Carmen Torres-Blanc

Negations on type-2 fuzzy sets

Abstract So far, the negation that usually has been considered within the type-2 fuzzy sets (T2FSs) framework, and hence T2FS truth values M (set of all functions from [ 0 , 1 ] to [ 0 , 1 ] ), was obtained by means of Zadehs extension principle and calculated from standard negation in [ 0 , 1 ] . But there has been no comparative analysis of the properties that hold for the above operation and the axioms that any negation in M should satisfy. This suggests that negations should be studied more thoroughly in this context. Following on from this, we introduce in this paper the axioms that an operation in M must satisfy to qualify as a negation and then prove that the usual negation on T2FSs, in particular, is antimonotonic in L (set of normal and convex functions of M ) but not in M . We propose a family of operations calculated from any suprajective negation in [ 0 , 1 ] and prove that they are negations in L . Finally, we examine De Morgans laws for some operations with respect to these negations.

An approach to automatic learning assessment based on the computational theory of perceptions

E-learning systems output a huge quantity of data on a learning process. However, it takes a lot of specialist human resources to manually process these data and generate an assessment report. Additionally, for formative assessment, the report should state the attainment level of the learning goals defined by the instructor. This paper describes the use of the granular linguistic model of a phenomenon (GLMP) to model the assessment of the learning process and implement the automated generation of an assessment report. GLMP is based on fuzzy logic and the computational theory of perceptions. This technique is useful for implementing complex assessment criteria using inference systems based on linguistic rules. Apart from the grade, the model also generates a detailed natural language progress report on the achieved proficiency level, based exclusively on the objective data gathered from correct and incorrect responses. This is illustrated by applying the model to the assessment of Dijkstras algorithm learning using a visual simulation-based graph algorithm learning environment, called GRAPHs.

Measuring contradiction on a-IFS defined in finite universes

The work outlined here aims to build and examine contradiction measures on Atanassov intuitionistic fuzzy sets (A-IFS) that are defined in this particular case in finite universes. The axiomatic definition of contradiction measure in the A-IFS framework was given in [7]. A number of axioms formalizing the concept of continuity for the above measures were also given. In this paper, Section 1, which briefly discusses the preliminaries required to develop the work, is followed by a section analysing how the restriction of the universe of discourse to a finite set influences the continuity axioms. The following three sections look at three types of specific contradiction measures. In Section 3, continuous t-norms and fuzzy negations are used to construct a large family of measures. These measures satisfy different types of continuity, which are examined at length. Then, Sections and take up other families introduced in , and , proving that their behaviour with respect to continuity is better than it was in the earlier articles because the universes considered here are finite.

Aggregation operators on type-2 fuzzy sets

Abstract Cubillo et al. in 2015 established the axioms that an operation must fulfill to be an aggregation operator on a bounded poset (partially ordered set), in particular on M (set of fuzzy membership degrees of T2FSs, which are the functions from [ 0 , 1 ] to [ 0 , 1 ] ). Previously, Takac in 2014 had applied Zadehs extension principle to obtain a set of operators on M which are, under some conditions, aggregation operators on L*, the set of strongly normal and convex functions of M. In this paper, we introduce a more general set of operators on M than were given by Takac, and we study, among other properties, the conditions required to satisfy the axioms of the aggregation operator on L (set of normal and convex functions on M), which includes the set L*.

OBTAINING CONTRADICTION MEASURES ON INTUITIONISTIC FUZZY SETS FROM FUZZY CONNECTIVES

In a previous paper1, we proposed an axiomatic model for measuring self-contradiction in the framework of Atanassov fuzzy sets. This way, contradiction measures that are semicontinuous and completely semicontinuous, from both below and above, were defined. Although some examples were given, the problem of finding families of functions satisfying the different axioms remained open. The purpose of this paper is to construct some families of contradiction measures firstly using continuous t-norms and t-conorms, and secondly by means of strong negations. In both cases, we study the properties that they satisfy. These families are then classified according the different kinds of measures presented in the above paper.

New Order on Type 2 Fuzzy Numbers

Since Lotfi A. Zadeh introduced the concept of fuzzy sets in 1965, many authors have devoted their efforts to the study of these new sets, both from a theoretical and applied point of view. Fuzzy sets were later extended in order to get more adequate and flexible models of inference processes, where uncertainty, imprecision or vagueness is present. Type 2 fuzzy sets comprise one of such extensions. In this paper, we introduce and study an extension of the fuzzy numbers (type 1), the type 2 generalized fuzzy numbers and type 2 fuzzy numbers. Moreover, we also define a partial order on these sets, which extends into these sets the usual order on real numbers, which undoubtedly becomes a new option to be taken into account in the existing total preorders for ranking interval type 2 fuzzy numbers, which are a subset of type 2 generalized fuzzy numbers.

Some geometrical methods for constructing contradiction measures on Atanassov's intuitionistic fuzzy sets

Trillas et al. (1999, Soft computing, 3 (4), 197–199) and Trillas and Cubillo (1999, On non-contradictory input/output couples in Zadehs CRI proceeding, 28–32) introduced the study of contradiction in the framework of fuzzy logic because of the significance of avoiding contradictory outputs in inference processes. Later, the study of contradiction in the framework of Atanassovs intuitionistic fuzzy sets (A-IFSs) was initiated by Cubillo and Castiñeira (2004, Contradiction in intuitionistic fuzzy sets proceeding, 2180–2186). The axiomatic definition of contradiction measure was stated in Castiñeira and Cubillo (2009, International journal of intelligent systems, 24, 863–888). Likewise, the concept of continuity of these measures was formalized through several axioms. To be precise, they defined continuity when the sets ‘are increasing’, denominated continuity from below, and continuity when the sets ‘are decreasing’, or continuity from above. The aim of this paper is to provide some geometrical construction methods for obtaining contradiction measures in the framework of A-IFSs and to study what continuity properties these measures satisfy. Furthermore, we show the geometrical interpretations motivating the measures.

New Negations on the Type-2 Membership Degrees.

Hernandez et al. [9] established the axioms that an operation must fulfill in order to be a negation on a bounded poset (partially ordered set), and they also established in [14] the conditions that an operation must satisfy to be an aggregation operator on a bounded poset. In this work, we focus on the set of the membership degrees of the type-2 fuzzy sets, and therefore, the set M of functions from [0, 1] to [0, 1]. In this sense, the negations on M respect to each of the two partial orders defined in this set are presented for the first time. In addition, the authors show new negations on L (set of the normal and convex functions of M) that are different from the negations presented in [9] applying the Zadeh’s Extension Principle. In particular, negations on M and on L are obtained from aggregation operators and negations. As results to highlight, a characterization of the strong negations that leave the constant function 1 fixed is given, and a new family of strong negations on L is presented.

Self-contradiction for type-2 fuzzy sets whose membership degrees are normal and convex functions

Abstract In order to detect contradictory information or to avoid conflicting outputs in processes of inference, the contradiction has been studied in the framework of fuzzy logic. It was found that a set A is N-self-contradictory with respect to a given negation N if A implies its negation N ( A ) . Further, A is self-contradictory if it is N-self-contradictory for some strong negation N. Similar definitions were found in the framework of the Atanassovs intuitionistic fuzzy sets, following the same idea: a set is contradictory if it involves its own negation. Nevertheless, in some systems or applications the information is given through type-2 fuzzy sets, where the degree in which an element belongs to the set is just a label of the linguistic variable “TRUTH”, that is, the degree is given by a fuzzy set in the universe [0,1]. Then, since in these systems contradictions could also appear, it might be wise to do a similar study in this case. The purpose of this article is to establish definitions of N-self-contradiction and self-contradiction in the framework of the type-2 fuzzy sets. It is also the intention to provide some criteria to verify these properties in the special case in which the membership degrees are in L, the set of the normal and convex functions from [0,1] to [0,1]. In order to do this, the strong negations in L given in previous papers, N n , associated with strong negations on [0,1], n, are used.

GLMP for automatic assessment of DFS algorithm learning

We describe how to use a Granular Linguistic Model of a Phenomenon (GLMP) to assess e-learning processes. We apply this technique to evaluate algorithm learning using the GRAPHs learning environment.