17,320 research outputs found

    Efficient state reduction methods for PLA-based sequential circuits

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    Experiences with heuristics for the state reduction of finite-state machines are presented and two new heuristic algorithms described in detail. Results on machines from the literature and from the MCNC benchmark set are shown. The area of the PLA implementation of the combinational component and the design time are used as figures of merit. The comparison of such parameters, when the state reduction step is included in the design process and when it is not, suggests that fast state-reduction heuristics should be implemented within FSM automatic synthesis systems

    Geometric and homological finiteness in free abelian covers

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    We describe some of the connections between the Bieri-Neumann-Strebel-Renz invariants, the Dwyer-Fried invariants, and the cohomology support loci of a space X. Under suitable hypotheses, the geometric and homological finiteness properties of regular, free abelian covers of X can be expressed in terms of the resonance varieties, extracted from the cohomology ring of X. In general, though, translated components in the characteristic varieties affect the answer. We illustrate this theory in the setting of toric complexes, as well as smooth, complex projective and quasi-projective varieties, with special emphasis on configuration spaces of Riemann surfaces and complements of hyperplane arrangements.Comment: 30 pages; to appear in Configuration Spaces: Geometry, Combinatorics and Topology (Centro De Giorgi, 2010), Edizioni della Normale, Pisa, 201

    Redundancy, Deduction Schemes, and Minimum-Size Bases for Association Rules

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    Association rules are among the most widely employed data analysis methods in the field of Data Mining. An association rule is a form of partial implication between two sets of binary variables. In the most common approach, association rules are parameterized by a lower bound on their confidence, which is the empirical conditional probability of their consequent given the antecedent, and/or by some other parameter bounds such as "support" or deviation from independence. We study here notions of redundancy among association rules from a fundamental perspective. We see each transaction in a dataset as an interpretation (or model) in the propositional logic sense, and consider existing notions of redundancy, that is, of logical entailment, among association rules, of the form "any dataset in which this first rule holds must obey also that second rule, therefore the second is redundant". We discuss several existing alternative definitions of redundancy between association rules and provide new characterizations and relationships among them. We show that the main alternatives we discuss correspond actually to just two variants, which differ in the treatment of full-confidence implications. For each of these two notions of redundancy, we provide a sound and complete deduction calculus, and we show how to construct complete bases (that is, axiomatizations) of absolutely minimum size in terms of the number of rules. We explore finally an approach to redundancy with respect to several association rules, and fully characterize its simplest case of two partial premises.Comment: LMCS accepted pape

    Ordered direct implicational basis of a finite closure system

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    Closure system on a finite set is a unifying concept in logic programming, relational data bases and knowledge systems. It can also be presented in the terms of finite lattices, and the tools of economic description of a finite lattice have long existed in lattice theory. We present this approach by describing the so-called D-basis and introducing the concept of ordered direct basis of an implicational system. A direct basis of a closure operator, or an implicational system, is a set of implications that allows one to compute the closure of an arbitrary set by a single iteration. This property is preserved by the D-basis at the cost of following a prescribed order in which implications will be attended. In particular, using an ordered direct basis allows to optimize the forward chaining procedure in logic programming that uses the Horn fragment of propositional logic. One can extract the D-basis from any direct unit basis S in time polynomial in the size of S, and it takes only linear time of the cardinality of the D-basis to put it into a proper order. We produce examples of closure systems on a 6-element set, for which the canonical basis of Duquenne and Guigues is not ordered direct.Comment: 25 pages, 10 figures; presented at AMS conference, TACL-2011,ISAIM-2012 and at RUTCOR semina

    Ordered direct implication basis of a finite closure system

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    Closure system on a nite set is a unifying concept in logic programming, relational data bases and knowledge systems. It can also be presented in the terms of nite lattices, and the tools of economic description of a nite lattice have long existed in lattice theory. We present this approach by describing the so-called D-basis and introducing the concept of ordered direct basis of an implicational system. A direct basis of a closure operator, or an implicational system, is a set of implications that allows one to compute the closure of an arbitrary set by a single iteration. This property is preserved by the D-basis at the cost of following a prescribed order in which implications will be attended. In particular, using an ordered direct basis allows to optimize the forward chaining procedure in logic programming that uses the Horn fragment of propositional logic. One can extract the D-basis from any direct unit basis in time polynomial in the size s( ), and it takes only linear time of the cardinality of the D-basis to put it into a proper order. We produce examples of closure systems on a 6-element set, for which the canonical basis of Duquenne and Guigues is not ordered direc

    Stability Yields Sublinear Time Algorithms for Geometric Optimization in Machine Learning

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    In this paper, we study several important geometric optimization problems arising in machine learning. First, we revisit the Minimum Enclosing Ball (MEB) problem in Euclidean space ?^d. The problem has been extensively studied before, but real-world machine learning tasks often need to handle large-scale datasets so that we cannot even afford linear time algorithms. Motivated by the recent developments on beyond worst-case analysis, we introduce the notion of stability for MEB, which is natural and easy to understand. Roughly speaking, an instance of MEB is stable, if the radius of the resulting ball cannot be significantly reduced by removing a small fraction of the input points. Under the stability assumption, we present two sampling algorithms for computing radius-approximate MEB with sample complexities independent of the number of input points n. In particular, the second algorithm has the sample complexity even independent of the dimensionality d. We also consider the general case without the stability assumption. We present a hybrid algorithm that can output either a radius-approximate MEB or a covering-approximate MEB, which improves the running time and the number of passes for the previous sublinear MEB algorithms. Further, we extend our proposed notion of stability and design sublinear time algorithms for other geometric optimization problems including MEB with outliers, polytope distance, one-class and two-class linear SVMs (without or with outliers). Our proposed algorithms also work fine for kernels
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