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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]

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]