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The selection of optimal scale has always been the essential problem of multi-scale system. However, most of the current studies only consider the consistency of the system, and ignore the cost information. Therefore, based on the ranking methods of single-valued neutrosophic number, this paper constructs a multi-scale dominant single-valued neutrosophic system. Furthermore, the test cost is defined according to the relative distance preference degree, and the risk cost of Bayesian theory introduced by decision-theoretic rough set is used as the decision cost of accepting, delaying and rejecting decisions. Therefore, we establish a multi-scale dominant single-valued neutrosophic decision-theoretic rough set based on test and decision cost. In addition, we also propose a scale updating algorithm to find out all consistent scales. Afterwards, we further raise an optimal scale selection algorithm based on the minimum total cost criterion. Finally, the algorithm is verified and analyzed by UCI data sets. The algorithm and model proposed in this paper further expand the application of single-valued neutrosophic rough set in multi-scale system, and provide a reference for subsequent research in this field. [ABSTRACT FROM AUTHOR]

We discuss the three-way rough set based approach for approximation of decision granules in Intelligent Systems (IS's). The novelty of the approach is in presenting a new concept of approximation space which is based on advanced reasoning tools. Many generalisations of the rough set approaches developed over the years are mainly concentrated around reasoning concerning (partial) inclusion of sets. However, such approximation spaces are not satisfactory to deal with important aspects of approximate reasoning by IS's aiming to construct of the high quality approximations of compound decision granules. We demonstrate a number of examples supporting this claim. In particular, in solving the considered in the paper problems are involved complex algorithmic optimization processes directed by reasoning tools supporting searching for (semi-)optimal approximations of decision granules in huge spaces. This paper is a step toward developing tools for derivation of granules supporting IS's in perceiving situations to a degree satisfactory for making the right decisions. [ABSTRACT FROM AUTHOR]

This paper addresses the preservation of numerous essential properties of a residuated lattice structure in extended algebras for partial fuzzy set theory and partial fuzzy logics. The preservation includes the residuated lattice axioms, the identities narrowing the classes of the residuated lattices, and some well-known additional properties. In this paper, we consider nine algebras for partial fuzzy logics which incorporate handling undefined values in a bit different way. In particular, we consider the Bochvar, the Bochvar external, the Sobociński, the Kleene, the McCarthy, the Nelson, and the Łukasiewicz algebras, and two recently developed ones, namely the Lower estimation and the Dragonfly algebras. We summarize the obtained results in a comprehensible form which allows readers to easily check the information for the preserved and non-preserved properties in a certain partial algebraic structure. The resulting shape of the contribution is a sort of "atlas book" that aims at providing researchers with a comfortable and comprehensible form of an overview of the (non)preservation of fundamental properties of residuated structures. [ABSTRACT FROM AUTHOR]

This paper investigates rules of proof for maximal entropy inference. It provides the first study of rules of proof for maximal entropy inference on infinite predicate languages. The main result of this paper is the construction of a set of rules, in which all rules are sound for maximal entropy inference on finite domains but are not sound for Williamson's Maximal Entropy Approach on infinite predicate languages. This elucidates differences between explications of the Maximum Entropy Principle on finite domains and on infinite predicate languages. [ABSTRACT FROM AUTHOR]

As a kind of emerging binary continuous aggregation operator that has been successfully applied in many practical application problems, overlap functions on the unit closed interval have been considered by scholars on different truth values sets lately. At the same time, studying aggregation operators on finite chains, especially for commonly used binary aggregation operators, is a meaningful and hot topic in the research field of aggregation operators. In this paper, we pay attention to overlap functions on finite chains, which are called discrete overlap functions. Specifically, first, we introduce the notions of discrete overlap functions on the finite chain L with n + 2 elements and its arbitrary subchains along with an extended form of them. Second, we study some basic properties of discrete overlap functions on L , especially for the idempotent property, Archimedean property and cancellation law. In particular, we obtain some new properties which are different from those of the overlap functions on other truth values sets, for instance, every discrete overlap function on L takes the greatest element on L as the neutral element. Third, we discuss the construction methods of discrete overlap functions on L. Finally, it is worth mentioning that the results obtained in this paper provide a theoretical basis and more possibilities for the potential applications of overlap functions in other fields besides their known applications, especially for the situation of that the reasoning of experts are described by linguistic terms or labels, such as in expert systems, fuzzy control and etc. [ABSTRACT FROM AUTHOR]

In this paper, we consider analogical proportions which are statements of the form " a → is to b → as c → is to d → ", understood as a comparative formulation between vectors of items described on the same set of attributes. Analogical proportions and analogical inference have been extensively studied in the last decade, in particular by the authors of this paper. Some important remarks have been made regarding these proportions on i) the role of ordered pairs in them; ii) the large number of them associated with taxonomic trees; and more recently iii) their close relationship with multi-valued dependencies. We offer a renewed presentation of these facts together with some new insights on analogical proportions, emphasizing the role of equivalence classes of ordered pairs. Moreover, not all consequences had been drawn for a better understanding of analogical inference. This is the main purpose of this paper. In particular, it is advocated that analogical proportions whose four members are equal on some attributes are better predictors in general for classification purposes than analogical proportions for which there does not exist such attribute. This is confirmed by experimental results also reported in the paper. Thus this paper can be read both as an introductory survey on recent advances on analogical proportions and as a study on the impact of particular patterns on analogical inference, a topic never considered before. [ABSTRACT FROM AUTHOR]

Explainable Artificial Intelligence (XAI) is widely considered to be critical for building trust into the deployment of systems that integrate the use of machine learning (ML) models. For more than two decades Shapley values have been used as the theoretical underpinning for some methods of XAI, being commonly referred to as SHAP scores. Some of these methods of XAI now rank among the most widely used, including in high-risk domains. This paper proves that the existing definitions of SHAP scores will necessarily yield misleading information about the relative importance of features for predictions. The paper identifies a number of ways in which misleading information can be conveyed to human decision makers, and proves that there exist classifiers which will yield such misleading information. Furthermore, the paper offers empirical evidence that such theoretical limitations of SHAP scores are routinely observed in ML classifiers. [ABSTRACT FROM AUTHOR]

Let A be a pairwise comparison matrix obtained from a consistent one by perturbing three entries above the main diagonal, x , y , z , and the corresponding reciprocal entries, in a way that there is a submatrix of size 2 containing the three perturbed entries and not containing a diagonal entry. In this paper we describe the relations among x , y , z with which A always has its principal right eigenvector efficient. Previously, and only for a few cases of this problem, R. Fernandes and S. Furtado (2022) proved the efficiency of the principal right eigenvector of A. In this paper, we continue to use the strong connectivity of a certain digraph associated with A and its principal right eigenvector to characterize the vector efficiency. For completeness, we show that the existence of a sink in this digraph is equivalent to the inefficiency of the principal right eigenvector of A. [ABSTRACT FROM AUTHOR]

Multidimensional fuzzy sets (MFS) is a new extension of fuzzy sets on which the membership values of an element in the discourse universe are increasingly ordered vectors on the set of real numbers in the interval [ 0 , 1 ]. This paper aims to investigate fuzzy negations and, mainly, fuzzy implications on the set of increasingly ordered vectors on [ 0 , 1 ] , i.e. on L ∞ ([ 0 , 1 ]) , MFN and MFI in short, with respect to some partial order. In this paper we study partial orders, giving special attention to admissible orders on L ∞ ([ 0 , 1 ]). In addition, some properties and methods to construct such operators from fuzzy negations and fuzzy implications, respectively, are provided and we demonstrate that the action of the group of automorphisms on fuzzy implications on L ∞ ([ 0 , 1 ]) preserves several original properties of the implication. Finally, through a specific type of representable MFI, we are able to generate a class of MFN called natural m-negations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend closure systems from being crisp sets with certain fuzzy properties to proper fuzzy sets. The presentation of the paper shows a thorough discussion on the different alternatives that could be taken to define the desired fuzzy closure systems. These plausible alternatives are discarded if they are proven impossible to be in a bijective correspondence with closure operators. Finally, a definition of fuzzy closure system is established and a one-to-one relation with closure operators is proved. [ABSTRACT FROM AUTHOR]

Elementary formulas for propagating information about means and variances through mathematical expressions have long been used by analysts. Yet the precise implications of such information are rarely articulated. This paper explores distribution-free techniques for risk analysis that do not require simulation, sampling or approximation of any kind. We describe best-possible bounds on risks that can be inferred given only information about the range, mean and variance of a random variable. These bounds generalise the classical Chebyshev inequality in an obvious way. We also collect in convenient tables several formulas for propagating range and moment information through calculations involving 7 binary convolutions (addition, subtraction, multiplication, division, powers, minimum, and maximum) and 9 unary transformations (scalar multiplication, scalar translation, exponentiation, natural and common logarithms, reciprocal, square, square root and absolute value) commonly encountered in risk expressions. These formulas are rigorous rather than approximate, and in most cases are either exact or mathematically best-possible. The formulas can be used effectively even when only interval estimates of the moments are available. Although most discussions of moment propagation assume stochastic independence among variables, this paper shows the assumption to be unnecessary and generalises formulas for the case when no assumptions are made about dependence, and when correlations are partially known. Along with partial means and variances, we show how interval covariances may be propagated and tracked through expressions. [ABSTRACT FROM AUTHOR]

Three-way decision (3WD) is a widely used and studied mathematical theory that generalizes the thinking norm of tri-level in cognitive learning, problem solving, and information processing. By utilizing the negative information contained in data, three-way concept lattice (3WCL) developed 3WD in Formal Concept Analysis and has been applied in various applications such as conflict analysis, role based access control, knowledge discovery, concept learning, and medical diagnose. However, the connections between 3WCL and classical concept lattices have not received its deserved attention. To this end, first, this paper proved that 3WCL is exactly the minimal closure system containing both concept lattice and complementary concept lattice, and classified three-way concepts into four categories. Second, this paper proved the structure theorem of 3WCL that characterizes mathematically the relationships between concept lattice, complementary concept lattice and 3WCL as two isomorphisms. Third, this paper presented several applications of the structure theorem to reveal its essentiality in discussing the properties of 3WCL. Finally, some problems that are not involved in the structure theorem are also discussed. [ABSTRACT FROM AUTHOR]