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Fuzzy rough approximation operators are the underlying concepts in fuzzy rough set theory . There are at least two approaches to develop these primary concepts, i.e., the constructive approach and the axiomatic approach . Single axiomatic characterizations of fuzzy rough approximation operators have got tons of attention. In this paper, considering L being a GL-quantale, we will develop the theory of L -valued rough sets with an L -set as the basic universe of defining L -valued rough approximation operators. Adopting the idea of single axiomatic characterizations of fuzzy rough sets , we will present the axiomatic characterizations of L -valued upper and lower rough approximation operators on an L -set with respect to reflexive, symmetric, transitive L -valued relations on an L -set as well as their compositions. Choosing an L -set as the universe will break the rules that adopting Zadeh’s fuzzy sets as the universe. By these results, we will further provide a new framework of axiomatic research of fuzzy rough set theory.

In rough set theory , lower and upper approximation operators are two primitive notions. Various fuzzy generalizations of lower and upper approximation operators have been introduced over the years. Considering L being a completely distributive De Morgan algebra, this paper mainly proposes a general framework of L -fuzzifying approximation operators in which constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper L-fuzzifying approximation operators is defined. The connections between L-fuzzy relations and L-fuzzifying approximation operators are examined. In the axiomatic approach, various types of L-fuzzifying rough sets are proposed and L-fuzzifying approximation operators corresponding to each type of L-fuzzy relations as well as their compositions are characterized by single axioms. Moreover, the relationships between L-fuzzifying rough sets and L-fuzzifying topological spaces are investigated. It is shown that there is a one-to-one correspondence between reflexive and transitive L-fuzzifying approximation spaces and saturated L-fuzzifying topological spaces.

For decades practitioners have been using the center-based partitional clustering algorithms like Fuzzy C Means (FCM), which rely on minimizing an objective function, comprising of an appropriately weighted sum of distances of each data point from the cluster representatives. Numerous generalizations in terms of choice of the distance function have been introduced for FCM since its inception. In a stark contrast to this fact, to the best of our knowledge, there has not been any significant effort to address the issue of convergence of the algorithm under the structured generalization of the weighting function. Here, by generalization we mean replacing the conventional weighting function uijm (where uij indicates the membership of data xi to cluster Cj and m is the real-valued fuzzifier with 1 m < ) with a more general function g(uij) which can assume a wide variety of forms under some mild assumptions. In this article, for the first time, we present a novel axiomatic development of the general weighting function based on the membership degrees. We formulate the fuzzy clustering problem along with the intuitive justification of the technicalities behind the axiomatic approach. We develop an Alternative Optimization (AO) procedure to solve the main clustering problem by solving the partial optimization problems in an iterative way. The novelty of the article lies in the development of an axiomatic generalization of the weighting function of FCM, formulation of an AO algorithm for the fuzzy clustering problem with the extension to this general class of weighting functions, and a detailed convergence analysis of the proposed algorithm.

Stochastic processes with semicontious covariances are introduced in a flexible way.Useful optimal designs are computed.Groundbreaking financial applications for probabilities of ruin, Famas hypothesis on EMH and IS/LM model. We are encountering deep financial crisis. One of the basic problems we face is how to predict financial developments. Current modelling is clearly insufficient since continuity of autocorrelation oversmoothes the empirical financial data. We prove that geometric probability to chose positive definite continuous covariance by financial data is zero. We also have the empirical evidence for this fact from real stock data. In order to overcome this gap we introduce the novel abc class of semicontinuous covariance functions, which integrates both continuous and discontinuous covariances. The abc class of covariances is much more flexible, powerful, and realistic than its continuous counterpart. It gives answer to the question why financial markets face negative interest rates, cyclic downbreaks, and other difficulties nowadays. As it is clearly illustrated in this paper, the experimenter (e.g.the financial institution) can choose only up to some extent whether abc class member will be positive definite everywhere. To the best knowledge of the authors this fits well to the own Kolmogorovs opinions on axiomatic theory of probability. The Kolmogorovs axiomatic theory of probability was developed as one possible alternative in order to utilize the flexibility of Lebesgue integral in probability theory, but in many experiments e.g.in physics and finance, one should go beyond this axiomatics. That means one should accept negative probabilities (and consequently negative definite covariance functions). In this paper we clarify that the modelers choice of positive/negative definite functions should be considered within broader Information Theory setup. The introduced abc class is purely topologically defined, with soft regularity conditions on covariances, which are still applicable for increasing and infill domain asymptotics for regression problems, kriging, and finance. These conditions are related to semicontinuous maps of Ornstein Uhlenbeck (OU) processes. We provide several novel results for optimal design of random fields with abc covariances, random walks in finance, and probabilities of ruins related to shocks (e.g.by earthquakes). In particular, we construct a random walk model with semicontinuous covariance. The novel abc class provides a proper environment for unifying several contradictions on IS/ML model in the economic literature. All mentioned novel economical applications are hardly visible from currently used continuous covariance framework.

The key strategy of the three-way decisions theory is to consider a decision-making problem as a ternary classification one (i.e. acceptance, rejection and non-commitment). Recently, this theory has been introduced into formal concept analysis for mining three-way concepts to support three-way decisions in formal contexts. That is, the three-way decisions have been performed by incorporating the idea of ternary classification into the design of extension or intension of a concept. However, the existing methods on the studies of three-way concepts are constructive, which means that the three-way concepts had been formed by defining certain concept-forming operators in advance. In order to reveal the essential characteristics of three-way concepts in making decisions from the perspective of cognition, it is necessary to reconsider three-way concepts under the framework of general concept-forming operators. In other words, axiomatic approaches are required to characterize three-way concepts. Motivated by this problem, this study mainly focuses on three-way concept learning via multi-granularity from the viewpoint of cognition. Specifically, we firstly put forward an axiomatic approach to describe three-way concepts by means of multi-granularity. Then, we design a three-way cognitive computing system to find composite three-way cognitive concepts. Furthermore, we use the idea of set approximation to simulate cognitive processes for learning three-way cognitive concepts from a given clue. Finally, numerical experiments are conducted to evaluate the performance of the proposed learning methods.

We proposed a pair of rough approximation operators on a complete completely distributive lattice (CCD lattice for short) in 2015. In this paper, we further discuss its properties and study the axiomatic approaches to the rough approximation operators. Through these axioms, fuzzy rough approximation operators can be seen as special cases of rough approximation operators on a CCD lattice. We also discuss the axiomatic approaches to generalized rough sets on CCD lattices.

The Smets-Magrez axiomatic is usually used to define the class of fuzzy continuous implications which are both S and R-implications (Lukasiewicz implications). Another approach is the construction of such class starting from a basic implication and applying automorphisms. Literature has shown that there is a harmony between those approaches, however in this paper we show that the extension of the Lukasiewicz implication defined on [0,1] for interval values cannot be applied in a direct way. We show that the harmony between the Smets-Magrez axiomatic approach and the one that comes from the generation by automorphisms is not preserved when such extension is done. One of the main consequences lies on the fact that the automorphism approach induces the loss of R-implications from the resulting class of implicators. More precisely, we show that the interval version of such approaches produce two disjunct classes of implicators, meaning that, unlike the usual case, the choice of the respective approach is an important step.

As a technique for granular computing, rough sets deal with the vagueness and granularity in information systems. Covering-based rough sets have been proposed to generalize this theory for wider application. Three types of covering-based rough sets have been studied for different situations. To make the theory more complete, this paper proposes a fourth type of covering-based rough sets. Compared with the existing ones, the new type shows its special characteristic in the interdependency between its lower and upper approximations. We carry out a systematical study of this new theory. First, we discuss basic properties such as normality, contraction, and monotone. Then we investigate the conditions for this type of covering-based rough sets to satisfy the properties of Pawlak's rough sets and study the interdependency between the lower and upper approximation operations. In addition, axiomatic systems for the lower and upper approximation operations are established. Lastly, we address the relationships between this type of covering-based rough sets and the three existing ones.

Similarity based rough set theory (RST) involving choice in the formation of approximations was recently introduced by the present author. Though the theory can be used to develop improved semantics and models of knowledge and belief with ontology, application requires a priori concepts of granules and granulation as opposed to the more common a posteriori or not a priori concepts of the same prevalent in the literature. In this research, we clarify the desirable semantic features of a context for seamless application of the theory to more general situations, formalise them and refine the semantics. A new axiomatic theory of granules in general RST (including hybrid versions involving fuzzy set theories) is also developed in the process. Interesting new applications to human learning are also illustrated in this paper.

Rough set theory is a useful tool for dealing with inexact, uncertain or vague knowledge in information systems. The classical rough set theory is based on equivalence relations and has been extended to covering based generalized rough set theory. This paper investigates three types of covering generalized rough sets within an axiomatic approach. Concepts and basic properties of each type of covering based approximation operators are first reviewed. Axiomatic systems of the covering based approximation operators are then established. The independence of axiom set for characterizing each type of covering based approximation operators is also examined. As a result, two open problems about axiomatic characterizations of covering based approximation operators proposed by Zhu and Wang in (IEEE Transactions on Knowledge and Data Engineering 19(8) (2007) 1131-1144, Proceedings of the Third IEEE International Conference on Intelligent Systems, 2006, pp. 444-449) are solved.

A multiset is a collection of objects in which repetition of elements is significant. In this paper an attempt is made to generalize the concepts of relation, function, composition and equivalence in the multiset context. As a pre-requisite a brief survey of the axiomatic approach to the multiset theory is also presented.

Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. This paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering-based rough sets and the generalized rough sets based on binary relation. Firstly, we present basic concepts and properties of this kind of rough sets. Then we investigate the relationships between this type of generalized rough sets and other five types of covering-based rough sets. The major contribution in this paper is that we establish the equivalency between this type of covering-based rough sets and a type of binary relation based rough sets. Through existing results in binary relation based rough sets, we present axiomatic systems for this type of covering-based lower and upper approximation operations. In addition, we explore the relationships among several important concepts such as minimal description, reduction, representative covering, exact covering, and unary covering in covering-based rough sets. Investigation of this type of covering-based will benefit to our understanding of other types of rough sets based on covering and binary relation.

This paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches. From the viewpoint of the constructive approach, the basic properties of generalized rough sets over fuzzy lattices are obtained. The matrix representation of the lower and upper approximations is given. According to this matrix view, a simple algorithm is obtained for computing the lower and upper approximations. As for the axiomatic approach, a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices.

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. This paper studies covering-based rough sets from the topological view. We explore the topological properties of this type of rough sets, study the interdependency between the lower and the upper approximation operations, and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation. Lastly, axiomatic systems for the lower approximation operation and the upper approximation operation are constructed.

In the axiomatic approach of rough set theory, rough approximation operators are characterized by a set of axioms that guarantees the existence of certain types of binary relations reproducing the operators. Thus axiomatic characterization of rough approximation operators is an important aspect in the study of rough set theory. In this paper, the independence of axioms of generalized crisp approximation operators is investigated, and their minimal sets of axioms are presented.

Several generalizations of the classical measure and integration theory are based on some generalizations of the standard arithmetical operations. The axiomatic approach to the pseudo-arithmetical operations of pseudo-addition and pseudo-multiplication is discussed. Some of required properties strongly influence the structure of these operations (and consequently the resulting measure and integral generalizations). So, e.g., the ⊕-idempotency of the ⊗-unit element u results to the idempotency of the pseudoaddition ⊕, i.e., ⊕ = v (sup). Several other properties of ⊕ and ⊗ and their consequences are discussed and illustrated by examples.

Relational data model has constituted an incontestable success in database history. In this context, a lot of attention has been paid to functional dependencies due to their paramount importance in the design of relational database. For about fifteen years, several attempts to formalize (soft) real world constraints imposed on the data has been made, leading to the emergence of the concept of fuzzy functional dependency. In this paper, an overview of the different proposals of the fuzzy functional dependency is presented. A new extension of classical functional dependency based on the Lukasiewicz implication is presented and called dynamic fuzzy functional dependency. The associated axiomatic system is introduced and proved to be sound.

A characterization of Onicescu's information energy [1] has been made by an axiomatic approach. Further, a generalization of the same measure has also been provided.

A similarity between finite state machines and directed graphs can be seen from the natural way a directed graph can be associated with a finite state machine to describe the state transition of the finite state machine. Likewise, there is a similarity between fuzzy finite state machines and fuzzy directed graphs. As a matter of fact, all four of these concepts, together with that of information retrieval systems and fuzzy systems, share this similarity, namely that of the notion of successor. This paper gives an axiomatic treatment of the notion of successor in such a way that all of the above systems fall under this axiomatic approach.

Various authors [1–4, 6] have developed different types of information measures through an axiomatic approach. By assuming a set of axioms in this paper, we have developed a new concept of weighted information energy that depends upon two parameters, α and β. Some special cases are also discussed.

An axiomatic basis for the concept of “fuzzy set” is established, following the classical class theory developed by von Neumann and Bernays (VNB). The set of axioms we exhibit is the minimal set that enables a formal definition of fuzzy sets, without assuming a universe of discourse, and requires fewer axioms than any other published formal definition of fuzzy sets.

Recently the multicriterion decision problem (MCDP) has attracted more and more attention, and many decision principles have been proposed for it, among which the linear weighted sum decision principle is the most widely used. This paper discusses an axiomatic characterization of that principle to find out its essential meaning. Firstly, we give an axiomatic system characterizing it. Secondly, we examine the meanings of the axioms. Thirdly, we try to give a logical explanation of why the linear weighted sum decision principle is so popular by comparing it with other decision principles for the MCDP and for decision making under uncertainty. Finally, we investigate whether or not decision principles for the MCDP are applicable to decision making under uncertainty and vice versa.

The concept of a Uniformly Reflexive Structure (U.R.S.) is developed as an abstract approach to computability. An axiomatic characterization of a Godelization is given such that any structure satisfying these axioms is a U.R.S. The fundamental lemmata of the subject and the relationships between U.R.S. and the set of computable functions is explored. Theorems analogous to the basic theorems of computability such as the iteration theorem, Kleene's second recursion theorem are established for U.R.S. Each U.R.S. contains a model N of the non-negative integers such that the restriction to N of the functions of U.R.S. contains all recursive functions. From the proof that the partial recursive functions form a U.R.S. we obtain a new Godelization of these functions.