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The paper introduces a new class of functions from [0,1]n to [0,n] called d-Choquet integrals. These functions are a generalization of the 'standard' Choquet integral obtained by replacing the difference in the definition of the usual Choquet integral by a dissimilarity function. In particular, the class of all d-Choquet integrals encompasses the class of all 'standard' Choquet integrals but the use of dissimilarities provides higher flexibility and generality. We show that some d-Choquet integrals are aggregation/pre-aggregation/averaging/functions and some of them are not. The conditions under which this happens are stated and other properties of the d-Choquet integrals are studied. This work was supported in part by the Spanish Ministry of Science and Technology under project TIN2016-77356-P (AEI/FEDER, UE), by the Public University of Navarra under project PJUPNA13 and by grant VEGA 1/0614/18 . Z. Takáč was supported by the project VEGA 1/0545/20. R. Mesiar was supported by the project of Grant Agency of the Czech Republic (GACR) no. 18-06915S and by the Slovak grant APVV-17-0066 . G.P. Dimuro was supported by Brazilian agency CNPq under the grant 301618/2019-4 and FAPERGS (Proc. 19/2551-0001660 ). B. Bedregal was supported by Brazilian agency CNPq under the grant 307781/2016-0 and Caixa y Fundación Caja Navarra.

Overlap functions are aggregation functions that express the overlapping degree between two values. They have been used both as a conjunction in several practical problems (e.g., image processing and decision making), and to generate overlap indices between two fuzzy sets, which can be used to construct fuzzy confidence values to be applied in fuzzy rule based classification systems. Some generalizations of overlap functions were recently proposed, such as n-dimensional and general overlap functions, which allowed their application in n-dimensional problems. More recently, the concept of interval-valued overlap functions was presented, mainly to deal with uncertainty in providing membership functions. In this paper, we introduce: (i) the concept of n-dimensional interval-valued overlap functions, studying their representability, (ii) the definition of general interval-valued overlap functions, providing their characterization and some construction methods. Moreover, we also define the concept of interval-valued overlap index, as well as some constructing methods. In addition, we show an illustrative example where those new concepts are applied.

In 2013, Barrenechea et al. used the Choquet integral as an aggregation function in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems. After that, starting from 2016, new aggregation-like functions generalizing the Choquet integral have appeared in the literature, in particular in the works by Lucca et al. Those generalizations of the Choquet integral, namely CT-integrals (by t-norm T), CF-integrals (by a fusion function F satisfying some specific properties), CC-integrals (by a copula C), CF1F2-integrals (by a pair of fusion functions (F1, F2) under some specific constraints) and their generalization gCF1F2-integrals, achieved excellent results in classification problems. The works by Lucca et al. showed that the aggregation task in a FRM may be performed by either aggregation, pre-aggregation or just ordered directional monotonic functions satisfying some boundary conditions, that is, it is not necessary to have an aggregation function to obtain competitive results in classification. The aim of this paper is to present and discuss such generalizations of the Choquet integral, offering a general panorama of the state of the art, showing the relations and intersections among such five classes of generalizations. First, we present them from a theoretical point of view. Then, we also summarize some applications found in the literature.

This paper introduces the theoretical framework for a generalization of C F 1 F 2 -integrals, a family of Choquet-like integrals used successfully in the aggregation process of the fuzzy reasoning mechanisms of fuzzy rule based classification systems. The proposed generalization, called by g C F 1 F 2 -integrals, is based on the so-called pseudo pre-aggregation function pairs ( F 1 , F 2 ) , which are pairs of fusion functions satisfying a minimal set of requirements in order to guarantee that the g C F 1 F 2 -integrals to be either an aggregation function or just an ordered directionally increasing function satisfying the appropriate boundary conditions. We propose a dimension reduction of the input space, in order to deal with repeated elements in the input, avoiding ambiguities in the definition of g C F 1 F 2 -integrals. We study several properties of g C F 1 F 2 -integrals, considering different constraints for the functions F 1 and F 2 , and state under which conditions g C F 1 F 2 -integrals present or not averaging behaviors. Several examples of g C F 1 F 2 -integrals are presented, considering different pseudo pre-aggregation function pairs, defined on, e.g., t-norms, overlap functions, copulas that are neither t-norms nor overlap functions and other functions that are not even pre-aggregation 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.

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))

This paper introduces the family of CF-integrals, which are pre-aggregations functions that generalizes the Choquet integral considering a bivariate function F that is left 0-absorbent. We show that CF-integrals are 1 → -pre-aggregation functions, studying in which conditions they are idempotent and/or averaging functions. This characterization is an important issue of our approach, since we apply these functions in the Fuzzy Reasoning Method (FRM) of a fuzzy rule-based classification system and, in the literature, it is possible to observe that non-averaging aggregation functions usually provide better results. We carry out a study with several subfamilies of CF-integrals having averaging or non-averaging characteristics. As expected, the proposed non-averaging CF-integrals obtain more accurate results than the averaging ones, thus, offering new possibilities for aggregating accurately the information in the FRM. Furthermore, it allows us to enhance the results of classical FRMs like the winning rule and the additive combination.

In this work, we extend to the interval-valued setting the notion of overlap functions, presenting a method which makes use of interval-valued overlap functions for constructing OWA operators with interval-valued weights. Some properties of interval-valued overlap functions and the derived interval-valued OWA operators are analyzed. We specially focus on the homogeneity and migrativity properties.

In this work we introduce the definition of interval-valued fuzzy implication function with respect to any total order between intervals. We also present different construction methods for such functions. We show that the advantage of our definitions and constructions lays on that we can adapt to the interval-valued case any inequality in the fuzzy setting, as the one of the generalized modus ponens. We also introduce a strong equality measure between interval-valued fuzzy sets, in which we take the width of the considered intervals into account, and, finally, we discuss a construction method for this measure using implication functions with respect to total orders.

In this paper, we point out several problems in the different definitions (and related results) of penalty functions found in the literature. Then, we propose a new standard definition of penalty functions that overcomes such problems. Some results related to averaging aggregation functions, in terms of penalty functions, are presented, as the characterization of averaging aggregation functions based on penalty functions. Some examples are shown, as the penalty functions based on spread measures, which happen to be continuous. We also discuss the definition of quasi-penalty functions, in order to deal with non-monotonic (or weakly/directionally monotonic) averaging functions.

Considering the important role played by overlap and grouping functions in several applications in which associativity is not demanded, in this paper we introduce the notion of QL-operations constructed from tuples ( O , G , N ) , where overlap functions O, grouping functions G and fuzzy negations N are used for the generalization of the implication p → q ≡ ¬ p ∨ ( p ∧ q ) , which is defined in quantum logic (QL). We also study under which conditions QL-operations constructed from tuples ( O , G , N ) are fuzzy implication functions, presenting a general form for obtaining QL-implication functions, and particular forms of such fuzzy implication functions according to specific properties of O and G. We analyze the main properties satisfied by QL-operations and QL-implication functions, establishing under which conditions of O, G and N, the derived QL-operations (implication functions) satisfy the different known properties for fuzzy implication functions. We show that QL-implication functions constructed from tuples ( O , G , N ) are richer than QL-implication functions constructed from t-norms and positive t-conorms. We provide a comparative study of QL-implication functions and other classes of fuzzy implication functions constructed from fuzzy negations, overlap and grouping functions, analyzing the intersections among such classes. Finally, we present the application of both QL-operations and QL-implication functions constructed from tuples ( O , G , N ) to the generation of fuzzy subsethood and derived entropy measures, which are useful for several practical applications.

This work presents a novel point interpolation algorithm that is derived from a simple weighted linear regression model. The resulting expression is similar to Inverse Distance Weighting (IDW), which is a widely adopted interpolation algorithm. The novel approach is compared to other methods on synthetic data and also over study cases related to solar radiation, surface elevation, well elevation, and precipitation. Relevant aspects of IDW are preserved while the novel algorithm achieves better results with statistical significance. Artifacts are alleviated in interpolated surfaces generated by the novel approach when compared to the respective surfaces from IDW. The novel method was also revealed, for some cases, as the best alternative among all methods tested in terms of root mean square error. Computational efficiency was shown as competitive or even superior to most of the alternatives under certain conditions. This work is an extended version of our previous conference paper [LNCS 12138, 576 (2020)].

In this paper, we generalize a formula for the discrete Choquet integral by replacing the standard product by a suitable fusion function. For the introduced fusion functions based discrete Choquet-like integrals we discuss and prove several properties and also show that our generalization leads to several new interesting functionals. We provide a complete characterization of the introduced functionals as aggregation functions. For n = 2 , several new aggregation functions are obtained, and if symmetric capacities are considered, our approach yields new generalizations of OWA operators. If n > 2, the introduced functionals are aggregation functions only if they are Choquet integrals with respect to some distorted capacity.

An important problem in open multiagent systems is that of the regulation of social exchanges, toward producing social equilibrium. This problem may be generalized to the regulation of autonomous agents' interactions when cooperating/competing in order to achieve their individual, collective objectives. In this paper, we take an abstract and generalizing approach to this issue. The problem is formalized as a regulation model for the sequential decision making of an agent, acting in an open partially observable stochastic environment, with the aim to induce another autonomous agent to interact in certain way, so as to lead both agents toward a target exchange state configuration. The regulation model is defined as a combination of a partially observable Markov decision process (POMDP), to structure the regulator agent decision process, with a Hidden Markov Model (HMM), to structure its exchange strategy learning process. The main challenge we face is the reciprocal conversion between POMDPs and HMMs. The solution we have found builds on the particular structures of the POMDPs and HMMs that arise in the context of the regulation of social exchanges, which allow for the establishment of a kind of isomorphism between the two models. This paper formally develops these ideas, stating and proving the conversion theorems, and shows their application to an example of regulation of social exchanges.

Overlap functions are aggregation operators specially introduced to be used in applications involving the overlap problem and/or when the associativity property is not strongly required for the aggregation operator, as in classification problems and decision making based on fuzzy preference relations. This paper considers the existent results on residual implication induced by fuzzy conjunctions to introduce the concept of residual implication derived from overlap functions O, denoted by R O -implication, preserving the residuation property. R O -implications are weaker than R-implications constructed from positive and continuous t-norms, in the sense that R O -implications do not necessarily satisfy certain properties satisfied by such R-implications, as the exchange principle, but only weaker versions of these properties. However, in general, such properties are not demanded for many applications. The objectives of this paper are: (a) to analyse the main properties satisfied by R O -implications, establishing under which conditions of an overlap function O the derived R O -implication satisfies the properties of fuzzy implications and (b) to provide two particular characterization of R O -implications derived from (i) the sub-class of overlap functions O that have 1 as neutral element and (ii) the more general sub-class of overlap functions O satisfying the condition O ( x , 1 ) ≤ x .

Overlap functions are a particular type of aggregation functions, given by increasing continuous commutative bivariate functions defined over the unit square, satisfying appropriate boundary conditions. Overlap functions are applied mainly in classification problems, image processing and in some problems of decision making based on some kind of fuzzy preference relations, in which the associativity property is not strongly required. Moreover, the class of overlap functions is reacher than the class of t-norms, 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 functions. This flexibility of overlap functions increases their applicability. The aim of this papers is to introduce the concept of Archimedean overlap functions, presenting a study about the cancellation, idempotency and limiting properties, and providing a characterization of such class of functions. The concept of ordinal sum of overlap functions is also introduced, providing constructing/representing methods of certain classes of overlap functions related to idempotency, cancellation, limiting and Archimedean properties.

Overlap and grouping functions are special kinds of non-necessarily associative aggregation operators recently proposed for applications in classification problems involving the overlap problem and/or when the associativity property is not strongly required, as in image processing and decision making based on fuzzy preference relations, respectively. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow their application in several different contexts. This paper introduces the concept of (G,N)-implication, for a grouping function G and a fuzzy negation N. (G,N)-implications are weaker then (S,N)-implications for positive and continuous t-conorms S, in the sense that (G,N)-implications do not necessarily satisfy certain properties, as the exchange and the left neutrality principles, which are not demanded for applications in decision making based on fuzzy preference relations. We analyze several related important properties, providing a characterization of (G,N)-implications.

Overlap functions and grouping functions are special kinds of aggregation operators that have been recently proposed for applications in classification problems, like, e.g., imaging processing. Overlap and grouping functions can also be applied in decision making based on fuzzy preference relations, where the associativity property is not strongly required and the use of t-norms or t-conorms as the combination/separation operators is not necessary. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow the application in several different contexts. This paper introduces new interesting results related to overlap and grouping functions, investigating important properties, such as migrativity, homogeneity, idempotency and the existence of generators. De Morgan triples are introduced in order to study the relationship between those dual concepts. In particular, we introduce important results related to the action of automorphisms on overlap and grouping functions, analyzing the preservation of those properties and also the Lipschitzianity condition.

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.

This paper presents an analysis of interval-valued S-implications and interval-valued automorphisms, showing a way to obtain an interval-valued S-implication from two S-implications, such that the resulting interval-valued S-implication is said to be obtainable. Some consequences of that are: (1) the resulting interval-valued S-implication satisfies the correctness property, and (2) some important properties of usual S-implications are preserved by such interval representations. A relation between S-implications and interval-valued S-implications is outlined, showing that the action of an interval-valued automorphism on an interval-valued S-implication produces another interval-valued S-implication.

The main contribution of this paper is to introduce an autonomous definition of the connective “fuzzy exclusive or” (fuzzy Xor, for short), which is independent from others connectives. Also, two canonical definitions of the connective Xor are obtained from the composition of fuzzy connectives, and based on the commutative and associative properties related to the notions of triangular norms, triangular conorms and fuzzy negations. We show that the main properties of the classical connective Xor are preserved by the connective fuzzy Xor, and, therefore, this new definition of the connective fuzzy Xor extends the related classical approach. The definitions of fuzzy Xor-implications and fuzzy E-implications, induced by the fuzzy Xor connective, are also studied, and their main properties are analyzed. The relationships between the fuzzy Xor-implications and the fuzzy E-implications with automorphisms are explored.