Basic T-norms Research Articles

On a new classification of triangular norms

In this paper, we introduce the notion of (k)-triangular norms and present a new classification of triangular norms (t-norms for short) splitting the set T of all t-norms into an infinite countable partition {T1⁎,T2⁎,…}∪{T⁎}. From the four basic t-norms, the minimum TM is the unique member of T1⁎, the drastic product TD belongs to T2⁎, the product TP and also the Łukasiewicz t-norm TL are members of T⁎. We show that t-norms from Tk⁎ are always (k+1)-contractive, but not vice-versa. We also provide some necessary and some sufficient conditions for t-norms to be (proper) (k)-t-norms. In the end, we give several examples and construction methods for t-norms belonging to particular classes of the proposed partition or to the set Tk=⋃i=1kTi⁎.

Learning rules for Sugeno ANFIS with parametric conjunction operations

The paper presents a Sugeno Adaptive Neuro-Fuzzy Inference System with parametric conjunction operations architecture, ANFIS-CX. The advantages of using parametric conjunction operations in fuzzy models are discussed, and learning rules for system identification with such operations are proposed. These learning strategies can include steepest descent gradient, differential evolution and least square estimation algorithms for tuning antecedent, conjunction, and consequent parameters, respectively. The results of system identification by parameter tuning of conjunction operations in addition to or instead of parameter tuning of the input membership functions are presented. Simulation results show that parameter training in conjunction operations, composed of four basic t-norms, significantly improves the approximation capability of fuzzy models.

Vector t-Norms With Applications

Some basic t-norms defined on [0, 1] are well known in many study areas and applications. However, more general extension of them into vector forms can be used in a lot of new decision-making realms. In this study, we first define preference vector on a linearly ordered set, which includes different special vectors that are mathematically equivalent but with different application backgrounds and practical meanings. As a reasoning rule, we provide a merging method for more than two preference vectors to be aggregated into a finally resultant preference vector. The merging method is based on a special vector value function, which can be seen as a reasonable counterpart of product t-norm, f ( x, y ) = xy ( x and y belong to [0, 1]). In addition, we use bilinear frame to define corresponding four types of basic bilinear vector t-norms, which are just counterparts of basic t-norms from many aspects. Mathematically, preference vectors are related to decreasing matrices, and we find the general entry relation for the decreasing matrices, by which we prove that a decreasing matrix is commutative and closed under matrix multiplication. Thus, we finally present the definition of preference multiplication commutative monoid, which is equivalent to product bilinear vector t-norm. We also show illustrative examples and applications of some results.

Hardware Design of Digital Parametric Conjunctors and t-Norms

This paper presents the hardware design and its implementation on FPGA of several parametric families of digital conjunctors and t-norms built from simple basic t-norms. The authors propose the method of unified presentation of the p-monotone sum, the simplified versions of the ordinal sum of t-norms and t-subnorms, and the method of extension of t-norms by the drastic t-norm. Such unification gives possibility to join several methods of construction of parametric digital conjunctors and t-norms in one scheme with the efficient FPGA implementation. The logic schemes of the proposed design are presented, and the comparative analysis of the latency time and the resources used for the implementation is given.

On the monotone sum of basic t-norms in the construction of parametric families of digital conjunctors for fuzzy systems with reconfigurable logic

A problem of hardware implementation of fuzzy rule-based systems with reconfigurable parametric families of operations is discussed. The methods of generation of parametric families of digital fuzzy conjunctors and t-norms by means of monotone sum and ordinal sum of basic t-norms are studied in the paper. The paper considers several methods of generation of such operations utilizing only the simplest operations, such as comparison, min, max, bounded sum and bounded difference, providing efficient hardware implementation of different reconfigurable parametric families of digital fuzzy conjunctors in one modular structure.

FPGA Implementation of Fuzzy Mamdani System with Parametric Conjunctions Generated by Monotone Sum of Basic t-Norms

The paper presents the results of FPGA implementation of fuzzy Mamdani system with parametric conjunctions generated by monotone sum of basic t-norms. The system is implemented on the DE2 Altera development board using VHDL language. The system contains reconfigurable fuzzy Mamdani model with parametric membership functions and parametric operations that gives possibility to adjust the system to specific application.

Only Smooth Rule Bases Can Generate Monotone Mamdani–Assilian Models Under Center-of-Gravity Defuzzification

At first sight, it seems that ordered linguistic values for all input variables and the output variable and a set of rules describing a monotone system are all that is needed for a monotone model. However, this is not the case. In this study, we show that the choice of the mathematical operators used when calculating the model output and the properties of the membership functions in the output domain are also of crucial importance to obtain a monotone input-output behavior. In the Mamdani-Assilian models considered in this study, the linguistic values of the input variables, as well as the output variable, are described by trapezoidal membership functions that form a fuzzy partition, the rule base is monotone, and the crisp output is obtained by the center-of-gravity (COG) defuzzification method. It is verified that for each of the three basic t-norms, i.e., the minimum T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> , the product T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> , and the Lukasiewicz t-norm T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> , a monotone input-output behavior is obtained for any monotone rule base, or at least for any monotone smooth rule base. The outcome of this study is a guideline for designers of monotone linguistic fuzzy models. For the t-norms T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> and T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> , models with a single input variable show a monotone input-output behavior for any monotone rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions with intervals of changing membership degrees of equal length. The latter restriction can easily be bypassed by an auxiliary interpolation procedure. For the t-norm T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> , models with a single input variable show a monotone input-output behavior for any monotone rule base and any fuzzy output partition. When designing a monotone model with more than one input variable, one should opt for the t-norm T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> and use a monotone smooth rule base. It is shown that monotonicity of models with two input variables that apply T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> is guaranteed for any monotone smooth rule base and any fuzzy output partition. Finally, it is proved that a monotone input-output behavior is always obtained for models with three input variables that apply T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> and a monotone smooth rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions of identical shape.

Filters in fuzzy class theory

A concept of filter is introduced in fuzzy class theory. Graded properties of filters, prime filters, and related constructions are investigated. In particular, the properties of filters relativized with respect to the three basic t-norms (Gödel, product, and Łukasiewicz) are studied. A naturally arising example of a graded filter is constructed and it is shown that the models of filters include well-known families of set functions.

DOMINATION OF AGGREGATION OPERATORS AND PRESERVATION OF TRANSITIVITY

Aggregation processes are fundamental in any discipline where the fusion of information is of vital interest. For aggregating binary fuzzy relations such as equivalence relations or fuzzy orderings, the question arises which aggregation operators preserve specific properties of the underlying relations, e.g. T-transitivity. It will be shown that preservation of T-transitivity is closely related to the domination of the applied aggregation operator over the corresponding t-norm T. Furthermore, basic properties for dominating aggregation operators, not only in the case of dominating some t-norm T, but dominating some arbitrary aggregation operator, will be presented. Domination of isomorphic t-norms and ordinal sums of t-norms will be treated. Special attention is paid to the four basic t-norms (minimum t-norm, product t-norm, Łukasiewicz t-norm, and the drastic product).

On the order of triangular norms—comments on “A triangular norm hierarchy” by E. Cretu

In Cretu (Fuzzy Sets and Systems 120 (2001) 371), the members of the families of Frank, Dubois–Prade, Yager, and Hamacher t-norms, respectively, are compared (in a pointwise way) with the minimum, the product, and the Łukasiewicz t-norm. All these results are well-known and trivial. Moreover, these families of t-norms cannot only be compared with the three basic t-norms above, but all these families are monotone with respect to their index, a fact which is also well known and straightforward to prove (with the exception of the family of Frank t-norms whose monotonicity has been first proven in Butnariu and Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer, Dordrecht, 1993).

Operations fitting triangular-norm-based biresiduation

Given a t-norm T and the corresponding biresiduation ↔ T . An operation K fits the latter (or shortly, is fitting) if ↔ T is respected by K in all its arguments. In this paper, the structure of fitting operations is investigated. Special attention is paid to the case of basic t-norms. The connection between the fitting property and the Lipschitz property is stressed. Some examples are given.