Showing papers on "Equivalence class published in 2004"

Recognition of shapes by editing their shock graphs

Abstract: This paper presents a novel framework for the recognition of objects based on their silhouettes. The main idea is to measure the distance between two shapes as the minimum extent of deformation necessary for one shape to match the other. Since the space of deformations is very high-dimensional, three steps are taken to make the search practical: 1) define an equivalence class for shapes based on shock-graph topology, 2) define an equivalence class for deformation paths based on shock-graph transitions, and 3) avoid complexity-increasing deformation paths by moving toward shock-graph degeneracy. Despite these steps, which tremendously reduce the search requirement, there still remain numerous deformation paths to consider. To that end, we employ an edit-distance algorithm for shock graphs that finds the optimal deformation path in polynomial time. The proposed approach gives intuitive correspondences for a variety of shapes and is robust in the presence of a wide range of visual transformations. The recognition rates on two distinct databases of 99 and 216 shapes each indicate highly successful within category matches (100 percent in top three matches), which render the framework potentially usable in a range of shape-based recognition applications.

Classification and Dynamics of Equivalence Classes in SU(N) Gauge Theory on the Orbifold S1/Z2

Abstract: The dynamical determination of the boundary conditions in SU(N) gauge theory on the orbifold S 1 /Z 2 is investigated. We classify the equivalence classes of the boundary conditions, and then the vacuum energy density of the theory in each equivalence class is evaluated at one loop order. Unambiguous comparison of the vacuum energy densities in the two theories in different equivalence classes becomes possible in supersymmetric theories. It is found that in the supersymmetric SU(5) models with the Scherk-Schwarz supersymmetry breaking, the theory with the boundary conditions yielding the standard model symmetry is in the equivalence class with the lowest energy density, though the low energy theory is not identically the minimal supersymmetric standard model. We also study how particular boundary conditions are chosen in the cosmological evolution of the universe.

Regular selections for multiple-valued functions

Abstract: Given a multiple-valued function f, we deal with the problem of selecting its single-valued branches. This problem can be stated in a rather abstract setting considering a metric space E and a finite group G of isometries of E. Given a function f, which takes values in the equivalence classes of E/G, the problem consists of finding a map g with the same domain as f and taking values in E, such that at every point t the equivalence class of g(t) coincides with f(t). If the domain of f is an interval, we show the existence of a function g with these properties which, moreover, has the same modulus of continuity of f. In the particular case where E is the product of Q copies of ℝ n and G is the group of permutations of Q elements, it is possible to introduce a notion of differentiability for multiple-valued functions. In this case, we prove that the function g can be constructed in such a way to preserve C k,α regularity. Some related problems are also discussed.

Structural similarity in graphs

Abstract: Standard methods for role assignment partition the vertex set of a graph in such a way that vertices in the same class can be considered to have equivalent roles in the graph Several classes of equivalence relations such as regular equivalence and equitable partitions have been proposed for role assignment, but they all suffer from the strictness of classifying vertices into being either equivalent or not It is an open problem how to allow for varying degrees of similarity Proposals include ad-hoc algorithmic approaches and optimization approaches which are computationally hard. In this paper we introduce the concept of structural similarity by relaxation of equitable partitions, thus providing a theoretical foundation for similarity measures which enjoys desirable properties with respect to existence, structure, and tractability.

Exploiting equivalence reduction and the sweep-line method for detecting terminal states

Abstract: State-space exploration is one of the main approaches to computer-aided verification and analysis of finite-state systems. It is used to reason about a wide range of properties during the design phase of a system, including system deadlocks. Unfortunately, state-space exploration needs to handle huge state spaces for most practical systems. Several state-space reduction methods have been developed to tackle this problem. In this paper, we develop algorithms for combining two of these methods: state equivalence class reduction and the sweep-line. The algorithms allow deadlocks to be detected by recording terminal states of the system on-the-fly during state-space exploration. We derive expressions for the complexity of the algorithms and demonstrate their usefulness with an industrial case study. Our results show that the combined method achieves at least a six-fold reduction of the state space for interesting parameter values compared with either method used in isolation while still proving the desired system property of the terminal states. The runtime performance of the combined method is almost the same as that of the equivalence class method over the chosen parameter range. Moreover, the improvement in space reduction increases with increased parameter values.

Soundness and completeness of formal logics of symmetric encryption

Abstract: In the last two decades, two major directions in cryptography have developed: formal and computational. The formal approach uses simple, manageable formal language to describe cryptographic protocols; it is amenable to automatization, suitable for computer tools, but its accuracy is often unclear. The computational approach is harder to handle mathematically, involves probability theory and considers limits in computing power; proofs are done by hand, but it is more accurate, hence widely accepted. Much effort has been done to bridge the gap between the two views starting with Martin Abadi and Philip Rogaway in 2000, and followed by many others. These approaches are inspiring, but are worked out only for specific settings, and lack generality. Our aim is to give a complete, general analysis of the original Abadi-Rogaway approach, including applications to specific settings. The AR approach has three important ingredients: a formal language along with an equivalence notion of formal expressions, a computational cryptosystem with the notion of computational equivalence of ensembles of random distributions, and an interpreting function that assigns to each formal expression an ensemble of distributions. We say that the interpretation satisfies soundness if equivalence of formal expressions implies computational equivalence of their interpretations, and satisfies completeness if computational equivalence of the interpretations requires equivalence of the expressions. The language of the AR logic uses a box as formal notation for indecipherable strings, through which formal equivalence is defined. We expand the logic by considering different kinds of boxes corresponding to equivalence classes of formal ciphers. We consider not only computational, but also purely probabilistic, information-theoretic interpretations. We establish soundness and completeness for specific interpretations not covered in earlier works: a purely probabilistic one that interprets formal expressions in One-Time Pad, and another one in the so-called type 2 (which-key revealing) cryptosystems based on computational complexity. Furthermore, we present a general, systematic treatment of expansions of the logic as well as general soundness and completeness theorems for the interpretations, and some applications for specific settings.

Learning coverage rules from incomplete data based on rough sets

Abstract: In this paper, we deal with the problem of producing a set of certain and possible rules for coverage of incomplete data sets based on rough sets. All the coverage rules gathered together can cover all the given training examples. Unknown values are first assumed to be any possible values and are gradually refined according to the incomplete lower and upper approximations derived from the given incomplete training examples. One of the best equivalence classes in incomplete lower or upper approximations is chosen according to some criteria. The objects covered by the incomplete equivalence class are then removed from the incomplete training set. The same procedure is repeated to find the coverage set of rules. The training examples and the approximations then interact on each other to find the maximally general coverage rules and to estimate appropriate unknown values. The rules derived can then be used to build a prototype knowledge base.

Cross layer optimization - an equivalence class approach

Abstract: A cross layer design method is presented which is built upon the idea of equivalence classes of key-parameters of several layers in the protocol stack. An equivalence class is composed of all key-parameter tuples which fulfill a desired quality of service. Since different parameter tuples are usually associated with different costs (e.g. transmit power), a cross-layer design can select the most cost-efficient parameter tuple

Feature Synthesis and Extraction for the Construction of Generalized Properties of Amino Acids

Abstract: Amino acid similarity matrices are used for protein sequence comparison. It has been shown previously that they can be reconstructed from equivalence classes between amino acids. The goal of the current study is to propose an algorithm for identification of the properties that generate these equivalence classes. An approximate reasoning method for feature extraction and synthesis is developed to this end. It is shown that these equivalence classes are related with the amino acid properties that are important for the formation of the protein structure. The algorithm presented in this study works best for bit-patterns, which are frequently encountered in bit-vector representations.

A primer on the evolution of equivalence classes of Bayesian-network structures

Abstract: Bayesian networks (BN) constitute a useful tool to model the joint distribution of a set of random variables of interest. To deal with the problem of learning sensible BN models from data, we have previously considered various evolutionary algorithms for searching the space of BN structures directly. In this paper, we explore a simple evolutionary algorithm designed to search the space of BN equivalence classes. We discuss a number of issues arising in this evolutionary context and provide a first assessment of the new class of algorithms.

Computations on distributed discrete-event systems

Abstract: This thesis explores computational issues related to the control and verification of systems with distributed structure. The framework of supervisory control theory and discrete-event systems is used where system modules are modelled as sets of finite state automata whose behavior coordinates on the occurrence of common events. It is shown that in general many problems related to the supervision of these systems are PSPACE-complete. There are methods for solving these problems that are more efficient in memory than the current state-of-the-art methods, but there are most likely no time-efficient general solution methods that would aid in the study of such “large-scale” systems. This thesis explores methods for avoiding the computational difficulty of solving these problems. For decentralized control situations a new state estimator is presented that accounts for past local control actions when calculating the set of estimated system states. The new state estimator is used to develop new decentralized control protocols with a common sufficient safety condition. It is also shown that it is difficult to approximate minimal solutions to a sensor selection problem for partial observation control situations. Heuristic methods for solving this approximation problem based on a type of edge-colored graph cutting problem are then discussed. It is also shown how to convert a type of communicating controller problem into this edge-colored graph cutting problem. A notion of state permutation symmetry that defines an equivalence class for the distributed system states is introduced. A method is shown to reduce the complexity of verifying μ-calculus propositions for systems with state permutation symmetry. A special class of symmetric distributed systems is also shown that allows for an even greater reduction in the difficulty of testing several fundamental system properties. Control and verification problems related to both local and global specifications for these special systems are then explored.

Mathematical Structure of Anomalous Dimensions and QCD Wilson Coefficients in Higher Order

Abstract: The alternating and non-alternating harmonic sums and other algebraic objects of the same equivalence class are connected by algebraic relations which are induced by the product of these quantities and which depend on their index class rather than on their value. We show how to find a basis of the associated algebra. The length of the basis l is found to be ⩽ 1 / d , where d is the depth of the sums considered and is given by the 2nd Witt formula. It can be also determined counting the Lyndon words of the respective index set. There are two further classes of relations: structural relations between Nielsen –type integrals and relations due to the specific structure of Feynman diagrams which lead to a considerable reduction of the set of basic functions. The relations derived can be used to simplify results of higher order calculations in QED and QCD. We also report on results calculating the 16th non–singlet moment of unpolarized structure functions at 3–loop order in the MS ¯ scheme.

A primer on the evolution of equivalence classes of Bayesian-network structures

Abstract: Bayesian networks (BN) constitute a useful tool to model the joint distribution of a set of random variables of interest. To deal with the problem of learning sensible BN models from data, we have previously considered various evolutionary algorithms for searching the space of BN structures directly. In this paper, we explore a simple evolutionary algorithm designed to search the space of BN equivalence classes. We discuss a number of issues arising in this evolutionary context and provide a first assessment of the new class of algorithms.

Equivalence classes for optimizing risk models in Markov decision processes

Abstract: We consider eight problems in which we maximize or minimize threshold probabilities in discounted Markov decision processes with bounded reward set. We show that such problems are classified to two equivalence classes and give a relationship between optimal values and optimal policies of problems in each equivalence class. Literatures relative to such problems deal with only first equivalence class (cf. White(1993), Wu and Lin(1999) and Ohtsubo and Toyonaga(2002)). We consider a problem of the second equivalence class in the same situation as Ohtsubo and Toyonaga and characterize optimal values in finite and infinite horizon cases, by using an argument of a dual problem. We also give two sufficient conditions for the existence of an optimal policy. Finally we give a relationship of optimal values between first and second equivalence classes.

Apparatus and methods for performing generational escape analysis in managed runtime environments

Abstract: Apparatus and methods for performing generational escape analysis in managed runtime environments are disclosed. The disclosed apparatus and methods determine the generational age of an equivalence class while performing escape analysis. Equivalence classes having generational ages are cloned if their generational ages are less than a threshold age.

Rough set model for discovering single-dimensional and multidimensional association rules

Abstract: In this paper, the mining of association rules with rough set technology is investigated as the algorithm RSASM. The RSASM algorithm is introduced for mining of single-dimensional association rules, which is constituted of three steps: (1) generalizing database to discretize quantitative attributes and decrease quantity of data; (2) finding candidate itemsets with the concept of equivalence class derived from indiscernibility relation in rough set theory; and (3) finding frequent itemsets with multiple minimum supports. The RSASM can be expanded to multidimensional association rules mining easily. It can be seen from experiments that the mining algorithm is elegant and efficient, which can obtain more rapid computing speed and sententious rules at the same time.

Branch-and-cut for symmetrical ILPs and combinatorial designs

Abstract: Many combinatorial problems can be formulated as a 0-1 integer linear program (0-1 ILP), which consists of an objective function subject to linear constraints over 0-1 variables. A method called branch-and-cut has been successfully used to attack ILPs, but ILPs are still difficult to solve in practice. The problems we investigate demonstrate large numbers of equivalent (isomorphic) solutions. We are only interested in generating one solution from each equivalence class. To accomplish this, we study and extend a method by Margot [22] that avoids considering unnecessary partial solutions by generating an isomorph-free search tree. We implement a branch-and-cut library, and solve combinatorial design problems involving t-designs, packings, coverings and intersecting set systems. We experimentally analyze the strengths and limitations of our algorithms and determine when it is efficient to use isomorph-free branch-and-cut. Our framework generates new results and reproduces, in competitive time, existing ones.

To Compare the Theory of Quotient Space with Rough Set

Abstract: The relationship between rough set and quotient space is discussed by comparing in the paper.After analyzing basic algorithms and complexity and expansibility in theory about rough set and quotient space,think that the same point of them is to represent granule by using equivalence relation and to depict concept by using granule.But the emphasis of both discussions is difference.The theory of quotient space researches transformed and depended relations between different granules mainly.It is the theory to represent space relationship.For granular computing today such as rough set,it studies to represent and depict granule and relation between granule and concept primarily.The greatest difference is that there is topological relation among domain elements in the theory of quotient space,i.e.the domain is a topological space.Whereas the domain of rough set is only simple point set,there is no topological relation among elements.So the theory of quotient space applies to not only data mining and knowledge discovery,but also restriction problem such as route layout and space state distributing etc.

Practically handling some configuration isomorphisms

Abstract: Configuring consists in simulating the realization of a complex product from a catalog of component parts, using known relations between types, and picking values for object attributes. A configuration can be viewed as a graph of interconnected components. An inherent difficulty in solving configuration problems is the existence of many structural isomorphisms. A practical way of dealing with isomorphisms is by isolating one configuration in each equivalence class called a canonical representative. Since no poly time algorithm is known for the general graph isomorphism problem, it is interesting to explore subproblems that can efficiently be exploited by backtrack search procedures. We describe a formalism independent approach to detect a large number of structure related configuration isomorphisms. The algorithm has the essential property that isomorphism detection is both incremental and can be performed in pseudo linear time, two essential conditions for backtrack search. Backtrack occurs as soon as a nonweakly canonical DAG structure is generated for a configuration, which allows to extend the range of practically tractable problems, as shown experimentally. Weakly canonical configurations explicitly expose their automorphism group, which are readily available thanks to the lexicographic ordering chosen. The efficiency of the approach is assessed both theoretically and by experimental results obtained for a range of realistic configuration problems.

Some quotients on a bck-algebra generated by a fuzzy set

Abstract: First we show that the cosets of a fuzzy ideal µ in a BCK-algebra X form another BCK-algebra X µ (called the fuzzy quotient BCK-algebra of X by µ). Also we show that X µ is a fuzzy partition of X and we prove several some isomorphism theorems. Moreover we prove that if the associated fuzzy similarity relation of a fuzzy partition P of a commutative BCK-algebra is compatible, then P is a fuzzy quotient BCK-algebra. Finally we define the notion of a coset of a fuzzy ideal and an element of a BCK-algebra and prove related theorems.

Finite equivalence relations on algebraic varieties and hidden symmetries

Abstract: This paper can be considered as a continuation of Miyanishi's paper which contains a theorem on existence of a quotient of an affine normal or a projective smooth variety by a finite equivalence relation such that every component of the relation projects onto the variety (we call such an equivalence relation a wide finite equivalence relation). Later papers of Kollar and Keel-Mori shed new light on the subject and can serve as a base for further studies. The results of the present paper are based on the fact that every wide finite equivalence relation on a normal variety V is determined by an action of a finite group on the normalization of V in some Galois extension of k(V). Hence, such an equivalence relation hides some symmetry of a (ramified) cover of V. One may find some analogy of the situation with the concept of a hidden symmetry considered in physics. An important part of the paper is examples described in Section 6 which show that the main result of the paper (Theorem 2.3) is valid neither in the seminormal case, nor under the additional assumptions that there exists a finite morphism whose fibers contain equivalence classes of a given finite relation. In the nonnormal case, identification of some points described by a finite wide equivalence relation may force identification of some other nonequivalent points. This seems to show that the class of normal varieties and wide equivalence relation is a proper frame for considering the general problems of quotients by finite equivalence relations.

Efficiency and accuracy trade-offs in process detection

Abstract: Hidden Discrete Event Systems Models (HDESM) are discrete event dynamical system models whose underlying internal state spaces are not directly observable. Observations on such systems are artifacts of the hidden, internal states and are not deterministically or uniquely associated with the hidden states. The distribution of an observation of a HDESM is typically given by a probability distribution conditioned on the hidden state of the system. Classical linear systems, Hidden Markov Models (HMM) and certain types of Petri Net models are examples of HDESM's. A major challenge in working with this type of model is the estimation of an HDESM's hidden states based on a sequence of observations. In some cases, well-known algorithms can be used to solve this problem. In many cases of practical interest, however, the complexity of those algorithms is too high to be practical. New ideas and algorithms are therefore needed for effective solutions to the state estimation problem. In this paper we will investigate sub-classes of HDESM's whose structure would allow efficient state estimation algorithms to exist. Such structures could be related to the sparsity and/or equivalence class structure of transition dynamics within the underlying discrete event system. Efficient algorithms that compute approximate solutions will be investigated with the goal of understanding the trade-offs between computational efficiency and estimation accuracy. Ideas on how to implement such trade-offs also are proposed.

Reduction of multiple harmonic sums and harmonic polylogarithms

Abstract: The alternating and non-alternating harmonic sums and other algebraic objects of the same equivalence class are connected by algebraic relations which are induced by the product of these quantities and which depend on their index class rather than on their value. We show how to find a basis of the associated algebra. The length of the basis l is found to be ⩽ 1 / d , where d is the depth of the sums considered and is given by the 2nd Witt formula. It can be also determined by counting the Lyndon words of the respective index set. The relations derived can be used to simplify results of higher-order calculations in QED and QCD.

An Algorithm for the Direct Generation of Set Class Representatives in Any Pitch Class Space

Abstract: When spaces with greater than 20 pitch classes are considered, the problem of generating equivalence classes (set classes) with respect to operations like transposition and inversion becomes increasingly difficult. The brute force approach of enumerating all possible pitch class sets and ignoring those that fall into already selected set classes becomes computationally intractable. Some improvements can be made using Read’s Orderly Algorithm, and the essential features are seen to be the binary coding of pitch class sets and an augmentation operation. A further refined algorithm is described that makes use of a stack technique to directly generate, straight to a text file, all equivalence class representatives of given cardinality within any pitch class universe. This supports the mathematical and compositional exploration of much larger pitch class spaces than hitherto.

An efficient reduction algorithm of high-dimensional decision tables based on rough sets theory

Abstract: After comparing several attribute significance reduction methods and analyzing the deep meanings and importance of indiscernibility relation defined by rough sets theory, according to the basic theory and the determining conditions on attribute reduction of RST, the paper brings forward the equivalence class maximum approximate match attribute significance reduction algorithm. Especially, the prophase reduction analysis plays a direct role in the reduction course. The availability and limited computation of the algorithm are proved by some examples and minimum reduction sets are obtained.

The SU3 Space and Its Quotient Spaces

Abstract: A metric description of symmetric Riemannian spaces is needed for constructing gauge fields with a symmetry. We describe the group SU 3 as a Riemannian space for two different parameterizations and develop a Hamiltonian technique for constructing quotient spaces. We construct the quotient spaces of the group SU 3, namely, the six-dimensional quotient space ( $$SU_3 /O_2^2 $$ ), the five-dimensional quotient space (SU 3/O 3), and the two four-dimensional quotient spaces ( $$SU_3 /O_2^4 $$ ) and (SU 3/O 3/O 2).

Data driven local coordinates for normalized state-space systems: orthoDDLC

Abstract: A new approach to the question of parametrizing state-space systems is considered. The approach consists of using input-normal state-space representations. These representations are unique up to an isometric state isomorphism. By decomposing the tangent space of the set of normalized controllable matrix pairs into the tangent space of the equivalence class and the orthogonal complement one obtains an interesting set of local coordinates with a number of remarkable properties. Here the approach is presented in the context of a separable least squares approach to maximum likelihood estimation of a linear system.

The Lattice Properties of Rough Set Quotient Spaces

Abstract: Let U denote a finite and nonempty set called the universe, and P(U) a power set. Suppose R is an equivalence relation on U. Consider three equivalence relations on the subsets of U, each of them is an equivalence relation on power set P(U), which induces a partition of P(U), thus we obtain three algebras of rough sets, and extend these results to the fuzzy sets of U.