Semi-divisible triangular norms

Semi-divisibility of left-continuous triangular norms is a weakening of the divisibility (i.e., continuity) axiom for t-norms. In this contribution we focus on the class of semi-divisible t-norms and show the following properties: Each semi-divisible t-norm with Ran(n _T ) = [0, 1] is nilpotent. Semi-divisibility of an ordinal sum t-norm is determined by the corresponding property of its first component (which can be a proper t-subnorm, too). Finally, negations with finite range derived from semi-divisible t-norms are studied.     

On prioritized weighted aggregation in multi-criteria decision making

This paper deals with multi-criteria decision making (MCDM) problems with multiple priorities, in which priority weights associated with the lower priority criteria are related to the satisfactions of the higher priority criteria. Firstly, we propose a prioritized weighted aggregation operator based on ordered weighted averaging (OWA) operator and triangular norms (t-norms). To preserve the tradeoffs among the criteria in the same priority level, we suggest that the degree of satisfaction regarding each priority level is viewed as a pseudo criterion. On the other hand, t-norms are used to model the priority relationships between the criteria in different priority levels. In particular, we show that strict Archimedean t-norms perform better in inducing priority weights. As Hamacher family of t-norms provide a wide class of strict Archimedean t-norms ranging from the product to weakest t-norm, Hamacher parameterized t-norms are used to induce the priority weight for each priority level. Secondly, considering decision maker (DM)’s requirement toward higher priority levels, a benchmark based approach is proposed to induce priority weight for each priority level. In particular, ?ukasiewicz implication is used to compute benchmark achievement for crisp requirements; target-oriented decision analysis is utilized to obtain the benchmark achievement for fuzzy requirements. Finally, some numerical examples are used to illustrate the proposed prioritized aggregation technique as well as to compare with previous research.     

Score level fusion of multimodal biometrics using triangular norms

A multimodal biometric system that alleviates the limitations of the unimodal biometric systems by fusing the information from the respective biometric sources is developed. A general approach is proposed for the fusion at score level by combining the scores from multiple biometrics using triangular norms (t-norms) due to Hamacher, Yager, Frank, Schweizer and Sklar, and Einstein product. This study aims at tapping the potential of t-norms for multimodal biometrics. The proposed approach renders very good performance as it is quite computationally fast and outperforms the score level fusion using the combination approach (min, mean, and sum) and classification approaches like SVM, logistic linear regression, MLP, etc. The experimental evaluation on three databases confirms the effectiveness of score level fusion using t-norms.     

Two classes of pseudo-triangular norms and fuzzy implications

Two kinds of extensions of triangular norms (t-norms) are proposed, and the relations between these extensions and fuzzy implications are discussed in this paper. First, two classes of pseudo-t-norms (ps-t-norms), called type-A and type-B ps-t-norms, and their respective residual operators are defined. Then, we prove that these residual operators are fuzzy implications and satisfy the left neutral property. For these two classes of pseudo-t-norms, we give a series of equivalent conditions of left-continuity with respect to their first or second variable. By combining the axioms of the two classes of pseudo-t-norms, we simply get the definition of the triangular seminorms. Furthermore, we define two classes of induced operators from fuzzy implications and give the conditions under which they are type-A ps-t-norms, type-B ps-t-norms and t-seminorms. For a fuzzy implication, a series of equivalent conditions of right-continuity with respect to its second variable are established. Finally, another method inducing type-A ps-t-norms, type-B ps-t-norms and t-seminorms by implications is proposed.     

On the representation of intuitionistic fuzzy t-norms and t-conorms

Intuitionistic fuzzy sets form an extension of fuzzy sets: while fuzzy sets give a degree to which an element belongs to a set, intuitionistic fuzzy sets give both a membership degree and a nonmembership degree. The only constraint on those two degrees is that their sum must be smaller than or equal to 1. In fuzzy set theory, an important class of triangular norms and conorms is the class of continuous Archimedean nilpotent triangular norms and conorms. It has been shown that for such t-norms T there exists a permutation /spl phi/ of [0,1] such that T is the /spl phi/-transform of the Lukasiewicz t-norm. In this paper we introduce the notion of intuitionistic fuzzy t-norm and t-conorm, and investigate under which conditions a similar representation theorem can be obtained.     

Characterization of T-measures

As a natural generalization of a measure space, Butnariu and Klement introduced T-tribes of fuzzy sets with T-measures. They made the first steps towards a characterization of monotonic real-valued T-measures for a Frank triangular norm T. Later on, Mesiar and the authors of this paper found independently two generalizations, one for vector-valued T-measures with respect to Frank t-norms (in particular for nonmonotonic ones) [3], the other for monotonic real-valued T-measures with respect to general strict t-norms [15]. Here we present a common generalization – a characterization of nonmonotonic T-measures with respect to an arbitrary strict t-norm. Moreover, we prove this for vector-valued T-measures. Using this characterization, we generalize Ljapunov Theorem to this context.The authors gratefully acknowledge the support of Ministero dell'Universit'91a e della Ricerca Scientifica e Tecnologica (Italy), grant 201/02/1540 of the Grant Agency of the Czech Republic, and the Czech Ministry of Education under Research Programme MSM 212300013 Decision Making and Control in Manufacturing.     

Uninorm-Based Logic Neurons as Adaptive and Interpretable Processing Constructs

Fuzzy logic neurons and networks as introduced more than a decade ago, seamlessly combine the transparency of logic and architectural underpinnings of neural networks. They dwell on logic connectives (operators) implemented in terms of t-norms and s-conorms along with some logic predicates of inclusion, dominance, similarity and difference. In this study, while adhering to the principle of fuzzy neural networks, we venture into the use of uninorms in place of triangular norms and co-norms. The paper offers new models of generic processing units, called AND and OR unineurons, discusses the underlying functionality of such neurons along with their numeric characteristics and develops comprehensive architectures of logic networks. Discussed is the use of Particle Swarm Optimization as a vehicle of the learning of the networks. Several illustrative numeric examples are included.     

A fuzzy mathematical morphology based on discrete t-norms: fundamentals and applications to image processing

In this paper, a new approach to fuzzy mathematical morphology based on discrete t-norms is studied. The discrete t-norms that have to be used in order to preserve the most usual algebraical and morphological properties, such as monotonicity, idempotence, scaling invariance, among others, are fully determined. In addition, the properties related to B-open and B-closed objects and the generalized idempotence are also studied. In fact, all properties satisfied by the approach based on continuous nilpotent t-norms hold in the discrete case. This is quite important since in practice we only work with discrete objects. In addition, it is proved that more discrete t-norms satisfying all the properties are available in this approach than in the continuous case, which reduces to the ?ukasiewicz t-norm. This morphology based on discrete t-norms can be considered embedded in more general frameworks, such as L-fuzzy sets or quantale modules, but all these frameworks have been studied only from a theoretical point of view. Our main contribution is the practical application of this discrete approach to image processing. Experimental results on edge detection, noise removal and top-hat transformations for some discrete t-norms and their comparison with the corresponding ones obtained by the umbra approach and the continuous ?ukasiewicz t-norm are included showing that this theory can be suitable to be used in a wide range of applications on image processing. In particular, a new edge detector based on the morphological gradient, non-maxima suppression and a hysteresis method is presented.     

The structure of n-contractive t-norms

Archimedean components of t-norms are shown to determine the degree of contractivity of the investigated t-norms. It is shown that Archimedean components play an important role also in the case of torsion t-norms. A special class of generated n-contractive left-continuous t-subnorms is introduced, thus allowing to construct left-continuous n-contractive t-norms by means of ordinal sums. Several examples of 3- and 4-contraction t-norms are given.     

New triangular operator generators for fuzzy systems

Triangular operators (t-operators) form an integral part in the design and analysis of fuzzy systems. Simple monotonic, continuous, nonconditional functions are used in an operator generator to generate t-operators. Depending on the operator generator and the function that it uses, it becomes easier to characterize and classify the families of t-operators. In this paper, the author proposes two operator generators that will extend the domain of triangular operators in the realm of fuzzy set theory. The conventional operator generators generate a t-norm and a t-conorm by using a decreasing function and an increasing function, respectively. In contrast, in this study, increasing functions generate t-norms, while decreasing functions generate t-conorms, respectively     

On complete fuzzy preorders and their characterizations

In the context of crisp or classical relations, one may find several alternative characterizations of the concept of a total preorder. In this contribution, we first discuss the way of translating those characterizations to the framework of fuzzy relations. Those new properties depend on t-norms. We focus on two important families of t-norms, namely those that do not admit zero divisors and those that are rotation invariant. For these families, we study whether or not the properties shown for fuzzy relations lead to characterizations of complete fuzzy preorders. Special attention is paid to the minimum operator, which shows the best behaviour in preserving most of the characterizations known for crisp relations.     

Interval additive generators of interval t-norms and interval t-conorms

The aim of this paper is to introduce the concepts of interval additive generators of interval t-norms and interval t-conorms, as interval representations of additive generators of t-norms and t-conorms, respectively, considering both the correctness and the optimality criteria. The formalization of interval fuzzy connectives in terms of their interval additive generators provides a more systematic methodology for the selection of interval t-norms and interval t-conorms in the various applications of fuzzy systems. We also prove that interval additive generators satisfy the main properties of additive generators discussed in the literature.     

Extended triangular norms

The paper is devoted to classical t-norms extended to operations on fuzzy quantities in accordance with the generalized Zadeh extension principle. Such extended t-norms are used for calculating intersection of type-2 fuzzy sets. Analytical expressions for membership functions of some extended t-norms are derived assuming special classes of fuzzy quantities, i.e., fuzzy truth intervals or fuzzy truth numbers. The possibility of applying these results in the construction of type-2 adaptive network fuzzy inference systems is illustrated on several examples.     

Theoretical analysis of crisp-type fuzzy logic controllers usingvarious t-norm sum-gravity inference methods

The input-output parametric relationship of a class of crisp-type fuzzy logic controllers (FLCs) using various t-norm sum-gravity inference methods is studied. Four most important t-norms are used to calculate the matching level of each control rule and the explicit mathematical forms of reasoning surfaces obtained by using these four t-norms are addressed. Reasoning surfaces of these crisp-type FLCs are proved to be composed of a two-dimensional multilevel relay no matter which t-norm is used and a local position-dependent nonlinear compensator with output pattern influenced by the t-norms is selected. By analyzing the intrinsic operation of the four t-norms, the authors find that both standard intersection and algebraic product are suitable operators to perform the inference of the FLC. However, bounded difference and drastic intersection are disqualified because they cannot satisfy some important criteria. A measure of relative degree-of-nonlinearity is defined to examine the output figures of these crisp-type FLCs. The ultimate behavior of these crisp-type FLCs as the number of linguistic terms approaches infinity is also explored. The local stability criteria for the proportional-integral (PI)-type fuzzy control systems and the natural global stability characteristic for the proportional-derivative (PD)-type fuzzy control systems are also examined     

Compatibility of t-norms with the concept of ?-partition

In this paper we study the behaviour of a kind of partitions formed by fuzzy sets, the ?-partitions, with respect to three important operations: refinement, union and product of partitions. In the crisp set theory, the previous operations lead to new partitions: every refinement of a partition is also a partition; the union of partitions of disjoint sets is a partition of the union set; the product of two partitions of two sets is a partition of the intersection of the partitioned sets. It has been proven that ?-partitions extend the three previous properties when the intersection of fuzzy sets is defined by the minimum t-norm and the union by the maximum t-conorm. In this paper we consider any t-norm defining the intersection of fuzzy sets and we characterize those t-norms for which refinements, unions and products of ?-partitions are ?-partitions. We pay special attention to these characterizations in the case of continuous t-norms.     

Basic Fuzzy Logic is the logic of continuous t-norms and their residua

In this paper we prove that Basic Logic (BL) is complete w.r.t. the continuous t-norms on [0, 1], solving the open problem posed by Hájek in [4]. In fact, Hájek proved that such completeness theorem can be obtained provided two new axioms, B1 and B2, were added to the original axioms of BL. The main result of the paper is to show that B1 and B2 axioms are indeed redundant. We also obtain an improvement of the decomposition theorem for saturated BL-chains as ordinal sums whose components are either MV, product or Gödel chains, in an analogous way as for continuous t-norms. Finally we provide equational characterizations of the variety of BL-algebras generated by the three basic BL subvarieties, as well as of the varieties generated by each pair of them, together with completeness results of the calculi corresponding to all these subvarieties.     

Continuous fuzzy conjunctions and disjunctions

Several functions have been used to implement conjunction and disjunction in fuzzy logic and to implement intersection and union in fuzzy set theory. Usually t-norms and t-conorms are used, but for some applications these may not be the best function classes. In this paper, I propose specific requirements which a function should satisfy if it is to be used to implement conjunction (or intersection) or disjunction (or union) and compare the functions proposed for conjunction to t-norms. In addition to stating basic properties of the resulting function classes, I give theorems on polynomial approximation within each class and on the existence of functions in either class with specified values at particular points     

正则蕴涵算子族G-λ-R0及其三Ⅰ支持算法

首先给出了一个新的蕴涵算子族:G-λ-R0(λ∈[0,1])(它包括Gsdel(简称RG)算子与R0算子).然后重点讨论了G-λ-R0(λ∈[0,1])族算子的伴随算子及其正则性.结果表明,在该算子族中,每一个算子都具有伴随算子且具有正则性.从而说明了此算子是较理想的蕴涵算子.最后讨论了基于此蕴涵算子族的三Ⅰ支持算法.