Fuzzy Partial Order Research Articles

Fuzzy order-sorted feature logic

Order-Sorted Feature (OSF) logic is a knowledge representation and reasoning language based on function-denoting feature symbols and set-denoting sort symbols ordered in a subsumption lattice. OSF logic allows the construction of record-like terms that represent classes of entities and that are themselves ordered in a subsumption relation. The unification algorithm for such structures provides an efficient calculus of type subsumption, which has been applied in computational linguistics and implemented in constraint logic programming languages such as LOGIN and LIFE and automated reasoners such as CEDAR. This work generalizes OSF logic to a fuzzy setting. We give a flexible definition of a fuzzy subsumption relation which generalizes Zadeh's inclusion between fuzzy sets. Based on this definition we define a fuzzy semantics of OSF logic where sort symbols and OSF terms denote fuzzy sets. We extend the subsumption relation to OSF terms and prove that it constitutes a fuzzy partial order with the property that two OSF terms are subsumed by one another in the crisp sense if and only if their subsumption degree is greater than 0. We show how to find the greatest lower bound of two OSF terms by unifying them and how to compute the subsumption degree between two OSF terms, and we provide the complexity of these operations.

Fuzzy preorders and generalized distances: The aggregation problem revisited

Transitive fuzzy relations play a central role in fuzzy research activity. A special type of this class of fuzzy relations is known as indistinguishability operators. Many works have focused their efforts on the study of the properties of such operators. One celebrated problem is to find out the class of functions that allows to aggregate a collection of indistinguishability operators into a new single one. Characterizations of this class of functions have been obtained and the relationship with those functions that aggregate a collection of extended pseudo-metrics into a single one has been revealed in the literature. Moreover, fuzzy preorders are a class of fuzzy transitive relations that extend the notion of indistinguishability operators to the asymmetric context. In this paper we show that there is an equivalence between functions that aggregate fuzzy preorders and those functions that merge extended quasi-pseudo-metrics. The new provided characterizations reveal that, in essence, such functions must be monotone and subadditive. Special attention is paid to the case of fuzzy partial orders (a special case of fuzzy preorder) showing that there is a correspondence between those functions aggregating fuzzy partial orders and those that aggregate extended quasi-metrics. Since all the aforesaid fuzzy relations are transitive we also provide new information about those functions that preserve the class of transitive fuzzy relations in terms of those that preserve the so-called ordinary triangular triplets and those that preserve the triangle inequality. The potential applicability of the exposed theory to multi-criteria decision making problems has been discussed.

Neutrosophic Fuzzy lattice Via Fuzzy Partial Ordering

In this paper, we introduce the concept of neutrosophic fuzzy lattice via fuzzy partial ordering. The definition of neutrosophic fuzzy lattice and equipotent are developed with suitable examples. Cartesian product and some equivalent properties of neutrosophic fuzzy lattice are discussed . Also some theorems on neutrosophic fuzzy lattice are developed here.

Fuzzy structures induced by fuzzy betweenness relations

In the setting of complete residuated lattices, we explore the relationships between the recently introduced fuzzy betweenness relations and three important mathematical notions: fuzzy interval operators, fuzzy partial orders and fuzzy Peano–Pasch spaces. After recalling the concept of a fuzzy betweenness relation w.r.t. a fuzzy equivalence relation, we prove that the resulting category is isomorphic to that of geometric fuzzy interval spaces w.r.t. the same fuzzy equivalence relation. Next, we construct a fuzzy partial order via a fuzzy betweenness relation w.r.t. a fuzzy equivalence relation and analyze their relationships in depth. Finally, taking a field as underlying set, we introduce the concept of a fuzzy betweenness field. Furthermore, in the setting of completely distributive lattices, we provide an interesting example showing that a vector space over a fuzzy betweenness field can yield a fuzzy Peano–Pasch space.

On generating fuzzy Pareto solutions in fully fuzzy multiobjective linear programmingviaa compromise method

In the present paper, it is unified and extended recent contributions on fully fuzzy multiobjective linear programming, and it is proposed a new method for obtaining fuzzy Pareto solutions of a fully fuzzy multiobjective linear programming problem. For its formulation, triangular fuzzy numbers and variables are combined with fuzzy partial orders and fuzzy arithmetic, and no ranking functions are required. By means of solving related crisp multiobjective linear problems, it is provided algorithms to generate fuzzy Pareto solutions; in particular, to generate compromise fuzzy Pareto solutions, what is a novelty in this field.

ADOPTION OF FRACTIONAL DIFFERENTIAL EQUATIONS UNDER IMPROVED VARIATIONAL ITERATIVE ALGORITHM COMBINED WITH DNA CODING ALGORITHM IN IMAGE ENCRYPTION

To explore the application of improved variational iterative method in solving fractional differential equations and the effect of DNA coding algorithm based on logistic chaos mapping in image encryption, combining Chebyshev polynomial with variational iterative method, a new algorithm for solving fractional differential equations is proposed. In order to solve the initial value problem, the simulation software is used to compare the approximate solution after the change of values of [Formula: see text] and [Formula: see text] in the solution process to investigate the influence of different parameters on the accuracy of fractional order system. Subsequently, the logistic chaos sequence is used to generate discrete binary sequences, and an image encryption algorithm based on DNA coding is proposed. In order to verify the encryption performance of the proposed algorithm, Matlab simulation software is used to simulate and verify the image encryption processing. The results show that when [Formula: see text] approaches 1, the solution is closer to the exact solution than that when [Formula: see text] approaches 0.5, 0.7, and 0.9; compared with 1, 3, 5, and 7, the constructed algorithm has the highest accuracy when [Formula: see text] value is 9. The simulation results of the fuzzy partial fractional order system show that the chaotic sequence is sensitive to the initial value, and the logistic mapping sequence is suitable for the information encryption of the secure communication system. The simulation results of the image encryption algorithm based on DNA coding show that the algorithm can effectively encrypt the image, and the image histogram after encryption is approximately a horizontal line; slight changes in the initial value will affect the decryption effect of the image; the correlation analysis results show that there is no correlation between adjacent pixels in the image after encryption, which indicates that combining Chebyshev polynomials with variational iteration method can reduce the computational burden and improve the computational accuracy, and the image encryption algorithm proposed in this study can improve the reliability and security of image encryption.

USER-ORIENTED INTELLIGENCE MINING UNDER THE EXISTENCE OF SOLUTIONS TO INTEGRAL BOUNDARY VALUE PROBLEMS FOR FUZZY PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

In order to accurately perceive user intention and improve the reliability of user intention information mining, the existence of solution of integral boundary value problem of fuzzy partial fractional differential equation is used to mine the information of user intention. Firstly, the periodic boundary value problem of fractional order fuzzy linear differential equation, periodic boundary value problem of fractional order fuzzy nonlinear differential equation and periodic boundary value problem of fractional order fuzzy coupled differential system are described in this research. The existence of the solution of the integral boundary value problem of the fuzzy partial fractional order differential equation is proved. Then a user-oriented information mining framework and an evaluation model based on the existence of solutions for the integral boundary value problem of fuzzy partial fractional differential equations are constructed. Finally, this research makes a case study and a comprehensive evaluation of user-oriented information mining based on the existence of solutions of integral boundary value problems for fuzzy partial fractional differential equations. The results show that the method based on the existence of the solution of the integral boundary value problem of fuzzy partial fractional order differential equation is feasible and scientific for the information mining of user intention. It is also concluded that this method is suitable for the modelling and solving of intelligence mining with complicated and unclear user intention. The research provides a good guidance for information mining oriented to user intention.

Various fuzzy connections and fuzzy concepts in complete co-residuated lattices

We introduce the notions of various fuzzy connections by using distance functions on complete co-residuated lattices instead of using fuzzy partial orders on complete co-residuated lattices. We show that (1) fuzzy residuated connections induce attribute-oriented fuzzy concept lattices and object-oriented fuzzy concept lattices, (2) fuzzy Galois (resp. dual Galois) connections induce formal (resp. dual formal) fuzzy concept lattices, (3) various fuzzy concept lattices are complete lattices and (4) the relations between various fuzzy concept lattices are isomorphic or anti-isomorphic. It is shown that fuzzy relation equations can be solved using the properties of various fuzzy connections. Moreover, the concepts of closure operators, interior operators and generalized fuzzy rough sets are defined on complete co-residuated lattices.

The aggregation of transitive fuzzy relations revisited

In this paper, we consider the problem of aggregating a collection of transitive fuzzy binary relations in such a way that the aggregation process preserves transitivity. Specifically, we focus our efforts on the characterization of those functions that aggregate a collection of fuzzy binary relations which are transitive with respect to a collection of t-norms preserving the transitivity. We characterize them in terms of triangular triplets. Further, the relationship between triangular triplets, the monotonicity of the aggregation function and an appropriate dominance notion is explicitly stated. Special attention is paid to a few classes of transitive fuzzy binary relations that are relevant in the literature. Concretely, we describe, in terms of triangular triplets, those functions that aggregate a collection of fuzzy pre-orders, fuzzy partial orders, relaxed indistinguishability relations, indistinguishability relations and equalities. A surprising relationship between functions that aggregate transitive fuzzy relations into a TM-transitive fuzzy relation and those that aggregate relaxed indistinguishability relations is shown. A few consequences of the new results are also provided for those cases in which all the t-norms of the given collection are the same. Some celebrated results are retrieved as a particular case from the exposed theory.

Predicting stock price trends based on financial news articles and using a novel twin support vector machine with fuzzy hyperplane

This study extracts the hidden topic model and emotional information from news articles. A novel fuzzy twin support vector machine is also developed to merge the large volume of information from on-line news, using this to predict stock price trends. Fuzzy set theory is very useful for this approach because the texts use fuzzy terminology, such as high/low and big/small, and the boundary between rising and falling categories is poorly defined. By using fuzzy partial ordering relation, the decision function of our approach is generalized such that the values assigned to the stocks fall within a specified range and indicate the membership grade of these stocks in the positive class (rising trends). The proposed approach is useful for dealing with outliers. Outliers only slightly increase the spread of the fuzzy decision boundary and the estimation of the mode of fuzzy hyperplane remains unchanged. Therefore, our model is more robust compared with other methods when the data contain some outliers. Besides, the membership grade estimated by the fuzzy decision boundary gives a better insight into the degree of confidence in the predicted outputs. The explicable characteristic for the degree of confidence is a vital aspect for decision-making applications.

Negations and aggregation operators based on a new hesitant fuzzy partial ordering

Based on a new hesitant fuzzy partial ordering proposed by Garmendia et al.~cite{GaCa:Pohfs}, in this paper a fuzzy disjunction ${D}$ on the set ${H}$ of finite and nonempty subsets of the unit interval and a t-conorm ${S}$ on the set $bar{{B}}$ of equivalence class on the set of finite bags of unit interval based on this partial ordering are introduced respectively. Then, hesitant fuzzy negations $N_n$ on ${H}$ and $mu_n$ on $bar{{B}}$ are proposed. Particularly, their De Morgan's laws are investigated with respect to binary operations ${C}$ and ${D}$ on ${H}$, as well as ${T}$ and {S} on $bar{{B}}$ respectively, where ${C}$ is a commutative fuzzy conjunction on $({H},leq_H)$ and ${T}$ is a t-norm on $(bar{{B}},leq_B)$. Finally, the new hesitant fuzzy aggregation operators are presented on ${H}$ and $bar{{B}}$ and their more general forms are given. Moreover, the validity of the aggregation operations is illustrated by a numerical example on decision making.

(1806-4519) Negations and aggregation operators based on a new hesitant fuzzy partial ordering

(1806-4519) Negations and aggregation operators based on a new hesitant fuzzy partial ordering

Heyting Almost Distributive fuzzy Lattices

In this paper, we introduce the concept of Heyting almost distributive fuzzy lattices (HADFL) using the concepts of Heyting almost distributive lattices (HADL), almost distributive fuzzy lattices, fuzzy partial order relation and fuzzy Heyting algebra. Using the properties of fuzzy Heyting algebra, we also give a necessary and sufficient condition for an HADFL to be fuzzy Heyting algebra (FHA).

Kinds of [formula omitted]-fuzzy Filters of Fuzzy Lattices

The starting point of this work comes from the results obtained by Inheung Chon in his paper “Fuzzy partial order relations and fuzzy lattices”. In this way, some characterizations of filters and prime filters are given. Also, we characterize fuzzy t-filters and fuzzy prime t-filters using their α-cuts. Finally, we give necessary and sufficient conditions under which a fuzzy set is a prime t-filter, presenting the notion of fuzzy lattices isomorphism.

Aggregation and consensus for preference relations based on fuzzy partial orders

We propose a framework for eliciting and aggregating pairwise preference relations based on the assumption of an underlying fuzzy partial order. We also propose some linear programming optimization methods for ensuring consistency either as part of the aggregation phase or as a pre- or post-processing task. We contend that this framework of pairwise-preference relations, based on the Kemeny distance, can be less sensitive to extreme or biased opinions and is also less complex to elicit from experts. We provide some examples and outline their relevant properties and associated concepts.

Estimation of a Fuzzy Regression Model Using Fuzzy Distances

Regression analysis is a powerful statistical tool that has many applications in different areas. The problem of regression analysis under a fuzzy environment has been treated in the literature from different points of view and considering a variety of input/output data (crisp or fuzzy). However, we realize that, in general, most research papers have a conflict between the solution of the fuzzy regression problem using crisp distances (minimizing a real error function) and the interpretation of fuzzy data as possibility distributions. The main aim of this paper is to develop a methodology to solve this problem introducing a fuzzy partial order and a family of fuzzy distance measures on the whole set of fuzzy numbers. The new approach allows us to obtain linear and nonlinear models that reach the lowest fuzzy error; the estimation process, in general, can be considered easier to apply in practice, and it is not limited to triangular fuzzy numbers. Numerical examples are provided to illustrate the usefulness and applicability of these results, and comparisons with existing methodologies show that the performance of the proposed solution is very satisfactory.

Formalization of Human Categorization Process Using Interpolative Boolean Algebra

Since the ancient times, it has been assumed that categorization has the basic form of classical sets. This implies that the categorization process rests on the Boolean laws. In the second half of the twentieth century, the classical theory has been challenged in cognitive science. According to the prototype theory, objects belong to categories with intensities, while humans categorize objects by comparing them to prototypes of relevant categories. Such categorization process is governed by the principles of perceived world structure and cognitive economy. Approaching the prototype theory by using truth-functional fuzzy logic has been harshly criticized due to not satisfying the complementation laws. In this paper, the prototype theory is approached by using structure-functional fuzzy logic, the interpolative Boolean algebra. The proposed formalism is within the Boolean frame. Categories are represented as fuzzy sets of objects, while comparisons between objects and prototypes are formalized by using Boolean consistent fuzzy relations. Such relations are directly constructed from a Boolean consistent fuzzy partial order relation, which is treated by Boolean implication. The introduced formalism secures the principles of categorization showing that Boolean laws are fundamental in the categorization process. For illustration purposes, the artificial cognitive system which mimics human categorization activity is proposed.

Fuzzy order congruences on fuzzy posets

Fuzzy order congruences play an important role in studying the categoricalproperties of fuzzy posets. In this paper, the correspondence between the fuzzyorder congruences and the fuzzy order-preserving maps is discussed. We focus onthe characterization of fuzzy order congruences on the fuzzy poset in terms ofthe fuzzy preorders containing the fuzzy partial order. At last, fuzzy completecongruences on fuzzy complete lattices are discussed.

A Fuzzy Partial Ordering Approach for QoS-based Selection of Web Services

As the development of Service-Oriented Computing (SOC), more and more functional similar Web services are deployed over the Internet. Nowadays, Web service selection becomes a crucial issue for making SOC more applicable. Troublesome Web services will affect the reliability of the whole SOC application which invokes the service. Therefore, when choosing Web services, not only the functional attributes but also the Quality of Service (QoS) should also be considered. In this work, we develop a framework to collect the indices of service quality, including runtime and non-runtime indices. A fuzzy partial-ordering approach, which takes both quality indices and their uncertainties into consideration, is proposed to evaluate Web services. A series of partial-ordering models has been developed to rank Web services according to their qualities. Case study shows that the proposed approach is effective for selecting the service having the highest synthetic quality from the collections of functional similar services.

Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed complete posets

This paper deals with quantitative domain theory via fuzzy sets. It examines the continuity of fuzzy directed complete posets (dcpos for short) based on complete residuated lattices. First, we show that a fuzzy partial order in the sense of Fan and Zhang and an L-order in the sense of Bělohlávek are equivalent to each other. Then we redefine the concepts of fuzzy directed subsets and (continuous) fuzzy dcpos. We also define and study fuzzy Galois connections on fuzzy posets. We investigate some properties of (continuous) fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy-double-downward-arrow-operator has a right adjoint. We define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy way-below relation is the smallest approximating fuzzy auxiliary relation.