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The aim of this note is to show that many papers on various kinds of filters (and related concepts) in (subreducts of) residuated structures are in fact easy consequences of more general results that have been known for a long time. [ABSTRACT FROM AUTHOR]

Identification of a multi-input/single-output system enables the creation of a mathematical model that as accurately as possible represents the input-output relation induced by a physical system. When there is little information about the physical laws related to the system or when the system is too complex, methods such as parametric identification are used to define the system model. In that case, preliminary assumptions can be made about the system leading to a parametric aggregation function based model. This parametric model may be learned by estimating the parameters of this aggregation function from a representative set of inputs/outputs. A major difficulty is to design a model that is relatively simple yet precise enough to meet the user's needs. Linear models are commonly used because they meet these two contradictory constraints. However, use of a linear model is often at the expense of the accuracy of input-output relationship description. In a recent paper, a new modeling approach was proposed, under the name of macsum modeling [31] , which aims at replacing the linear model concept by a set of linear models. The obtained aggregation function leads to an interval-valued output. This represents the lack of accuracy of the model to predict the system output when the inputs are known. An interesting feature of this model is that it is ruled by a single precise parametric vector whereas the output is imprecise. Moreover, the vector dimension is equal to that of the input space. In this paper, we propose a method to learn such a model so that it best reflects the input-output relationship of the considered system. This method is based, as in the case of linear modeling, on a regression method. This is particularly new approach because the macsum aggregation is based on a Choquet integral and very few authors have proposed learning of an aggregation function based on the Choquet integral. [ABSTRACT FROM AUTHOR]

In a recent paper by the authors, Jensen's inequality for Choquet integral was given, and a wrong assertion—"Jensen's inequality does not hold for asymmetric Choquet integral" was made. This paper can be viewed as a continuation of the previous one, Jensen's inequality for asymmetric Choquet integral is proved, the error is corrected. As its generalization, Jensen's inequality for generalized asymmetric Choquet integral is obtained. [ABSTRACT FROM AUTHOR]

The research on fuzzy implication functions has exponentially grown in the last decades becoming one of the main topics studied in the Fuzzy Logic community. These efforts have led to significant advances on the theoretical (mainly) and applied aspects, but the overwhelming number of papers hinder the tracking of the new results and the problems that remain unsolved. The goal of this paper is to determine the valuable practices in the research on classes of fuzzy implication functions and the desirable characteristics that a paper on this topic should have to be useful for a reader. Clearly adopting these features can accelerate the pace of the future research and be constructive for new researchers. [ABSTRACT FROM AUTHOR]

This paper discusses the finite-time stabilization of a class of discontinuous fuzzy neutral-type neural networks with multiple time-varying delays. For the purpose of achieving this stabilization, first of all, by using the differential inclusions and set-valued mapping, the differential inclusion system is established in order to cope with the discontinuities. Then, by means of the introduced sign function, delayed discontinuous control strategies are given. By constructing a modified Lyapunov-Krasovskii functional concerning with the mixed delays, delay-dependent criteria formulated by algebraic inequalities are derived for guaranteeing the finite-time stabilization. Moreover, the settling time is estimated. Finally, numerical examples are carried out to verify the effectiveness of the established results. Since this kind of stabilization analysis of such neural system in this work has not been given much attention, the delay-dependent criteria derived in this paper can be regarded as a leading stabilization result for fuzzy neutral-type neural networks with D operator including multiple time and multiple neutral delays. [ABSTRACT FROM AUTHOR]

Interpretability in intelligent models becomes a challenge in academic research and approaches that facilitate understanding the responses obtained in models based on artificial intelligence and machine learning. This paper presents a new logical fuzzy neuron based on the concept of null-uninorm, thus called null-unineuron to compose the architecture of an evolving neuro-fuzzy model. This new structural neuron can extract advanced fuzzy rules allowing AND and OR-connections of antecedents to better interpret and understand the analyzed problem. This three-layer model uses an evolving weighted fuzzification approach based on incremental data partitioning concepts for knowledge extraction through null-unineurons whose training procedure suits the classification of binary and multiclass patterns in an online and incremental way. The weights integrated in the evolving data partitioning algorithm belong to feature importance levels and achieve an automatic shrinkage of distance calculations along unimportant input directions (features), which in turn accounts for a soft dimension reduction and the likelihood to decrease over-fitting. The new architecture proposed in this model was, subject to pattern classification tests, being more efficient compared to related (evolving) neuro-fuzzy models in the literature. Finally, experiments on various real-world data sets proved that the evolving neuro-fuzzy model proposed in this paper can act in a simplified way in the extraction of knowledge from data while providing answers with a high degree of accuracy for pattern classification problems. [ABSTRACT FROM AUTHOR]

Möbius transforms of capacities or games were considered as a tool for extensions of capacities to particular aggregation functions in several papers. This is, for example, the case of the Lovász extension coinciding with the Choquet integral, or the Owen extension that is also known as multilinear extension. In this paper, a much deeper study of the links between the Möbius transforms of real-valued functions defined on finite bounded posets and aggregation functions is performed. We introduce a Möbius product of any two real-valued functions defined on an arbitrary finite bounded poset, and then, fixing the poset (2 N , ⊆) , N = { 1 , ... , n } , we propose and discuss a construction method for n -ary aggregation functions based on the Möbius product of any capacity on N and a real-valued function g x , x ∈ [ 0 , 1 ] n , defined on 2 N and determined by some appropriate n -ary aggregation function. We provide some necessary and some sufficient conditions for the introduced construction to yield an aggregation function for any capacity. For the binary case, a complete characterization of conditions under which our approach results in an aggregation function for each capacity m is given. [ABSTRACT FROM AUTHOR]

Graph theory has been a useful tool for logic programming in many aspects. In this paper, we propose an equivalent representation of multi-adjoint logic programs using hypergraphs, which are a generalization of classical graphs that allows the use of hypergraph theory in logic programming. Specifically, this representation has been considered in this paper to increase the level and flexibility of different termination results of the computation of the least model of fuzzy logic programs via the immediate consequence operator. Consequently, the least model of more general and versatile fuzzy logic programs can be obtained after finitely many iterations, although infinite programs or programs with loops and general aggregators will be considered. [ABSTRACT FROM AUTHOR]

The algebraic theory of quasi-Nelson logic, a non-involutive generalization of Nelson's constructive logic with strong negation, has been shown to be surprisingly rich in a series of recent papers. In the present paper we bring quasi-Nelson logic into the fuzzy setting by adding the prelinearity axiom to it. We observe that the resulting system is an extension of the well-known Weak Nilpotent Minimum logic, as well as a rotation logic in the sense of recent work by P. Aglianò and S. Ugolini. We characterize the algebraic models of prelinear quasi-Nelson logic as twist-structures over Gödel algebras endowed with a nucleus operator and use the insight thus gained to look at subvarieties corresponding to extensions of well-known fuzzy systems. Our study of the quasi-Nelson negation in a prelinear setting also allows us to show that the variety of prelinear quasi-Nelson algebras is generated by a single standard algebra, thus obtaining a single chain completeness theorem for the logic. [ABSTRACT FROM AUTHOR]

Uncertainty extensively exists in data and knowledge intensive applications, in which fuzzy information processing plays a crucial role. Fuzzy sets have been extensively used to enhance various database models for managing fuzzy data or flexibly querying crisp data. This has resulted in numerous contributions in this research area. This paper pays attention to three crucial issues in fuzzy techniques for data management: modeling fuzzy data , querying fuzzy data , and fuzzy queries over crisp data , and provides a full up-to-date survey on the current state of the art in fuzzy data modeling and querying. The paper identifies fuzzy conceptual data models, fuzzy (relational and object-oriented) database models and fuzzy XML model as well as the relationships among these fuzzy data models. For each type of fuzzy data models, the paper summarizes its query processing. The paper also reviews fuzzy querying over classical data models. In addition to providing a generic overview of the approaches for fuzzy data modeling and querying, this survey paper serves for identifying possible research opportunities in the area of fuzzy data processing. [ABSTRACT FROM AUTHOR]

In the paper Jónás et al. (2022) [1] we asserted a sufficient condition for the concavity (convexity) of the tau function, which is a generator function-based parametric mapping. In our original definition, the generators of the tau function are all the additive generators of the strict triangular norms and strict triangular conorms. We realized that with the original definition, our assertion regarding the concavity (convexity) of the tau function is not valid for all its generators. In this erratum, we present a corrected definition for the tau function such that with this new definition the original assertion regarding the concavity (convexity) of the tau function is valid. With the new definition for the tau function, on the one hand, we narrow the class of its generators, but on the other hand, this class still remains quite versatile and all the other results of the referenced paper remain valid. [ABSTRACT FROM AUTHOR]

There have been developed several approaches to a quantale-valued quantitative domain theory. If the quantale Q is integral and commutative, then Q -valued domains are Q -enriched, and every Q -enriched domain is sober in its Scott Q -valued topology, where the topological «intersection axiom» is expressed in terms of the binary meet of Q (cf. D. Zhang, G. Zhang, Fuzzy Sets and Systems (2022)). In this paper, we provide a framework for the development of Q -enriched dcpos and Q -enriched domains in the general setting of unital quantales (not necessarily commutative or integral). This is achieved by introducing and applying right subdistributive quasi-magmas on Q in the sense of the category Cat (Q). It is important to point out that our quasi-magmas on Q are in tune with the «intersection axiom» of Q -enriched topologies. When Q is involutive, each Q -enriched domain becomes sober in its Q -enriched Scott topology. This paper also offers a perspective to apply Q -enriched dcpos to quantale computation. [ABSTRACT FROM AUTHOR]

Three basic issues of granular computing are construction or definition of granules, measures of granules, and computation or reasoning with granules. This paper reviews the main theories of granular computing and introduces the definition of spatial granules. A granule is composed of one or more atomic granules. The rationality of this definition is explained from the four aspects: simplicity, applicability, measurability and visualization. A one-to-one correspondence is established between the granules and the points in the unit hypercube, and the coarsening and refining of the granules are the descending and ascending dimensions of the points, respectively. The weak fuzzy tolerance relation and weak fuzzy equivalence relation are defined so as to study on all fuzzy binary relations. The notion of layer granularity/fineness is introduced and each granule can be easily denoted by two numbers, which can be used to pre-process macro knowledge space and greatly improve the search speed. This paper also discusses the main properties of granules including the necessary and sufficient conditions of coarse-fine relation and the main principles of granular space. • Introduce the definition of spatial granules and explain its rationality. • Define coarse-fine relation and all the granules can be hierarchicalized. • Adopt a one-to-one mapping between the granules and the points. • Discuss the main properties of spatial granules. • Layer granularity/fineness is used to pre-process macro knowledge space. [ABSTRACT FROM AUTHOR]

This paper presents a novel hybrid architecture, denoted as EFNN-Nul0 (evolving neuro-fuzzy system based on null-unineurons), meticulously crafted for stress identification within the realm of pattern classification. The model seamlessly integrates neural networks, fuzzy systems, and n-uninorms to cultivate knowledge within an adaptive, real-time learning context. Neuro-fuzzy rules, encapsulating interdependencies among input features through IF-THEN type relationships, are formulated utilizing null-unineurons derived from n-uninorms. This construction enables the articulation of both AND- and OR-connections, thereby augmenting the interpretability of the generated rules. The evolution of neurons is facilitated by an extended adaptation of the autonomous data partition method (ADPA). To dynamically interpret the evolution of rules, the paper introduces (i) a mechanism that tracks changes in rules over data stream samples, providing insights into the process dynamics as a structural active learning component, and (ii) a concept for incrementally updating feature weights. These weights encapsulate the varying impact levels of features on stress identification, enabling the reduction of rule length by effectively masking out less significant features. The outcomes of the rules are represented by certainty vectors, recursively updated through an indicator-based recursive weighted least squares (I-RWLS) approach. Neuron activation levels play a pivotal role in determining the weights, ensuring stable local learning. The effectiveness of the proposed model is substantiated through a comprehensive comparison with existing hybrid and evolving approaches in the literature, focusing on binary and multi-class pattern classification. Aspects that identify special characteristics of neuro-fuzzy models related to interpretability will also be demonstrated in this work. The results demonstrate the model's superior performance, characterized by consistently higher accuracy trend lines over time. Furthermore, the model's interpretability is underscored by the coherent neuro-fuzzy rules, contributing to its remarkable accuracy (the EFNN-Nul0 achieved approximately 99% accuracy in stress identification), establishing its efficacy in addressing stress identification within classification problems. [ABSTRACT FROM AUTHOR]

Fuzzifying the crisp functions is a well-known methodology to study the problems under fuzzy uncertainty. The traditional way for fuzzifying the crisp functions is based on the extension principle. In this paper, by referring to the expression in decomposition theorem, we present a new methodology to fuzzify the crisp functions for non-normal fuzzy sets. The main purpose of this paper is to establish the equivalence between the fuzzy functions obtained from the extension principle and the expression in decomposition theorem under the proposed concept of compatibility. The practical cases regarding the equivalences are also studied in order to be used for the practical problems. [ABSTRACT FROM AUTHOR]

In this paper we study some fuzzy order in fuzzy sets based on t-norm fuzzy arithmetic. The definition of the order comes from the extension principle for interval order: a > b iff a − b > 0 and from measurement sciences. In measurement sciences the order is given by a comparator whose operation is based on empirical determination of the difference of two input signals. Fuzzy comparison based on fuzzy sets subtraction is considered as an extension of substraction operation, namely a fuzzy set B is greater than a fuzzy set A if B − A is greater than zero in fuzzy arithmetic. In the paper we show that this fuzzy order is irreflexive, transitive, asymmetric and compact, subhomothetic, Archimedean and semi-Ferrers. Our idea refers to the work of M. K. Urbański (Modeling the measurement in algebraic fuzzy structures, Warsaw 2003, in polish) and to the fundamental problems in the measurement theory. [ABSTRACT FROM AUTHOR]

In a series of papers Klement et al. investigated discrete integrals such as the Choquet and Sugeno integral and their axiomatization. As part of their study they showed that universal integrals are based on semicopulas, and they provided lower and upper bounds of the integral operations based on a given semicopula. These real-valued resp. unit interval valued integrals can be considered as proper aggregation tool in the context of fuzzy sets. The aim of the current paper is to generalize this approach to so-called L -fuzzy sets and relations, i.e., fuzzy sets and relations that use an arbitrary Heyting algebra L as membership degree instead of the unit interval. Furthermore, we present the theory within arrow categories, i.e., we abstract from concrete sets and relations and work within a suitable algebraic framework. The current paper also shows that the results of the previous work can be proven without referring to the real numbers and specific measures such as the Lebesque measure and the induced measurable spaces. [ABSTRACT FROM AUTHOR]

The research purpose of this paper is to explore some new fuzzy rough set models in the fuzzy β -covering group approximation space (F β CGAS) and study the related properties, thereby provide new thinking directions for the theoretical development and application expansion of fuzzy rough set models. Firstly, in the fuzzy β -covering approximation space (F β CAS), in light of some existing fuzzy covering rough set (FCRS) models which do not satisfy the inclusion relations between the fuzzy upper and lower approximations, we define the concepts of parameterized fuzzy β -neighborhoods and parameterized fuzzy complementary β -neighborhoods to eliminate part of the noise data and establish four types of parameterized FCRS models that satisfy the inclusion relationship between the fuzzy lower and upper approximations. Meanwhile, the properties of the models are discussed. Further, in the F β CGAS, we define the concepts of the parameterized pessimistic (optimistic) fuzzy β -fitting neighborhoods and the parameterized pessimistic (optimistic) fuzzy complementary β -fitting neighborhoods and construct the parameterized pessimistic and optimistic FCRS models that satisfy the inclusion relationship between the fuzzy lower and upper approximations. At the same time, we likewise explore the properties of the models. Besides, we further use the newly proposed parameterized fuzzy neighborhoods to study the variable precision multi-granulation fuzzy decision-theoretic rough set models in the F β CGAS. Finally, for all the models mentioned above, we recommend them for solving the criteria weight determination problem, ranking problem, optimal selection problem and qualitative classification problem in multi-criteria decision-making. Meanwhile, we complete the relationships of all the new models mentioned in this paper and discuss them. [ABSTRACT FROM AUTHOR]

We know from a previous paper that the reticulation of a coherent quantale A is a bounded distributive lattice L (A) whose prime spectrum is homeomorphic to m -prime spectrum of A. This paper studies how the reticulation can be used for transferring some properties of bounded distributive lattices to quantales and vice versa. We shall illustrate this thesis by proving several results on the pure and w –pure elements of the quantale A by means of the reticulation L (A). In particular, we shall investigate how the properties of σ –ideals of L (A) can be transferred to pure and w –pure elements of A. Then the obtained transfer properties are used to prove new algebraic and topological results and characterization theorems for some important classes of quantales: normal quantales, mp –quantales, PF –quantales, purified quantales and PP –quantales. [ABSTRACT FROM AUTHOR]

There are two cases of the (conditional) distributivity of semi-t-operators over S -uninorms that have not been discussed by Fang and Hu in 2019. This paper further characterizes the necessary condition for the left (resp. right) distributivity of semi-t-operators over S -uninorms in these two cases. The sufficient conditions for the left (resp. right) distributivity of semi-t-operators over S -uninorms are also discussed. The outcome of one case is different from the conclusion obtained by Fang and Hu, whereby it is impossible to obtain the necessary and sufficient conditions for semi-t-operators distributed over S -uninorms as usual. In particular, this paper points out that the distributivity of semi-t-operators over S -uninorms is equivalent to their conditional distributivity over S -uninorms under the two cases mentioned above. Some examples are also presented to support our conclusions. [ABSTRACT FROM AUTHOR]

The characterization of (S , N) -implications when N is a non-continuous negation has remained one of the most significant open problems in fuzzy logic for the last decades. This paper constitutes the first progress in this topic. Namely, a general characterization of this family of fuzzy implication functions is presented, in which the central property is the existence of a completion of a binary function defined on a certain subregion of [ 0 , 1 ] 2 to a t-conorm. In this paper, the dual problem of finding a completion of a binary function defined on a subregion of [ 0 , 1 ] 2 to a continuous t-norm is studied and solved for the minimum and a cancellative function. These results are the basis for the novel axiomatic characterizations of (S , N) -implications in the case when N has one point of discontinuity and S is equal to the maximum t-conorm in a certain subregion of [ 0 , 1 ] 2 or a strict t-conorm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method to judge the degree of the relationship closeness between system input variables and theoretical output through independence test, and apply this method to construct a hierarchical interval type-2 Takagi-Sugeno-Kang (TSK) fuzzy inference system. Interval type-2 TSK fuzzy inference systems have been applied in many fields. An interval type-2 fuzzy inference system, which is a special case of generalized type-2 fuzzy inference systems, can reduce the computational complexity of type-reduction and save computational cost. An hierarchical fuzzy system can further reduce the complexity of systems. However, it is found that, with the increase in data dimensions, the computational complexity and error of the system will increase. The system accuracy of a hierarchical fuzzy system is also found related to the order of its input variables. Variable selection can select meaningful input variables to decrease input space dimensions, and sort the input variables in some way. This paper proposes a variable selection and sorting method based on independence test. Firstly, input variables data and theoretical output data are classified and contingency tables are established. Chi-square statistics of the contingency tables are calculated for independence test, and input variables that are not independent of theoretical output are selected. Then, the total inertia of the contingency tables is calculated to represent the degree of the relationship closeness between input variables and theoretical output. According to the total inertia, the input variables are sorted and entered the system in order. The proposed method and the model built by the method are tested by dealing with regression problems. By comparing the proposed method on several types of systems, the hierarchical interval type-2 TSK inference system shows a better performance. Then the model is compared with other advanced fuzzy models and non-fuzzy models. Experiments show that its performance is better than the rest. [ABSTRACT FROM AUTHOR]

In data sets some attributes may have higher or lower importance. One of the tools used for data analysis of such datasets are subspace neuro-fuzzy systems. They elaborate fuzzy rules to describe data sets. In subspace neuro-fuzzy systems fuzzy rules exist in subspaces defined with subsets of attributes. In the paper we propose a novel fuzzy biclustering algorithm that groups both objects and attributes in fuzzy clusters. In that way we create a subspace fuzzy rule base for a subspace fuzzy system. The paper is accompanied with numerical examples that show this approach can lead to better generalisation (and thus lower data prediction errors) with preserved interpretation of fuzzy models. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive fuzzy tracking control with filter-based event-triggered mechanism for stochastic nonlinear uncertain system suffered input hysteresis is presented. Considering stochastic disturbances and input hysteresis, Fuzzy Logic Systems (FLSs) are adopted to handle such issues. A time-varying vector contained weight vector and approximation error is established. Then, a class of smooth function is introduced to develop the adaptive control laws. It is aiming to avoid the chattering issue. Further, to achieve the desired tracking performance, this paper considers the discontinuous control input caused by the event-triggered control strategy. And the filter-based event-triggered strategy is presented. With the aid of Lyapunov function and Backstepping technique, the fuzzy event-triggered tracking control method is established for the considered system. It is testified that all signals are bounded under the action of the proposed method. Finally, the theoretical results are verified by the presented simulation examples. [ABSTRACT FROM AUTHOR]

This paper studies the adaptive event-based H ∞ control problem for a class of T-S fuzzy distributed parameter system with parameter uncertain and actuator faults. To overcome the drawback of the period control scheme, an event-triggered control scheme with adaptive threshold is proposed to minimize the number of unnecessary sampled data transmission and to reduce the update frequency of the controller. Furthermore, a T-S fuzzy controller based on the adaptive event-triggered sampled data is designed to ensure the stochastic exponential stability of the distributed parameter system with H ∞ disturbance attenuation performance. Based on a new Lyapunov functional and inequality technique, the stochastic exponential stability criterion of the closed-loop system is obtained, and the controller parameters are designed. The new developed inequality can reduce the conservatism of the stability criterion. Finally, the effectiveness of the theoretical calculation results is verified by numerical simulation, and the results are compared with the relevant literature in the simulation, showing that the methods and results in this paper are less conservative. [ABSTRACT FROM AUTHOR]

The present paper extends some previous works studying automorphisms in type-2 fuzzy sets. The framework for the analysis is the set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ] (fuzzy truth values). The paper concentrates on those automorphisms that, in this framework, leave the constant function 1 fixed. This function is quite important since it defines the boundary between the functions that represent "TRUE" (increasing functions) and those that represent "FALSE" (decreasing functions), being at the same time the only normal function that is simultaneously increasing and decreasing. While C.L. Walker, E.A. Walker and J. Harding introduced in 2008 a family of functions leaving the constant function 1 fixed, the main goal of this paper is to prove that the functions of that family are in fact automorphisms, and moreover, that they are the only automorphisms (in the mentioned set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ]) that preserve the function 1. [ABSTRACT FROM AUTHOR]

Localization Relations are symmetric and T -transitive fuzzy relations that are representable as suprema of conjunctions of fuzzy subsets. This paper shows that not all symmetric and T -transitive fuzzy relations may be obtained in such way, provides an easy condition for characterizing those that may, and states a representation theorem for this class of relations. The logical and geometric implications of such representation are analyzed. The paper also characterizes those (just) symmetric fuzzy relations that are representable as suprema of conjunctions of fuzzy subsets. [ABSTRACT FROM AUTHOR]

Adaptive-network-based fuzzy inference system (ANFIS) is a well-known neuro-fuzzy system, which is widely implemented in nonlinear system approximation, prediction, control, and pattern classification. However, either too many fuzzy sets or fuzzy rules are indispensable to establish an accurate ANFIS for a high-dimensional system, which increases the complexity and limits the widespread application of the ANFIS. In this paper, an approach to simplify the ANFIS is developed based on importance-confidence-similarity (ICS) measures. The simplification process consists of selection of fuzzy rules with large importance-confidence (IC) values and reduction of similar fuzzy rules. The importance is defined as the average normalized firing strength of the rule. The confidence which reflects the approximation performance of a T-S fuzzy rule is estimated by the weighted orthogonal distance from the output data to the rule hyperplane (the consequent of a linear T-S fuzzy rule). The IC measures provide a comprehensive basis for the selection of the appropriate fuzzy rules. Furthermore, to the best of our knowledge, an approach to measure similarities of the T-S fuzzy rules including the rule premises and consequents is first proposed in this paper. We also develop a direct and simple method to remove the similar fuzzy rules based on a hierarchical tree and the IC measures. The proposed simplification approach is applied to Mackey-Glass chaotic time series prediction problem and the strip temperature increment prediction problem of the indirect-fired furnace of the annealing heating line. The results show that our proposed method achieves a balance between simplicity and accuracy of the ANFIS for the high-dimensional system. [ABSTRACT FROM AUTHOR]

This paper concentrates on the performance recovery-based fuzzy robust control of networked nonlinear systems via a deferred actuator-switching approach. The main design procedure of this paper is divided into three parts: 1) nominal controller design; 2) monitoring function and shifting function design; 3) reconfigurable controller design. With the aid of low-computation method and fuzzy logic systems, a structurally and computationally efficient nominal controller is designed for fault-free. Then, an actuator-switching rule is established based on monitoring the stabilization error. Furthermore, a new error transformation is constructed via designing the shifting function, by which tangent barrier Lyapunov functions are effective at the replacement instant. Finally, the stabilization error recovers into the prescribed performance within a specified period of time, and a reconfigurable controller is designed for the backup actuator. By employing the event-triggering mechanism with the relative threshold strategy, communication loads from controllers to actuators are reduced. Simulation examples are given to confirm the effectiveness of the proposed control approach. [ABSTRACT FROM AUTHOR]

Fuzzy c -means (FCM) clustering is a clustering method based on fuzzy theory. This method shows good adaptability by assigning membership values to each sample to represent the degree of membership of the sample to each cluster. However, when dealing with fuzzy boundary data, FCM also generates uncertainty and randomness, which in turn affects the accuracy of clustering results and the number of iterations required for algorithm convergence. In order to solve this problem, fuzzy rough set, as a method of dealing with uncertain data, provides a more accurate and strict description method for the processing of boundary data. Considering this advantage, this paper proposes a new fuzzy rough c -means (FRCM) clustering algorithm to improve the performance and iteration efficiency of FCM. Specifically, in this paper, the similarity based on the Gaussian kernel and the membership information of the object for each cluster are firstly used to construct a fuzzy rough set model to describe the fuzzy roughness between the object and the cluster center, which is used to more accurately represent the relationship between the object and the cluster. Secondly, based on the fuzzy rough model, the fuzzy rough degree of the object is calculated, which is used to describe the approximation degree of the object to the center of each cluster. This fuzzy expression can better handle the fuzzy boundary problem between the object and the cluster center, thereby improving the clustering results and enhancing interpretability. Finally, based on fuzzy rough degree information, the FRCM algorithm is designed. The experimental results show that our proposed method has better performance compared to other comparative clustering methods on both synthetic and real datasets. Specifically, compared to FCM, this algorithm exhibits higher iteration efficiency. [ABSTRACT FROM AUTHOR]

The computation of (if-then) rules relating the most representative variables of a given dataset is a relevant goal in different areas of data analysis such as stock market prediction, disease diagnosis and census data analysis, among others. In Formal Concept Analysis (FCA), attribute implications follow the philosophy of the if-then rules establishing relationships between sets of attributes. This paper will focus on the theoretical development of attribute implications through the tools provided by the multi-adjoint concept lattice framework. We will show different notions to compute the validity of attribute implications, analyzing their properties and comparing the proposed alternatives. The use of linguistic hedges, and truth-stressing hedges in particular, is an interesting way of incorporating truth functions into attribute implications. Moreover, they have been considered in the computation of the concepts of a given context. This paper also introduces and studies the main properties of the multi-adjoint concept lattices framework with truth-stressing hedges. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of fuzzy measure and fuzzy integration which has a dynamic position and is different from previous approaches. Our definition of a new type of fuzzy measure deals with distance functions (special L -fuzzy numbers) and is based on continuous triangular norms. By this concept, we construct a new version of measure theory and integration which is more flexible since the measure of the set both depends on the set itself and on the other parameter named by time. Our approach is related to the idea of fuzzy metric spaces. We study some fuzzy measures induced by classical measures. An integral based on the introduced measures is proposed and studied, too. To complete our paper, we prove some limits and convergence theorems. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that Jensen's inequality for Choquet integral given by R. Wang ten years ago is incorrect, then it is revisited, the modified Jensen's and reverse Jensen's inequalities for Choquet integral are proved. Then Jensen's inequality for generalized Choquet integral, obtained by the authors in a recent paper, is modified accordingly. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of characterizing all functions f : [ 0 , 1 ] → R + such that C f (x , y) = x y f ((1 − x) (1 − y)) , x , y ∈ [ 0 , 1 ] is a bivariate copula. We provide a complete characterization for the two cases: (i) when C f is, in addition, totally positive of order 2 (TP 2) and (ii) when f is twice continuously differentiable. In general, the function f need only be twice differentiable Lebesgue almost everywhere as shown by investigating necessary conditions for C f to be a copula. The paper also contains numerous examples illustrating obtained results and connections to known facts from the literature. Moreover, several properties of such copulas are described. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish restrictions on the parameters under which a function in the form of a natural logarithm of a linear fractional function is a decreasing generator. In addition to the general case, all special cases of generators of this type are considered. A triangular norm is constructed for each generator. In fact, a new parametric class of decreasing generators that generalizes and systematizes known generators is proposed. Such generators generate triangular norms represented by rational functions. The results of this paper can be used to construct aggregation operations. [ABSTRACT FROM AUTHOR]

In this paper, by using nonnegative regular summability methods we improve and generalize the approximation properties of max-min operators which have been investigated systematically in our recent paper published in 2020 in this journal. We also discuss the rate of convergence in the approximation. Applications and concluding remarks at the end of the paper explain why we need such summability methods. [ABSTRACT FROM AUTHOR]

This paper introduces a logical characterization of multi-adjoint algebras with a twofold contribution. On the one hand, the study of multi-adjoint algebras, from a logical perspective, will allow us to discover both the core and new features of these algebras. On the other hand, the axiomatization of multi-adjoint algebras will be useful to take advantage of the properties of the logical connectives considered in the corresponding deductive system. The mechanism considered to carry out the mentioned axiomatization follows the one given by Petr Hájek for residuated lattices. Specifically, the paper presents the bounded poset logic (BPL) as an axiomatization of a bounded poset, since this algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. In the following, the language of BPL is enriched with a family of pairs, composed of a conjunctor and an implication, and its axiomatic system is endowed with new axioms, giving rise to the multi-adjoint logic (ML). The soundness and completeness of BPL and ML are proven. Finally, a comparison between the axiomatization of the multi-adjoint logic and the one given for the BL logic is introduced. [ABSTRACT FROM AUTHOR]

Coherent lower probabilities are one of the most general tools within Imprecise Probability Theory, and can be used to model the available information about an unknown or partially known precise probability. In spite of their generality, coherent lower probabilities are sometimes difficult to deal with. For this reason, in previous papers we studied the problem of outer approximating a given coherent lower probability by a more tractable model, such as a 2- or completely monotone lower probability. Unfortunately, such an outer approximation is not unique in general, even if we restrict our attention to those that are undominated by other models from the same family. In this paper, we investigate whether a number of approaches may help in selecting a unique undominated outer approximation. These are based on minimising a distance with respect to the initial model, maximising the specificity, or preserving the same preferences as the original model. We apply them to 2- and completely monotone approximating lower probabilities, and also to the particular cases of possibility measures and p -boxes. [ABSTRACT FROM AUTHOR]

This paper considers fuzzy multi-state systems (MSSs) with performance sharing between adjacent units, in which each fuzzy multi-state unit has a random performance and a random demand. If the performance of a unit exceeds its individual demand (performance > demand), its surplus performance can be transmitted to its adjacent unit that experiences performance deficiency (performance < demand). The performance/demand of a unit is represented by the performance/demand levels and the corresponding state probabilities. It is usually difficult to estimate the precise state probabilities due to the inaccuracy and insufficiency of data, therefore, fuzzy state probabilities are given instead of precise state probabilities. This paper proposes a fuzzy universal generating function (FUGF)-based method to evaluate the fuzzy reliability of series systems with performance sharing between adjacent units considering the parametric uncertainty related to the state probabilities of the performance/demand levels of units. The originality of this work lies in two aspects: i) the surplus performance of a unit can only be shared with its adjacent units which experience deficiency; ii) the parametric uncertainty related to the state probabilities is considered. An illustrative example is provided to validate the proposed method. [ABSTRACT FROM AUTHOR]

Models of Fuzzy Multi-Criteria Decision Analysis (FMCDA) are based, as a rule, on different approaches to fuzzy extension of source MCDA methods. For this, simplified models are used to approximate the functions of fuzzy variables with propagation of parametric fuzzy numbers (FNs) through all calculations. In this paper, authors suggest a novel approach to fuzzy extension of MCDA methods, for PROMETHEE-I/II, through development of fuzzy PROMETHEE-I/II (FPOMETHEE-I/II) models of different complexity: in addition to simplified models, the standard fuzzy arithmetic (SFA), and transformation methods (TMs) are implemented for assessing functions of FNs corresponding to these models. For ranking of alternatives, two defuzzification based, and one pairwise comparison ranking methods are implemented within the developed models. Special attention is paid to analysis of the overestimation problem, which can occur when using SFA in the presence of dependent variables in corresponding expressions, and to "proper fuzzy extensions" of PROMETHEE-I/II (i.e., results of all functions of FNs within the model are in accordance with the extension principle) based on TMs and, for some models, on the SFA. One of the key goals of this contribution is comparison of the distinctions in ranking alternatives by different FPROMETHEE-II models. It is demonstrated by evaluating a large number of scenarios based on Monte Carlo simulation that the probability of distinction in ranking alternatives by "proper" and "approximated" FPROMETHEE-II models may be considered as significant for ranking multicriteria problems. Another goal of this paper is analysis of the correctness of FPROMETHEE-I/II models with respect to the basic MCDA axiom related to ranking of dominated and dominating alternatives. Authors demonstrate that the basic axiom can be violated, in the general case, by all developed FPROMETHEE-I/II models and suggest an approach to fix this problem. [ABSTRACT FROM AUTHOR]

This paper discusses a class of uncertain optimization problems, in which unknown parameters are modeled by fuzzy intervals. The membership functions of the fuzzy intervals are interpreted as possibility distributions for the values of the uncertain parameters. It is shown how the known concepts of robustness and light robustness, for the traditional interval uncertainty representation of the parameters, can be generalized to choose solutions that optimize against plausible parameter realizations under the assumed model of uncertainty in the possibilistic setting. Furthermore, these solutions can be computed efficiently for a wide class of problems, in particular for linear programming problems with fuzzy parameters in constraints and objective function. Thus the problems under consideration are not much computationally harder than their deterministic counterparts. In this paper a theoretical framework is presented and results of some computational tests are shown. [ABSTRACT FROM AUTHOR]

In the first part of this paper we consider the α -cut of a sum of two fuzzy sets. We show that the α -cut of such a sum can be written as one closed interval determined by the minimum and maximum of some special functions, over a set uniquely determined by the considered t -norm. In the second part of the paper we show that not only α -cut of a sum of two fuzzy intervals has this property, but also α -cut of a continuous function of two variables, strictly increasing in each argument as well. Two examples of the presented theory are considered. [ABSTRACT FROM AUTHOR]

The t -norm fuzzy sets is a class of fuzzy sets equipped with a pair of algebraic operation, a notion of T -convexity and a metric, where all these notions are based on a strict triangular norm T. In this paper we introduce in this class a notion of T -convex hull of a fuzzy set. We prove theorem that binds the T -convex hull of an upper semicontinuous fuzzy set with the convex hull of a (crisp) set. We further show two applications of this result. First, we prove that the operation of forming T -convex hull behaves well with respect to algebraic operations. Second, we show an analogue of Shapley-Folkman theorem. Shapley-Folkman theorem is a well-known result that provides an upper bound on the distance between Minkowski sum of sets and the convex hull of this sum. In this paper we show that the distance between the sum of upper semicontinuous fuzzy subsets of a finite dimensional Euclidean space and the T -convex hull of this sum has an upper bound. As a consequence of this result, we present an iterative procedure for forming T -convex hull of an upper semicontinuous fuzzy set. As an application of the results regarding t -norm fuzzy sets we show an example from a game theoretic setting, where t -norm fuzzy sets were used to handle uncertainty in a repeated two player game. • A notion of T -convex hull of a fuzzy set based on a strict t-norm is studied. • We show result that binds T -convex hull of a fuzzy set with convex hull of a (crisp) set. • There is an upper bound on the distance between a sum of fuzzy sets and the T -convex hull of this sum. • We present a simple iterative procedure for forming T -convex hull of a fuzzy set. • We show an example from a game theoretic setting, where fuzzy sets are used to handle uncertainty in a game. [ABSTRACT FROM AUTHOR]

In this paper, the consensus problem of fractional-order multi-agent systems (FOMASs) with uncertain topological structure via designed Takagi-Sugeno fuzzy event-triggered control is investigated. First, Takagi-Sugeno fuzzy event-triggered control for FOMASs is designed to effectively alleviate the congestion of communication capacity. Then, by utilizing the tools including Lyapunov function method, the fractional inequality technique and Laplace transform, some sufficient conditions for the consensus of FOMASs with uncertain topological structure are obtained, and a rigorous proof is given to exclude Zeno behavior for the designed Takagi-Sugeno fuzzy event-triggered control. Finally, two examples are given to not only illustrate the availability of the proposed approach but also show that the designed Takagi-Sugeno fuzzy event-triggered control in this paper is more effective than the previous methods. [ABSTRACT FROM AUTHOR]

In this paper we study the maximal possible difference N of values of a quasi-copula at two different points of the unit square. This study enables us to give upper and lower bounds, called constrained bounds, for quasi-copulas with fixed value at a given point in the unit square, thus extending an earlier result from copulas to quasi-copulas. It turns out that the two bounds are actually copulas. Difference N is also the main tool in exhibiting two new characterizations of quasi-copulas, a major result of this paper, which sheds new light on the subject of copulas as well. Significant applications of our results are also given in the imprecise probability theory, one of the more important non-standard approaches to probability. After a full-scale bivariate Sklar's theorem has been proven under this approach, we want to establish the tightness of its background before moving to the more general multivariate scene. We present an extension of the partial order on quasi-distributions used in the said theorem, i.e., pointwise order with fixed margins, using again the difference N as a main tool. A careful study of the interplay between the order on quasi-distributions and the order on corresponding quasi-copulas that represent them is also given. Due to a recent result that the quasi-copulas obtained via Sklar's theorem in the imprecise setting are exactly the same as the ones in the standard setting, it is not surprising that results on quasi-copulas can shed some light both on open questions in the standard probability theory and in the imprecise probability theory at the same time. [ABSTRACT FROM AUTHOR]

Several simplified uncertainty models are derived from a given probability P 0 of which they are a perturbation. Among these, we introduced in previous work Nearly-Linear (NL) models. They perform a linear affine transformation of P 0 with barriers, obtaining a couple of conjugate lower/upper probabilities, and generalise several well known neighbourhood models. We classified NL models, partitioning them into three subfamilies, and established their basic consistency properties in [5]. In this paper we investigate how to extend NL models that avoid sure loss by means of their natural extension, a basic, although operationally not always simple, inferential procedure in Imprecise Probability Theory. We obtain formulae for computing directly the natural extension in a number of cases, supplying a risk measurement interpretation for one of them. The results in the paper also broaden our knowledge of NL models: we characterise when they avoid sure loss, express some of them as linear (or even convex) combinations of simpler models, and explore relationships with interval probabilities. [ABSTRACT FROM AUTHOR]

In this paper we discuss integral inequalities for collection integrals that are a special subclass of decomposition integrals introduced as a general framework for many non-linear integrals, including the Choquet integral, the Shilkret integral, the PAN integral, and the concave integral. We give a full characterization of collection integrals that are comonotone additive and for which Chebyshev's, Jensen's, Cauchy's, and Hölder's integral inequalities hold. Interestingly, all these classes of collection integrals coincide and thus we introduce a special subclass of collection integrals, called PCC integrals. The paper is complemented with some examples and remarks for collection and decomposition integrals. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the synchronization control of a class of discontinuous fuzzy Cohen-Grossberg neural networks (DFCGNNs) with uncertain external perturbations and mixed time delays (discrete and distributed time-delays). By using functional differential inclusions theory, inequality technique and the non-smooth analysis of Lyapunov-Krasovskii functional, a simple switching adaptive control is designed and some new criteria are obtained to achieve the finite-time synchronization of the proposed drive-response systems. Besides, the upper bound of the settling time is estimated. Based on the finite-time synchronization, we further present a novel discontinuous state-feedback controller to the response neural system in order to derive the fixed-time synchronization criteria and estimate the settling time. Compared with the previous results which can only fixed-timely synchronize or finite-timely synchronize some simple delayed fuzzy neural network systems with continuous activation functions and without uncertain external perturbations, the DFCGNNs with uncertain external perturbations and mixed time delays addressed and the theoretical results of this paper are more generalized and inclusive. Finally, several simulation examples and remarks are provided to verify the correctness and advantages of the main results. [ABSTRACT FROM AUTHOR]

This paper mainly studies the fixed-time synchronization of Markovian jump fuzzy cellular neural networks with stochastic perturbations, and time-varying delays in the leakage term. By designing delay-dependent controllers with or without fuzzy terms, constructing a suitable stochastic Lyapunov functional and using matrix analysis techniques, this paper derives some novel and useful sufficient conditions to guarantee the fixed-time synchronization of the addressed drive-response systems, and the conditions are delay-dependent, which has less conservative results. The finite time is also independent of the initial states. Finally, numerical examples are given to illustrate the effectiveness of the proposed main results. [ABSTRACT FROM AUTHOR]