Set-theoretic Operations Research Articles

Using one axiom to characterize L-fuzzy rough approximation operators based on residuated lattices

Axiomatic characterization of approximation operators plays an important role in the study of rough set theory. Different axiom sets of abstract operators can illustrate different classes of rough set systems. In this paper, we are devoted to searching for a single axiom to characterize L-fuzzy rough approximation operators based on residuated lattices. Axioms of L-fuzzy set theoretic operators make sure of the existence of certain types of L-fuzzy relations which produce the same operators. We demonstrate that the lower (upper) L-fuzzy rough approximation operators generated by a generalized L-fuzzy relation can be characterized by only one axiom. Furthermore, we also use one axiom to characterize L-fuzzy rough approximation operators produced by the L-fuzzy serial, reflexive, symmetric and T-transitive relations as well as any of their compositions.

Context-sensitive data-dependence analysis via linear conjunctive language reachability

Many program analysis problems can be formulated as graph reachability problems. In the literature, context-free language (CFL) reachability has been the most popular formulation and can be computed in subcubic time. The context-sensitive data-dependence analysis is a fundamental abstraction that can express a broad range of program analysis problems. It essentially describes an interleaved matched-parenthesis language reachability problem. The language is not context-free, and the problem is well-known to be undecidable. In practice, many program analyses adopt CFL-reachability to exactly model the matched parentheses for either context-sensitivity or structure-transmitted data-dependence, but not both. Thus, the CFL-reachability formulation for context-sensitive data-dependence analysis is inherently an approximation. To support more precise and scalable analyses, this paper introduces linear conjunctive language (LCL) reachability, a new, expressive class of graph reachability. LCL not only contains the interleaved matched-parenthesis language, but is also closed under all set-theoretic operations. Given a graph with n nodes and m edges, we propose an O ( mn ) time approximation algorithm for solving all-pairs LCL-reachability, which is asymptotically better than known CFL-reachability algorithms. Our formulation and algorithm offer a new perspective on attacking the aforementioned undecidable problem - the LCL-reachability formulation is exact, while the LCL-reachability algorithm yields a sound approximation. We have applied the LCL-reachability framework to two existing client analyses. The experimental results show that the LCL-reachability framework is both more precise and scalable than the traditional CFL-reachability framework. This paper opens up the opportunity to exploit LCL-reachability in program analysis.

Possibility Interval-Valued Vague Soft Expert Sets and Its Similarity Measure

In this paper, the authors aim to extend the notion of interval-valued vague soft sets to establish the concept of possibility interval-valued vague soft expert sets. The set theoretic operations of this concept and other related concepts are introduced. The algebraic properties of this notion such as the laws of commutativity, associativity and De Morgan are established and verified. Lastly, the similarity measure between this set is introduced and illustrated using a hypothetical example related to texture synthesis.

Precise Estimates for the Time Complexity of Implementing Algorithms of Set-Theoretic Operations in Table Algebras

Algorithms implementing intersection, union, and difference in table algebras are investigated. Modifications of the most widespread algorithms reducing the amount of computations are proposed. Based on the complexities evaluated in the worst case and on the average for the modified algorithms, the fastest algorithm is found for each operation. A program that experimentally confirms theoretical estimates is developed.

Оценки сложности алгоритмов реализации теоретико-множественных операций в табличных алгебрах

Оценки сложности алгоритмов реализации теоретико-множественных операций в табличных алгебрах

A hybrid model of single valued neutrosophic sets and rough sets: single valued neutrosophic rough set model

Smarandache initiated neutrosophic sets (NSs) as a tool for handling undetermined information. Wang et al. proposed single valued neutrosophic sets (SVNSs) that is an especial NSs and can be used expediently to deal with real-world problems. In this paper, we propose single valued neutrosophic rough sets by combining single valued neutrosophic sets and rough sets. We study the hybrid model by constructive and axiomatic approaches. Firstly, by using the constructive approach, we propose the lower/upper single valued neutrosophic approximation operators and illustrate the connections between special single valued neutrosophic relations (SVNRs) and the lower/upper single valued neutrosophic approximation operators. Then, by using the axiomatic approach, we discuss the operator-oriented axiomatic characterizations of single valued neutrosophic rough sets. We obtain that different axiom sets of the lower/upper single valued neutrosophic set-theoretic operators guarantee the existence of different classes of SVNRs which produce the same operators. Finally, we introduce single valued neutrosophic rough sets on two-universes and an algorithm of decision making based on single valued neutrosophic rough sets on two-universes, and use an illustrative example to demonstrate the application of the proposed model.

A method for exploring implicit concept relatedness in biomedical knowledge network.

BackgroundBiomedical information and knowledge, structural and non-structural, stored in different repositories can be semantically connected to form a hybrid knowledge network. How to compute relatedness between concepts and discover valuable but implicit information or knowledge from it effectively and efficiently is of paramount importance for precision medicine, and a major challenge facing the biomedical research community.ResultsIn this study, a hybrid biomedical knowledge network is constructed by linking concepts across multiple biomedical ontologies as well as non-structural biomedical knowledge sources. To discover implicit relatedness between concepts in ontologies for which potentially valuable relationships (implicit knowledge) may exist, we developed a Multi-Ontology Relatedness Model (MORM) within the knowledge network, for which a relatedness network (RN) is defined and computed across multiple ontologies using a formal inference mechanism of set-theoretic operations. Semantic constraints are designed and implemented to prune the search space of the relatedness network.ConclusionsExperiments to test examples of several biomedical applications have been carried out, and the evaluation of the results showed an encouraging potential of the proposed approach to biomedical knowledge discovery.

Configuration logics: Modeling architecture styles

We study a framework for the specification of architecture styles as families of architectures involving a common set of types of components and coordination mechanisms. The framework combines two logics: 1) interaction logics for the specification of architectures as generic coordination schemes involving a configuration of interactions between typed components; and 2) configuration logics for the specification of architecture styles as sets of interaction configurations. The presented results build on previous work on architecture modeling in BIP. We show how propositional interaction logic can be extended into a corresponding configuration logic by adding new operators on sets of interaction configurations. In addition to the usual set-theoretic operators, configuration logic is equipped with a coalescing operator + to express combination of configuration sets. We provide a complete axiomatization of propositional configuration logic as well as decision procedures for checking that an architecture satisfies given logical specifications. To allow genericity of specifications, we study first-order and second-order extensions of the propositional configuration logic. First-order logic formulas involve quantification over component variables. Second-order logic formulas involve additional quantification over sets of components. We provide several examples illustrating the application of the results to the characterization of various architecture styles. We also provide an experimental evaluation using the Maude rewriting system to implement the decision procedure for the propositional flavor of the logic.

A unified view of consistent functions

In order to compare the structures and properties of two generalized information systems, a class of special mappings, called consistent functions in some literature, have been extensively studied over the past years. Most recently, consistent functions have been unified and extended into the framework of neighborhood systems which have general binary relations, dominance relations, and coverings as instances. In this paper, we further extend and investigate the notion of consistent functions for fuzzy neighborhood systems. After introducing the definition of extended consistent functions and showing their relationships with related functions, we present some basic properties of the new consistent functions with respect to set-theoretic operations and fuzzy neighborhoods, respectively. As an application, we consider the attribute reduction based on consistent functions. In doing so, we contribute to a unified view of consistent functions and attempt to develop a general theory for investigating the invariant properties of fuzzy neighborhood systems under consistent functions.

On some similarity measures and entropy on quadripartitioned single valued neutrosophic sets

A notion of Quadripartitioned Single Valued Neutrosophic Sets (QSVNS) is introduced and a theoretical study on various set-theoretic operations on them has been carried out. The definitions of distance, similarity measure and entropy have been proposed. Finally an application of the proposed similarity measure in a problem pertaining to pattern recognition has been shown.

Correlation coefficients of single-valued neutrosophic refined soft sets and their applications in clustering analysis

Neutrosophic set theory based on neutrosophy, which is a branch of philosophy, is an important tool for dealing with uncertainty and inconsistency data. Concept of single-valued neutrosophic set is defined as a generalization of the concept of single-valued neutrosophic set, which is useful to model some engineering applications. In this study, the concept of single-valued neutrosophic refined soft set is defined as an extension of single-valued neutrosophic refined set. Also, set theoretical operations between two single-valued neutrosophic refined soft sets are defined and some basic properties of these operations are investigated. Furthermore, two methods to calculate correlation coefficient between two single-valued neutrosophic refined soft sets are proposed, and a clustering analysis application of one of proposed methods is given.

Complex neutrosophic set

Complex fuzzy sets and complex intuitionistic fuzzy sets cannot handle imprecise, indeterminate, inconsistent, and incomplete information of periodic nature. To overcome this difficulty, we introduce complex neutrosophic set. A complex neutrosophic set is a neutrosophic set whose complex-valued truth membership function, complex-valued indeterminacy membership function, and complex-valued falsehood membership functions are the combination of real-valued truth amplitude term in association with phase term, real-valued indeterminate amplitude term with phase term, and real-valued false amplitude term with phase term, respectively. Complex neutrosophic set is an extension of the neutrosophic set. Further set theoretic operations such as complement, union, intersection, complex neutrosophic product, Cartesian product, distance measure, and ź-equalities of complex neutrosophic sets are studied here. A possible application of complex neutrosophic set is presented in this paper. Drawbacks and failure of the current methods are shown, and we also give a comparison of complex neutrosophic set to all such methods in this paper. We also showed in this paper the dominancy of complex neutrosophic set to all current methods through the graph.

Method of face recognition with the use of scalar perturbation functions and a set-theoretic operation of subtraction

The problem of face recognition is discussed. For this purpose, a method based on scalar perturbation functions and a set-theoretic operation of subtraction is proposed. It is shown that all surface points and the mask volume are used in the process of sample testing for more accurate identification.

Dual hesitant fuzzy rough set and its application

In this paper, we propose a general framework for the study of dual hesitant fuzzy rough sets integrating rough set theory with dual hesitant fuzzy set theory in which both constructive and axiomatic approaches are considered. In a constructive approach, by using a dual hesitant fuzzy relation, lower and upper dual hesitant fuzzy rough approximation operators with respect to a dual hesitant fuzzy approximation space are first defined and some properties of this model are further discussed. Adopting an axiomatic approach, dual hesitant fuzzy rough approximation operators are defined by axioms. And different axiom sets of dual hesitant fuzzy set-theoretic operators guarantee the existence of different types of dual hesitant fuzzy relations producing the same operators. We then give an approach of decision-making in uncertainty environment by using the dual hesitant fuzzy rough sets. Finally, a practical application in medical diagnosis is provided to illustrate the validity of this approach.

Axiomatic characterizations of (S, T)-fuzzy rough approximation operators

Axiomatic characterizations of approximation operators are of importance in the study of rough set theory. In this paper axiomatic characterizations of relation-based (S, T)-fuzzy rough approximation operators are investigated. By employing a triangular conorm S and a triangular norm T on [0, 1], we first introduced the constructive definitions of S-lower and T-upper fuzzy rough approximation operators with their essential properties. We then propose an operator-oriented characterization of (S, T)-fuzzy rough sets, that is, fuzzy set-theoretic operators defined by axioms guarantee the existence of different types of fuzzy relations which produce the same operators. We show that the S-lower (and, respectively, T-upper) fuzzy rough approximation operators generated by a generalized fuzzy relation can be described by only one axiom. We further show that (S, T)-fuzzy rough approximation operators corresponding to special types of fuzzy relations, such as serial, reflexive, symmetric, and T-transitive ones as well as any of their compositions, can also be characterized by single axioms.

Type 2 fuzzy multisets and its applications in decision making problems

The purpose of this paper is to introduce the concept of type 2 fuzzy multiset by integrating the type 2 fuzzy set and multiset models. Basic set operations of union, intersection and complement are extended to type 2 fuzzy multisets and made a comparative study on various set theoretic laws satisfied by these operations with set theoretic operations on type 1 and type 2 fuzzy sets. The notion of (α, λ) cut set is defined and some properties of cut sets are studied. Furthermore, we investigate the decision making based on type 2 fuzzy multisets. By means of (α, λ) cut sets, we propose an adjustable approach to type 2 fuzzy multiset based decision making. Moreover, we introduce the weighted type 2 fuzzy multiset and examine its application to decision making. Finally, by means of some illustrative examples we show the validity of type 2 fuzzy multiset, weighted type 2 fuzzy multiset in decision making problems.

On some similarity measures and entropy on quadripartitioned single valued neutrosophic sets

A notion of Quadripartitioned Single Valued Neutrosophic Sets (QSVNS) is introduced and a theoretical study on various set-theoretic operations on them has been carried out. The definitions of distance, similarity measure and entropy have been proposed. Finally an application of the proposed similarity measure in a problem pertaining to pattern recognition has been shown.

Closure properties and complexity of rational sets of regular languages

The test specification language FQL describes relevant test goals as regular expressions over program locations, such that each matching test case has an execution path matching this expression. To specify not only test goals but entire suites, FQL describes families of related test goals by regular expressions over extended alphabets: Herein, each symbol corresponds to a regular expression over program locations, and thus, a word in an FQL expression corresponds to a regular expression describing a single test goal. In this paper we provide a systematic foundation for FQL test specifications, which are in fact rational sets of regular languages (RSRLs). To address practically relevant problems like query optimization, we tackle open questions about RSRLs: We settle closure properties of general and finite RSRLs under common set theoretic operations. We also prove complexity results for checking equivalence and inclusion of star-free RSRLs, and for deciding whether a regular language is a member of a general or star-free RSRL.

Design and Implementation of Spatial Operators and Energy-Efficient Query Processing Strategy in Wireless Sensor Network Database System

Database applications in wireless sensor networks very often demand data collection from sensor nodes of specific target regions. Design and development of spatial query expressions and energy-efficient query processing strategy are important issues for sensor network database systems. The existing sensor network database systems lack the needed sophistication for the space calculation of the target sensor nodes; hence, unnecessary query/data transmissions are required between the sensor nodes and the server. This paper describes our spatial operations and energy-efficient query processing methods that are designed and implemented in our sensor network database system called [Formula: see text]. With a set of spatial operators based on geometric parameters, such as Envelope, NearBy, Distance, Direction, and set theoretic operators, [Formula: see text] allows sensor network applications to easily specify the target space of interest. Our energy-efficient query processing strategy implements an in-network query management based on the lowest common ancestor (LCA) algorithm, so that the query processing cost for calculating the target spaces is greatly reduced by avoiding the need of heavy query/data transmissions between the base-station and target nodes. Performance evaluation shows that our proposed design and implementation of spatial query expressions and processing strategy achieve improved energy efficiency for database operations in the wireless sensor network.

On asymmetric matching between sets

A comparison of two objects may be viewed as an attempt to determine the degree to which they are similar or different in a given sense. Defining a good measure of proximity, or else similarity or dissimilarity between objects is very important in practical tasks as well as theoretical achievements. Each object is usually represented as a point in Cartesian coordinates, and therefore the distance between points reflects similarities between respective objects. In general, the space is assumed to be Euclidean, and a distance between points assigns a nonnegative number. From another point of view the concept of symmetry underlies essentially all theoretical treatments of similarity. Tversky (1977) provided empirical evidence of asymmetric similarities and argued that similarity should not be treated as a symmetric relation. According to Tversky’s consideration, an object is described by sets of features instead of geometric points in a metric space. In this paper we propose a new measure of remoteness between sets of nominal values. Instead of considering distance between two sets, we introduce the measures of perturbation type 1 of one set by another. The consideration is based on set-theoretic operations and the proposed measure describes changes of the second set after adding the first set to it, or vice versa. The measure of sets’ perturbation returns a value from [0,1], and it must be emphasized that this measure is not symmetric in general. The difference between 1 and the sum of these two measures of perturbation of a pair of sets can be understood as Jaccard’s extended similarity measure. In this paper several mathematical properties of the measure of sets’ perturbation are studied, and interpretation of proximity is explained by the comparison of selected measures.