New partial orderings =o=o, =p=p and =H=H are defined, studied and compared on the set HH of finite subsets of the unit interval with special emphasis on the last one. Since comparing two sets of the same cardinality is a simple issue, the idea for comparing two sets A and B of different cardinalities n and m respectively using =H=H is repeating their elements in order to obtain two series with the same length. If lcm(n,m)lcm(n,m) is the least common multiple of n and m we can repeat every element of A lcm(n,m)/mlcm(n,m)/m times and every element of B lcm(n,m)/nlcm(n,m)/n times to obtain such series and compare them (Definition 2.2).
(H,=H)(H,=H) is a bounded partially ordered set but not a lattice. Nevertheless, it will be shown that some interesting subsets of (H,=H)(H,=H) have a lattice structure. Moreover in the set BB of finite bags or multisets (i.e. allowing repetition of objects) of the unit interval a preorder =B=B can be defined in a similar way as =H=H in HH and considering the quotient set View the MathML sourceB¿=B/~ of BB by the equivalence relation ~ defined by A~BA~B when A=BBA=BB and B=BAB=BA, View the MathML source(B¿,=B) is a lattice and (H,=H)(H,=H) can be naturally embedded into it.
The modification (relaxation or intensification) of the antecedent or the consequent in a fuzzy “If, Then” conditional is an important asset for an expert in order to agree with it. The usual method to modify fuzzy propositions is the use of Zadeh's quantifiers based on powers of t-norms. However, the invariance of the truth value of the fuzzy conditional would be a desirable property when both the antecedent and the consequent are modified using the same quantifier. In this paper, a novel family of fuzzy implication functions based on powers of continuous t-norms which ensure the aforementioned property is presented. Other important additional properties are analyzed and from this study, it is proved that they do not intersect the most well-known classes of fuzzy implication functions.
This paper models the assessments of a group of experts when evaluating different magnitudes, features or objects by using linguistic descriptions. A new general representation of linguistic descriptions is provided by unifying ordinal and fuzzy perspectives. Fuzzy qualitative labels are proposed as a generalization of the concept of qualitative labels over a well-ordered set. A lattice structure is established in the set of fuzzy-qualitative labels to enable the introduction of fuzzy-qualitative descriptions as L-fuzzy sets. A theorem is given that characterizes finite fuzzy partitions using fuzzy-qualitative labels, the cores and supports of which are qualitative labels. This theorem leads to a mathematical justification for commonly-used fuzzy partitions of real intervals via trapezoidal fuzzy sets. The information of a fuzzy-qualitative label is defined using a measure of specificity, in order to introduce the entropy of fuzzy-qualitative descriptions. (C) 2016 Elsevier Inc. All rights reserved.
A qualitative approach to decision making under uncertainty has been proposed in the
setting of possibility theory, which is based on the assumption that levels of certainty
and levels of priority (for expressing preferences) are commensurate. In this setting,
pessimistic and optimistic decision criteria have been formally justified. This approach has
been transposed into possibilistic logic in which the available knowledge is described by
formulas which are more or less certainly true and the goals are described in a separate
prioritized base. This paper adapts the possibilistic logic handling of qualitative decision
making under uncertainty in the Answer Set Programming (ASP) setting. We show how
weighted beliefs and prioritized preferences belonging to two separate knowledge bases
can be handled in ASP by modeling qualitative decision making in terms of abductive logic
programming where (uncertain) knowledge about the world and prioritized preferences are
encoded as possibilistic definite logic programs and possibilistic literals respectively. We
provide ASP-based and possibilistic ASP-based algorithms for calculating optimal decisions
and utility values according to the possibilistic decision criteria. We describe a prototype
implementing the algorithms proposed on top of different ASP solvers and we discuss the
complexity of the different implementations.
Fuzzy equality relations or indistinguishability operators generalize the concepts of crisp equality and equivalence relations in fuzzy systems where inaccuracy and uncertainty is dealt with. They generate fuzzy granularity and are an essential tool in Computing with Words (CWW). Traditionally, the degree of similarity between two objects is a number between 0 and 1, but in many occasions this assignment cannot be done in such a precise way and the use of indistinguishability operators valued on a finite set of linguistic labels such as small, very much, etc. would be advisable. Recent advances in the study of finite-valued t-norms allow us to combine this kind of linguistic labels and makes the development of a theory of finite-valued indistinguishability operators and their application to real problems possible.
Lipschitzian and kernel aggregation operators with respect to natural T-indistinguishability operators ET and their powers are studied. A t-norm T is proved to be ET-lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an archimedean t-norm T with additive generator t, the quasi-arithmetic mean generated by t is proved to be the most stable aggregation operator with respect to T.
The aim of this research is to develop a new methodology called UNFIR (uncertainty in FIR) as an extension of the fuzzy inductive reasoning (FIR) technique. The main idea behind UNFIR is to expand the modeling capacity of the FIR methodology allowing it to work with classical fuzzy rules. On the one hand, UNFIR is able to automatically construct fuzzy rules starting from a set of pattern rules obtained by FIR. On the other hand, UNFIR affords the prediction of systems behavior by using a mixed pattern/fuzzy inference system that takes advantage of the uncertainty inherent to the data. The pattern rule base that the FIR methodology generates can be very large, obstructing the prediction process and reducing its efficiency. The new methodology preserves as much as possible the knowledge of the pattern rules in a compact fuzzy rule base. In this process some precision is lost but the robustness is considerably increased.
The performance of UNFIR methodology as a systems’ prediction tool is also studied in this work. Three different applications are used for this purpose, i.e., a linear system, a non-linear system and an industrial process.
Each theory or model implicitly defines its inherent notion of equality for the objects in question. In turn, this equality, and its counterpart, the mathematical concept of equivalence, provides the basis on which to establish classification mechanisms for the domain at hand.
Current learning methods for general causal networks are basically data-driven. Exploration of the search space is made by resorting to some quality measure of prospective solutions. This measure is usually based on statistical assumptions. We discuss the interest of adopting a different point of view closer to machine learning techniques. Our main point is the convenience of using prior knowledge when it is available. We identify several sources of prior knowledge and define their role in the learning process. Their relation to measures of quality used in the learning of possibilistic networks are explained and some preliminary steps for adapting previous algorithms under these new assumptions are presented.
The aim of this paper is to analyze Approximate Reasoning (AR) through extensionality with respect to the natural T-indistinguishability operator, by considering the indistinguishability level between fuzzy sets as a formal measure of its degree of similarity, resemblance or closeness, having in all these terms an intuitive meaning.
Among the several representations of uncertainty, possibility theory allows also for the management of imprecision coming from data. Domain models with inherent uncertainty and imprecision can be represented by means of possibilistic causal networks that, the possibilistic counterpart of Bayesian belief networks. Only recently the definition of possibilistic network has been clearly stated and the corresponding inference algorithms developed. However, and in contrast to the corresponding developments in Bayesian networks, learning methods for possibilistic networks are still few. We present here a new approach that hybridizes two of the most used approaches in uncertain network learning: those methods based on conditional dependency information and those based on information quality measures. The resulting algorithm, POSSCAUSE, admits easily a parallel formulation. In the present paper POSSCAUSE is presented and its main features discussed together with the underlying new concepts used.
A definition for similarity between possibility distributions is introduced and discussed as a basis for detecting dependence between variables by measuring the similarity degree of their respective distributions. This definition is used to detect conditional independence relations in possibility distributions derived from data. This is the basis for a new hybrid algorithm for recovering possibilistic causal networks. The algorithm POSS-CAUSE is presented and its applications discussed and compared with analogous developments in possibilistic and probabilistic causal networks learning.
This paper presents a first approach to a system based on fuzzy logic for the design of curves and surfaces in the context of computer aided geometric design. Bézier curves and surfaces can be seen as particular cases of this system.