Showing papers on "Equivalence class published in 2002"

Learning equivalence classes of bayesian-network structures

Abstract: Two Bayesian-network structures are said to be equivalent if the set of distributions that can be represented with one of those structures is identical to the set of distributions that can be represented with the other. Many scoring criteria that are used to learn Bayesian-network structures from data are score equivalent; that is, these criteria do not distinguish among networks that are equivalent. In this paper, we consider using a score equivalent criterion in conjunction with a heuristic search algorithm to perform model selection or model averaging. We argue that it is often appropriate to search among equivalence classes of network structures as opposed to the more common approach of searching among individual Bayesian-network structures. We describe a convenient graphical representation for an equivalence class of structures, and introduce a set of operators that can be applied to that representation by a search algorithm to move among equivalence classes. We show that our equivalence-class operators can be scored locally, and thus share the computational efficiency of traditional operators defined for individual structures. We show experimentally that a greedy model-selection algorithm using our representation yields slightly higher-scoring structures than the traditional approach without any additional time overhead, and we argue that more sophisticated search algorithms are likely to benefit much more.

Complex dynamics and the structure of small neural networks

Abstract: The discrete-time dynamics of small neural networks is studied empirically, with emphasis laid on non-trivial bifurcation scenarios. For particular two- and three-neuron networks interesting dynamical properties like periodic, quasi-periodic and chaotic attractors are observed, many of them co-existing for one and the same set of parameters. An appropriate equivalence class of networks is defined, describing them as parametrized dynamical systems with identical dynamical capacities. Combined symmetries in phase space and parameter space are shown to generate different representatives of such a class. Moreover, conditions on the connectivity structure are suggested, which guarantee the existence of complex dynamics for a considered equivalence class of network configurations.

System and method of employing efficient operators for Bayesian network search

Abstract: Methods and systems are disclosed for learning Bayesian networks The approach is based on specifying a search space that enables searching over equivalence classes of the Bayesian network A set of one or more operators are applied to a representation of the equivalence class A suitable search algorithm searches in the search space by scoring the operators locally with a decomposable scoring criteria To facilitate application of the operators and associated scoring, validity tests can be performed to determine whether a given operator is valid relative to the current state representation

On the construction of the inclusion boundary neighbourhood for markov equivalence classes of bayesian network structures

Abstract: The problem of learning Markov equivalence classes of Bayesian network structures may be solved by searching for the maximum of a scoring metric in a space of these classes. This paper deals with the definition and analysis of one such search space. We use a theoretically motivated neighbourhood, the inclusion boundary, and represent equivalence classes by essential graphs. We show that this search space is connected and that the score of the neighbours can be evaluated incrementally. We devise a practical way of building this neighbourhood for an essential graph that is purely graphical and does not explicitely refer to the underlying independences. We find that its size can be intractable, depending on the complexity of the essential graph of the equivalence class. The emphasis is put on the potential use of this space with greedy hillclimbing search.

Mathematical quotients and quotient types in Coq

Abstract: This note studies quotient types in the Calculus of Inductive Constructions (CIC), implemented in the proof assistant coq, and compares their expressivity to that of mathematical quotients. In [Hof95], Martin Hofmann proposes an extension of the Calculus of Constructions (CC) with quotient types which he shows consistent, but notices that they are not sufficient to account for the natural isomorphism θ which exists in set theory between functional spaces E → F/R and E→F/S where fSg iff ∀x ∈ F, f(x)Rg(x). One can thus ask the question to know if it is possible to extend these quotient types to be able to show injectivity and surjectivity of this morphism. We show here that any extension of this kind in Coq with the impredicative sort Set would be contradictory.

A Nested Depth First Search Algorithm for Model Checking with Symmetry Reduction

Abstract: We present an algorithm for the verification of properties of distributed systems, represented as B?chi automata, which exploits symmetry reduction. The algorithm is developed in the more general context of bisimulation preserving reductions along the lines of Emerson, Jha and Peled. Our algorithm is a modification of the nested depth first search (NDFS) algorithm by Courcoubetis, Yannakakis, Vardi and Wolper. As such, it has the standard advantages (memory and time efficiency) that NDFS shows over the state space exploration algorithms based on maximal strongly connected components in the state space graph. In addition, a nice feature of the presented algorithm is that it works also with multiple (non-canonical) representatives for the symmetry equivalence classes. Also, instead of an abstract counter-example, our algorithm is capable of reproducing a counter-example which exists in the original unreduced state space, which is an important feature for debugging.

Equivalence classes of non-local unitary operations

Abstract: We study when a multipartite non--local unitary operation can deterministically or probabilistically simulate another one when local operations of a certain kind -in some cases including also classical communication- are allowed. In the case of probabilistic simulation and allowing for arbitrary local operations, we provide necessary and sufficient conditions for the simulation to be possible. Deterministic and probabilistic interconversion under certain kinds of local operations are used to define equivalence relations between gates. In the probabilistic, bipartite case this induces a finite number of classes. In multiqubit systems, however, two unitary operations typically cannot simulate each other with non-zero probability of success. We also show which kind of entanglement can be created by a given non--local unitary operation and generalize our results to arbitrary operators.

Checking equivalence for circuits containing incompletely specified boxes

Abstract: We consider the problem of checking whether an implementation which contains parts with incomplete information is equivalent to a given full specification. We study implementations which are not completely specified, but contain boxes which are associated with incompletely specified functions (called Incompletely Specified Boxes or IS-Boxes). After motivating the use of implementations with Incompletely Specified Boxes we define our notion of equivalence for this kind of implementations and present a method to solve the problem. A series of experimental results demonstrates the effectiveness and feasibility of the methods presented.

Lipschitz quotient mappings with good ratio of constants

Abstract: It is shown that Lipschitz quotient mappings between finite dimensional spaces behave nicely (e.g., are bijective in the case of equal dimensions) if the Lipschitz and co-Lipschitz constants are close to each other. For Lipschitz quotient mappings of the plane, a bound for the cardinality of the pre-image of a point in terms of the ratio of the constants is obtained.

Three New Methods for Evaluating Reference Resolution

Abstract: Reference resolution on extended texts (several thousand references) cannot be evaluated manually. An evaluation algorithm has been proposed for the MUC tests, using equivalence classes for the coreference relation. However, we show here that this algorithm is too indulgent, yielding good scores even for poor resolution strategies. We elaborate on the same formalism to propose two new evaluation algorithms, comparing them first with the MUC algorithm and giving then results on a variety of examples. A third algorithm using only distributional comparison of equivalence classes is finally described; it assesses the relative importance of the recall vs. precision errors.

System and method for sorting data

Abstract: A method for comparing data, and in particular character data, is disclosed. Two pieces of data are compared to determine if they are within an equivalence class based on using a dictionary sort order table with a non-unique collating sequence. If so, the pieces of data are compared using a dictionary sort order table with a unique collating sequence. The comparison method may be implemented within a sorting module that receives an input data set and then uses the comparison to compare two pieces of data in the input data set at a time. The sorting module uses the result of the comparison method to sort the input data set into equivalence classes. The results of a second comparison provides data sorted within equivalence classes. The sorting module may provide sorting services to a database management system or to a calling program.

Matrix computation for data cleaning and rule extraction in knowledge system

Abstract: The concept of equivalence matrix, which expresses an equivalence relation, is introduced. The relations between the equivalence matrix and equivalence classes are discussed. The proposed algorithms for data cleaning and rule extraction in the knowledge system based on matrix computation and its complexity of computation are analyzed. All rules hidden in the knowledge system can be obtained efficiently via this methodology. The algorithms emphasize the practicability of rule generation and a case study indicated their effectiveness.

Smooth quiver quotient varieties

Abstract: In this paper we classify all the quivers and corresponding dimension vectors having a smooth space of semisimple representation classes. The result is that these quiver settings can be reduced via some specific reduction steps to 3 simple types.

Characterizations of Some Classes of Quotient BCI-Algebras

Abstract: By using Fuzzy ideals of BCI-algebra,a binary relation is definited,and that being a congruent relation is proved,thus,quotient BCI-algebras are constructed.The correspondent quotient BCI-algebras are completely characterized by using Fuzzy q-ideal,Fuzzy p-ideal,Fuzzy a-ideal,respectively.Namely,if μ is a Fuzzy ideal of BCI-algebra X ,then x /μ is a pseudo-united(p-semisimple,united)BCI-algebra,only when the μ is a Fuzzy q-ideal (p-ideal,a-ideal).

Closeness of Performance Map Information Granules: A Rough Set Approach

Abstract: This article introduces a rough set approach to measuring of information granules derived from performance maps. A performance map employs intuitive color-coding to visualize the behavior of system dynamics resulting from variations in system parameters. The resulting image is developed algorithmically via digital computation. With only moderate a priori knowledge, mathematical analysis of a performance map provides an immediate wealth of information. This study is motivated by an interest in measuring the separation between "islands" (collections of pixels with the same color) representing normal (e.g., black pixels) and potentially chaotic (e.g., red pixels) system behavior. A performance map island or sector is identified with groupings of cells in a mesh resulting from the partition of a performance map into equivalence classes. The information granules considered in this paper are associated with a feature set in an information system. The contribution of this article is the application of a measures of granule closeness based on an indistinguishability relation that partitions performance maps intervals into subintervals (equivalence classes). Such partitions are useful in measuring closeness of map cells containing color-coded pixels used to visualize dynamical system behavior.

The subsystem of the fuzzed rough sets based on equivalence class

Abstract: A method for rough sets fuzzification by using the equivalence relation is given. The definitions of the set of equivalence class and rough set being fuzzified are presented and the fact that fuzzified rough sets can constitute the fuzzy subsystem is proved.

Transducers with Set Output

Abstract: We consider transducers with set output, ie, finite state machines which produce a set of output symbols upon reading any input symbol When a word consisting of input symbols is read, the union of corresponding output sets is produced Such transducers are instrumental in some important data classification tasks, such as multi-field packet classification Two transducers are called equivalent if they produce equal output upon reading any input word In practical data classification applications, it is important to store in memory only one transducer of every equivalence class, in order to save memory space This yields the need of finding, in any equivalence class, one transducer, called canonical which is easy to compute, given any transducer from this class One of the results of this paper is the construction of an algorithm which completes this task Assuming that the input and output alphabets are of bounded size, for a given n-state transducer T, our algorithm finds the canonical transducer ?(T) equivalent to T in time O(n log n)

On the Existence of Factor Sets by External Equivalences in IST

Abstract: We study the possibility of defining the factor sets of the real axis by external equivalences through external formulas on using the IST axiomatics of nonstandard analysis. We consider the case of an additive convex equivalence whose equivalence classes are defined by a formula with external universal quantifiers. We show that in this case there exists an external function selecting one representative from each equivalence class if and only if the relation in question coincides up to translation and dilation with the relation of infinite proximity.