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The (OP)-polynomial implication I N p obtained from N (x) = 1 − x+x 2 2 .
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In this work, the class of fuzzy polynomial implications is introduced as those fuzzy implications whose expression is given by a polynomial of two variables. Some properties related to the values of the coefficients of the polynomial are studied in order to obtain a fuzzy implication. The polynomial implications with degree less or equal to 3 are...
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This paper mainly intends to offer a new method for finding a real solution of fuzzy polynomials (if exist) of the form A1x+...+Anxn=Ao with fuzzy coefficients Ai (for i= 1,..., n)and fuzzy target output A0 that based on the use of fuzzified feed-forward neural networks. The inputoutput relation of each unit of the fuzzy neural network is defined b...
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... We know that we can generate fuzzy implications from aggregation functions and fuzzy negations [7][8][9][10][11][12][13][14][15]. Other methods of generating fuzzy implications can be achieved using additive generating functions or by some initial implications [16][17][18][19][20][21][22]. Thus, fuzzy implications are useful in fuzzy relational equations and fuzzy mathematical morphology, fuzzy measures and image processing [23], data mining [24], and computing with words and fuzzy partitions. ...
In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form.
... 4. Classes generated according to their final expression: In comparison with the other strategies, in this method the important feature is not the construction method by itself but the final expression of the operator. This strategy is quite new and it started in 2014, when polynomial fuzzy implications were presented in [12]. ...
... Other classes introduced according to its final expression have been the class of (OP)-polynomial implications [48] and the class of rational implications [49]. ...
It is well known that fuzzy implication functions have a very flexible definition that encompasses lots of different potential families. Therefore, the additional properties that the family members fulfill define the behavior these operators will have when modeling fuzzy conditionals or applied to any other application. In this way, in this paper, we study the additional properties of fuzzy polynomial implications of degree 4, a recent family of fuzzy implications which has been recently introduced as a family whose members have a simple expression, appealing for applied researchers. It is proved that members of this family fulfill some of the additional essential properties of fuzzy implication functions.KeywordsFuzzy implication functionFuzzy polynomial implicationExchange principleFuzzy connectives
Aggregation functions play an essential role in many fields where it is necessary at some point to aggregate several input data into a representative output value. Due to this great number of applications, it is also necessary to investigate this type of functions from a theoretical point of view, with the intention of finding out which different families exist and which properties they satisfy. In this sense, aggregation functions that have simple expressions can be interesting for computational purposes. For this reason, in this work we study rational aggregation functions, that is, those whose expression is given by the quotient of two bivariate polynomial functions. A characterization of the binary rational aggregation functions of degree one (in both numerator and denominator) is presented. Moreover, specific characterizations of those that are symmetric and idempotent are also investigated.KeywordsAggregation functionsRational FunctionsIdempotenceSymmetry
In the last decades, more than a hundred families of fuzzy implication functions have been proposed. These families are generated by adequately combining other logical connectives or univariate functions. However, no attention has been given to the final expression of the fuzzy implication function, a crucial feature for any application. In this paper, fuzzy polynomial implications are introduced as those fuzzy implication functions whose expression is given by a polynomial of two variables. Polynomials present advantages with respect to other types of functions making them interesting for applications. Several results are proved for polynomials of any degree and the characterisation of all fuzzy polynomial implications of degree less or equal to 4 is achieved.
From the beginnings of fuzzy logic, the Sheffer stroke operation has been overlooked and the efforts of the researchers have been devoted to other logical connectives. In this paper, the Sheffer stroke operation is introduced in fuzzy logic generalizing the classical operation when the truth values are restricted to {0,1}2. Similar to what happens in Boolean logic, the fuzzy Sheffer stroke is functionally complete and it can be used to generate any other fuzzy logical connective by combinations of itself. Two construction methods are presented and the close connection of this operation with a pair of fuzzy conjunction and negation is analysed.
There exist a great quantity of aggregation functions at disposal to be used in different applications. The choice of one of them over the others in each case depends on many factors. In particular, in order to have an easier implementation, the selected aggregation is required to have an expression as simple as possible. In this line, aggregation functions given by polynomial expressions were investigated in [22]. In this paper we continue this investigation focussing on binary aggregation functions given by polynomial expressions only in a particular sub-domain of the unit square. Specifically, splitting the unit square by using the classical negation, the aggregation function is given by a polynomial of degree one or two in one of the sub-domains and by 0 (or 1) in the other sub-domain. This is done not only in general, but also requiring some additional properties like idempotency, commutativity, associativity, neutral (or absorbing) element and so on, leading to some families of binary polynomial aggregation functions with a non-trivial 0 (or 1) region.
In the last decade, several new construction methods of fuzzy implication functions from one or more given ones have been proposed. Following this line of research, in our paper some construction methods based on polynomial functions of three variables are presented. Concretely, these methods provide a (possibly new) fuzzy implication function from a given one and a quadratic polynomial function. A complete characterization of those quadratic functions adequate to obtain a fuzzy implication function through this strategy is presented. Moreover, the invariant fuzzy implication functions with respect to this method are determined. Finally, some quadratic construction methods preserving some specific additional properties are analysed.
