Abstract:
In practice, some of information systems are based on dominance relations, and values of decision attribute are fuzzy. So, it is meaningful to study attribute reductions in ordered decision tables with fuzzy decision. In this paper, upper and lower approximation reductions are proposed in this kind of complicated decision table, respectively. Some important properties are discussed. The judgement theorems and discernibility matrices associated with two reductions are obtained from which the theory of attribute reductions is provided in ordered decision tables with fuzzy decision. Moreover, rough set approach to upper and lower approximation reductions is presented in ordered decision tables with fuzzy decision as well. An example illustrates the validity of the approach, and results show that it is an efficient tool for knowledge discovery in ordered decision tables with fuzzy decision. 1. Introduction Rough set theory, which was first proposed by Pawlak in the early 1980s [1], can describe knowledge via set-theoretic analysis based on equivalence classification for the universe of discourse. It provides a theoretical foundation for inference reasoning about data analysis and has extensive applications in areas of artificial intelligence and knowledge acquisition. A primary use of rough set theory is to reduce the number of attributes in databases thereby improving the performance of applications in a number of aspects including speed, storage, and accuracy. For a data set with discrete attribute values, this can be done by reducing the number of redundant attributes and find a subset of the original attributes that are the most informative. As is well known, an information system may usually has more than one reduct. It means that the set of rules derived from knowledge reduction is not unique. In practice, it is always hoped to obtain the set of the most concise rules. Therefore, people have been attempting to find the minimal reduct of information systems, which means that the number of attributes contained in the reduction is minimal. Unfortunately, it has been proven that finding the minimal reduct of an information system is an NP-hard problem. Recently, some new theories and reduction methods have been developed. Many types of knowledge reduction have been proposed in the area of rough sets [2–8]. Possible rules and reducts have been proposed as a way to deal with inconsistence in an inconsistent decision table [9]. Approximation rules [10] are also used as an alternative to possible rules. On the other hand, generalized decision rules and

Abstract:
We aim to investigate intuitionistic fuzzy ordered information systems. The concept of intuitionistic fuzzy ordered information systems is proposed firstly by introducing an intuitionistic fuzzy relation to ordered information systems. And a ranking approach for all objects is constructed in this system. In order to simplify knowledge representation, it is necessary to reduce some dispensable attributes in the system. Theories of rough set are investigated in intuitionistic fuzzy ordered information systems by defining two approximation operators. Moreover, judgement theorems and methods of attribute reduction are discussed based on discernibility matrix in the systems, and an illustrative example is employed to show its validity. These results will be helpful for decisionmaking analysis in intuitionistic fuzzy ordered information systems.

Abstract:
Incomplete space and fuzzy decision information system exist in many applications, and their models and methods for knowledge reduction are not equivalent to incomplete Pawlak space information system or fuzzy decision inT formation system in complete space. This paper proposes the concept of incomplete and fuzzy decision information sys- tems based on incomplete approximation space and fuzzy decision information systerrL, and gives the rough set model of incomplete and fuzzy decision information systems by the tolerance relation. The model generalizes rough set model of complete fuzzy decision information systems and traditional decision information systems. In this paper the concept of precision reduction and the simple algorithm are presented.

Abstract:
It is well known that most of information systems are based on tolerance relation instead of the classical equivalence relation because of various factors in real-world. To acquire brief decision rules from the information systems, lower approximation reduction is needed. In this paper, the lower approximation reduction is proposed in inconsistent information systems based on tolerance relation. Moreover, the properties are discussed. Furthermore, judgment theorem and discernibility matrix are obtained, from which an approach to lower reductions can be provided in the complicated information systems.

Abstract:
The precision of classification rule is decided by the construction of classification algorithm.By the concepts and attribute reduction algorithm of basic rough set,a data mining algorithm based on the ordered character of attribute in decision system is proposed in this paper.First,the aggregation expression in decision system with ordered character of attribute is briefly introduced.Then,based on the basic characterization of criteria sets and attribute sets in decision system with ordered attributes,the upper and lower approximation expansion models are constructed to obtain the four relative parameters in decision system with ordered attributes.Thirdly,the corresponding data mining and classification rule extracting algorithm is constructed by using the proposed approach.Finally the rationality of the ordered attribute reduction method is validated by simulation example,and the result shows the rules mined by the method are concise and reliable.

Abstract:
State minimization is a fundamental problem in automata theory. The problem is also of great importance in the study of fuzzy automata. However, most work in the literature considered only state reduction of fuzzy automata, whereas the state minimization problem is almost untouched for fuzzy automata. Thus in this paper we focus on the latter problem. Formally, the decision version of the minimization problem of fuzzy automata is as follows: \begin{itemize} \item Given a fuzzy automaton $\mathcal{A}$ and a natural number $k$, that is, a pair $\langle \mathcal{A}, k\rangle$, is there a $k$-state fuzzy automaton equivalent to $\mathcal{A}$? \end{itemize} We prove for the first time that the above problem is decidable for fuzzy automata over totally ordered lattices. To this end, we first give the concept of systems of fuzzy polynomial equations and then present a procedure to solve these systems. Afterwards, we apply the solvability of a system of fuzzy polynomial equations to the minimization problem mentioned above, obtaining the decidability. Finally, we point out that the above problem is at least as hard as PSAPCE-complete.

Abstract:
We introduce the concept of -fuzzy left (right) ideals in ordered semigroups and characterize ordered semigroups in terms of -fuzzy left (right) ideals. We characterize left regular (right regular) and left simple (right simple) ordered semigroups in terms of -fuzzy left (-fuzzy right) ideals. The semilattice of left (right) simple semigroups in terms of -fuzzy left (right) ideals is discussed.

Abstract:
We re-examine a practical aspect of combinatorial fuzzy problems of various types, including search, counting, optimization, and decision problems. We are focused only on those fuzzy problems that take series of fuzzy input objects and produce fuzzy values. To solve such problems efficiently, we design fast fuzzy algorithms, which are modeled by polynomial-time deterministic fuzzy Turing machines equipped with read-only auxiliary tapes and write-only output tapes and also modeled by polynomial-size fuzzy circuits composed of fuzzy gates. We also introduce fuzzy proof verification systems to model the fuzzification of nondeterminism. Those models help us identify four complexity classes: Fuzzy-FPA of fuzzy functions, Fuzzy-PA and Fuzzy-NPA of fuzzy decision problems, and Fuzzy-NPAO of fuzzy optimization problems. Based on a relative approximation scheme targeting fuzzy membership degree, we formulate two notions of "reducibility" in order to compare the computational complexity of two fuzzy problems. These reducibility notions make it possible to locate the most difficult fuzzy problems in Fuzzy-NPA and in Fuzzy-NPAO.

Abstract:
Fuzzy set theory, rough set theory, and soft set theory are three effective mathematical tools for dealing with uncertainties and have many wide applications both in theory and practise. Meng et al. (2011) introduced the notion of soft fuzzy rough sets by combining fuzzy sets, rough sets, and soft sets all together. The aim of this paper is to study the parameter reduction of fuzzy soft sets based on soft fuzzy rough approximation operators. We propose some concepts and conditions for two fuzzy soft sets to generate the same lower soft fuzzy rough approximation operators and the same upper soft fuzzy rough approximation operators. The concept of reduct of a fuzzy soft set is introduced and the procedure to find a reduct for a fuzzy soft set is given. Furthermore, the concept of exclusion of a fuzzy soft set is introduced and the procedure to find an exclusion for a fuzzy soft set is given. 1. Introduction Soft set theory [1], firstly proposed by Molodtsov, is a general mathematical tool for dealing with uncertainty. Since its introduction, soft set theory has been successfully applied in many fields such as functions smoothness, game theory, riemann integration, and theory of measurement [1]. In recent year, soft set theory has received much attention [2–19]. It is worth noting that all of above works are based on the classical soft set theory. A lot of extensions of soft sets to uncertain environments have been proposed recently, such as fuzzy soft sets [20], generalised fuzzy soft sets [21], interval-valued fuzzy soft sets [22], vague soft sets [23], intuitionistic fuzzy sets [24–26], and interval-valued intuitionistic fuzzy soft sets [27]. More importantly, some approaches to these extended soft sets-based decision makings were also developed [21, 28–32]. Rough set theory was originally introduced by Pawlak [33] to deal with vagueness and granularity in information systems. The equivalence relation is the key in Pawlak’s rough set model. However, the equivalence relation is too restrictive for many practical applications. Therefore, some extensions of Pawlak’s rough sets have been developed by replacing the equivalence relations by some more general concepts. For example, by using arbitrary binary relaions, fuzzy relations, and intuitionistic fuzzy relations to granulate the universe of discourse, the concepts of variable precision rough sets, fuzzy rough sets, and intuitionistic fuzzy rough sets have been presented, respectively. But all these rough sets have their inherent difficulties, which are caused by the inadequacy of the parametrization

Abstract:
We consider the fuzzification of the notion of a positive implicative ordered filter in implicative semigroups. We show that every fuzzy positive implicative ordered filter is both a fuzzy ordered filter and a fuzzy implicative ordered filter. We give examples that a fuzzy (implicative) ordered filter may not be a fuzzy positive implicative ordered filter. We also state equivalent conditions of fuzzy positive implicative ordered filters. Finally, we establish an extension property for fuzzy positive implicative ordered filters.