T-overlap functions: a generalization of bivariate overlap functions by t-norms

Trabajo presentado a la 17th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2018. Cádiz, 11-15 junio 2018This paper introduces a generalization of overlap functions by extending one of the boundary conditions of its definition. More specifically, instead of requiring that 'the considered function is equal to zero if and only if some of the inputs is equal to zero' , we allow the range in which some t-norm is zero. We call such generalization by a t-overlap function with respect to such t-norm. Then we analyze the main properties of t-overlap function and introduce some construction methods.Supported by Caixa and Fundacion Caja Navarra of Spain, the Brazilian National Counsel of Technological and Scientific Development CNPq (Proc. 307781/2016-0), the Spanish Ministry of Science and Technology (TIN2016-77356-P)

General grouping functions

Trabajo presentado a la 18th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2020. Lisboa, junio de 2020Some aggregation functions that are not necessarily associative, namely overlap and grouping functions, have called the attention of many researchers in the recent past. This is probably due to the fact that they are a richer class of operators whenever one compares with other classes of aggregation functions, such as t-norms and t-conorms, respectively. In the present work we introduce a more general proposal for disjunctive n-ary aggregation functions entitled general grouping functions, in order to be used in problems that admit n dimensional inputs in a more flexible manner, allowing their application in different contexts. We present some new interesting results, like the characterization of that operator and also provide different construction methods.Supported by CNPq (233950/2014-1, 307781/2016-0, 301618/2019-4), FAPERGS (17/2551 - 0000872-3, 19/2551 - 0001279-9, 19/ 2551 - 0001660-3), PNPD/CAPES (464880 /2019-00) and the Spanish Ministry of Science and Technology (PC093 - 094TFIPDL, TIN2016-81731-REDT, TIN2016-77356-P (AEI/FEDER, UE))

(SS,NN,TT)-Implications

In this paper we introduce a new class of fuzzy implications called ($S$,$N$,$T$)-implications inspired in the logical equivalence $p\rightarrow q \equiv \neg(p\wedge\neg q)\vee\neg p$ and present a brief study of some of the main properties that characterize this class. We present methods of obtaining $t$-norms and $t$-conorms from an ($S$,$N$,$T$)-implication and a fuzzy negation

General admissibly ordered interval-valued overlap functions

Overlap functions are a class of aggregation functions that measure the verlapping degree between two values. They have been successfully applied in several problems in which associativity is not required, such as classification and image processing. Some generalizations of overlap functions were proposed for them to be applied in problems with more than two classes, such as - dimensional and general overlap functions. To measure the overlapping of interval data, interval-valued overlap functions were defined, and, later, they were also generalized in the form of -dimensional and general interval-valued overlap functions. In order to apply some of those concepts in problems with interval data considering the use of admissible orders, which are total orders that refine the most used partial order for intervals, -dimensional admissibly ordered interval-valued overlap functions were recently introduced, proving to be suitable to be applied in classification problems. However, the sole construction method presented for this kind of function do not allow the use of the well known lexicographical orders. So, in this work we combine previous developments to introduce general admissibly ordered interval-valued overlap functions, while also presenting different construction methods and the possibility to combine such methods, showcasing the flexibility and adaptability of this approach, while also being compatible with the lexicographical orders.Supported by the Spanish Ministry of Science and Technology (PC093-094 TFIPDL, TIN2016-81731-REDT, TIN2016-77356-P (AEI/FEDER, UE)), Spanish Ministry of Economy and Competitiveness through the Spanish National Research (project PID2019-108392GB-I00 / AEI / 10.13039/501100011033), UPNA (PJUPNA1926), CNPq (311429/2020-3, 301618/2019-4) and FAPERGS (19/2551-0001660)

General overlap functions

As a generalization of bivariate overlap functions, which measure the degree of overlapping (intersection for non-crisp sets) of n different classes, in this paper we introduce the concept of general overlap functions. We characterize the class of general overlap functions and include some construction methods by means of different aggregation and bivariate overlap functions. Finally, we apply general overlap functions to define a new matching degree in a classification problem. We deduce that the global behavior of these functions is slightly better than some other methods in the literature.The work has been supported by the Research Services of the Universidad Publica de Navarra, the research projects TIN2016-77356-P (AEI/FEDER, UE) and TIN2015-66471-P from the Government of Spain and by the Brazilian National Counsel of Technological and Scientific Development CNPq (Proc. 233950/2014-1, 306970/2013-9, 307781/2016-0) and by Caixa and Fundación Caja Navarra of Spain

Fuzzy implication functions constructed from general overlap functions and fuzzy negations

Fuzzy implication functions have been widely investigated, both in theoretical and practical fields. The aim of this work is to continue previous works related to fuzzy implications constructed by means of non necessarily associative aggregation functions. In order to obtain a more general and flexible context, we extend the class of implications derived by fuzzy negations and t-norms, replacing the latter by general overlap functions. We also investigate their properties, characterization and intersections with other classes of fuzzy implication functions

On fuzzy implications derived from general overlap functions and their relation to other classes

There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used in many applications, such as decision making, classification and image processing, the idea of this work is to continue previous studies related to fuzzy implication functions derived from general overlap functions. In order to obtain a more general and flexible context, we extend the class of implications derived by fuzzy negations and t-norms, replacing the latter by general overlap functions, obtaining the so-called (GO, N)-implication functions. We also investigate their properties, the aggregation of (GO, N)-implication functions, their characterization and the intersections with other classes of fuzzy implication functions.This research was funded by CNPq (grant numbers: 312053/2018-5, 301618/2019-4, 311429/2020-3), FAPERGS (grant number: 19/2551-0001660-3), CAPES-Print (grant number: 88887.363001/2019-00), Spanish Ministry Science and Tech. (grant numbers: TIN2016-77356-P, PID2019-108392GB I00 (AEI/10.13039/501100011033)), and Fundación “La Caixa” (grant number: LCF/PR/PR13/51080004)

The law of O-conditionality for fuzzy implications constructed from overlap and grouping functions

Overlap and grouping functions are special kinds of non necessarily associative aggregation operators proposed for many applications, mainly when the associativity property is not strongly required. The classes of overlap and grouping functions are richer than the classes of t-norms and t-conorms, respectively, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions. In previous works, we introduced some classes of fuzzy implications derived by overlap and/or grouping functions, namely, the residual implications R-0-implications, the strong implications (G, N)-implications and the Quantum Logic implications QL-implications, for overlap functions O, grouping functions G and fuzzy negations N. Such implications do not necessarily satisfy certain properties, but only weaker versions of these properties, e.g., the exchange principle. However, in general, such properties are not demanded for many applications. In this paper, we analyze the so-called law of O-Conditionality, O(x, 1(x, y)) <= y, for any fuzzy implication I and overlap function O, and, in particular, for Ro-implications, (G, N)-implications, QL-implications and D-implications derived from tuples (O, G, N), the latter also introduced in this paper. We also study the conditional antecedent boundary condition for such fuzzy implications, since we prove that this property, associated to the left ordering property, is important for the analysis of the O-Conditionality. We show that the use of overlap functions to implement de generalized Modus Ponens, as the scheme enabled by the law of O-Conditionality, provides more generality than the laws of T-conditionality and U-conditionality, for t-norms T and uninorms U, respectively.This work was partially supported by the Spanish Ministry of Science and Technology under the project TIN2016-77356-P (AEI/FEDER, UE), and by the Brazilian funding agency CNPQ under Processes 305882/2016-3, 481283/2013-7, 306970/2013-9, 232827/2014-1 and 307681/2012-2

Pseudo overlap functions, fuzzy implications and pseudo grouping functions with applications

Overlap and grouping functions are important aggregation operators, especially in information fusion, classification and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we find overlap functions as well as grouping functions are required to be commutative (or symmetric), which limit their wide applications. For the above reasons, this paper expands the original notions of overlap functions and grouping functions, and the new concepts of pseudo overlap functions and pseudo grouping functions are proposed on the basis of removing the commutativity of the original functions. Some examples and construction methods of pseudo overlap functions and pseudo grouping functions are presented, and the residuated implication (co-implication) operators derived from them are investigated. Not only that, some applications of pseudo overlap (grouping) functions in multi-attribute (group) decision-making, fuzzy mathematical morphology and image processing are discussed. Experimental results show that, in many application fields, pseudo overlap functions and pseudo grouping functions have greater flexibility and practicability.This research was funded by National Natural Science Foundation of China (No. 12271319) and research project No. PID2019-108392GB-I00 (AEI/10.13039/501100011033). The Major Program of the National Social Science Foundation of China under Grant No. 20&ZD047

Towards interval uncertainty propagation control in bivariate aggregation processes and the introduction of width-limited interval-valued overlap functions

Overlap functions are a class of aggregation functions that measure the overlapping degree between two values. Interval-valued overlap functions were defined as an extension to express the overlapping of interval-valued data, and they have been usually applied when there is uncertainty regarding the assignment of membership degrees. The choice of a total order for intervals can be significant, which motivated the recent developments on interval-valued aggregation functions and interval-valued overlap functions that are increasing to a given admissible order, that is, a total order that refines the usual partial order for intervals. Also, width preservation has been considered on these recent works, in an intent to avoid the uncertainty increase and guarantee the information quality, but no deeper study was made regarding the relation between the widths of the input intervals and the output interval, when applying interval-valued functions, or how one can control such uncertainty propagation based on this relation. Thus, in this paper we: (i) introduce and develop the concepts of width-limited interval-valued functions and width limiting functions, presenting a theoretical approach to analyze the relation between the widths of the input and output intervals of bivariate interval-valued functions, with special attention to interval-valued aggregation functions; (ii) introduce the concept of $(a,b)$-ultramodular aggregation functions, a less restrictive extension of one-dimension convexity for bivariate aggregation functions, which have an important predictable behaviour with respect to the width when extended to the interval-valued context; (iii) define width-limited interval-valued overlap functions, taking into account a function that controls the width of the output interval; (iv) present and compare three construction methods for these width-limited interval-valued overlap functions.Comment: submitte