification and search by working directly with Markov equivalence classes of ADGs rather than with ADGs themselves. A computer program was written to enumerate the equivalence classes of ADG models as specified by Pearl & Verma's equivalence criterion. The program counted equivalence classes for models up to and including 10 vertices. The ratio of numbers of classes to ADGs appears to approach an asymptote of about 0.267. Classes were analyzed according to number of edges and class size. By edges, the distribution of number of classes approaches a Gaussian shape. By class size, classes of size 1 are most common, with the proportions for larger sizes initially decreasing but then following a more irregular pattern. The maximum number of classes generated by any undirected graph was found to increase approximately factorially. The program also includes a new variation of orderly algorithm for generating undirected graphs.