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Pawlak conflict analysis focuses on three-valued ratings of a set of agents on a set of issues, in which the three values +, 0, and − indicate, respectively, that an agent is positive, neutral, and negative about an issue. According to their shared ratings, we can form different types of agent coalitions and, similarly, different types of issue bundles (i.e., families of issues to be considered together). The main objective of this paper is to introduce and investigate connections of agent coalitions and issue bundles from Wille formal concept analysis perspectives. By interpreting a three-valued rating table as four different formal contexts, we introduce four types of agent coalitions, namely, support, non-opposition, opposition, and non-support coalitions, the corresponding four types of issue bundles, and four lattices of coalition-bundle couplings. The lattices reveal structural information of agents and issues in a conflict situation. To demonstrate the usefulness of the proposed model, we analyze the problem of development planning of the Gansu Province of China. [ABSTRACT FROM AUTHOR]

Pedrycz shadowed sets are three-way approximations of fuzzy sets by transforming the infinite levels of fuzzy set membership grades in the unit interval [ 0 , 1 ] into three levels. The three levels represent qualitatively the sets of the white, grey, and black members of a shadowed set. In this paper, we generalize the notion of shadowed sets to the case of L-fuzzy sets by making three new contributions. First, we consider two representations of a shadowed set. One is a three-valued L-fuzzy set and the other is three pairwise disjoint sets. Second, we introduce two methods for constructing a shadowed set. One divides a finite lattice based on the notion of a pair of a set of designated core membership grades and a set of designated null membership grades. The other uses a pair of threshold sets, which generalizes the method that uses a pair of thresholds. We study formal properties of the two methods and show that they are equivalent. Finally, based on a distance function on a lattice, we present a simple method to build the sets of designated core and null membership grades. [ABSTRACT FROM AUTHOR]

By applying the principles of three-way decision as thinking in threes, in this paper I introduce a conceptual model of data science in three steps. First, I examine examples of triadic thinking in general and trilevel thinking in specific in data science. Then, based on Weaver's trilevel categorization of communications problems, I propose the concept of the symbols-meaning-value (SMV) space and discuss three perspectives on the SMV space from the viewpoints of information science and management science, cognitive science, and computer science. I label the operations on the SMV three levels metaphorically as seeing, knowing, and doing. Finally, I put forward a SMV-space-based conceptual model of data science, in which data are a resource, the power of data is the knowledge embedded in data, and the value of data is the wise decision and the best course of action supported by data. The goals and functions of data science at the SMV three levels are, respectively, making data available, making data meaningful, and making data valuable. To demonstrate the potential contributions of the conceptual model, I comment on some of its practical values and implications. [ABSTRACT FROM AUTHOR]

A Pawlak approximation space is a pair of a ground set/space and a quotient set/space of the ground set induced by an equivalence relation on the ground set. The quotient space is a simple granulation of the ground space such that an equivalence class is a granule of objects in the ground space and, at the same time, a single granular object in the quotient space. The new two-space view leads to more insights into and a deeper understanding of rough set theory. In this paper, we revisit results from rough sets from the two-space perspective and introduce the notions of granular rough sets and probabilistic granular rough sets in the quotient space, as three-way approximations of sets in the ground space. We propose a concept of granular shadowed sets in the quotient space, as three-way approximations of fuzzy sets in the ground space. We formulate a cost-sensitive method to construct a granular shadowed set from a fuzzy set. We show that, when the costs satisfy some conditions, the three granular approximations become the same for the special case where a fuzzy set is in fact a set. [ABSTRACT FROM AUTHOR]

By combining ideas from three-way decision and Pedrycz shadowed sets, this paper investigates two possible solutions to the problem of constructing a shadowed set from an Atanassov intuitionistic fuzzy set. The first solution treats the pair of membership and nonmembership functions as two evaluations and directly applies the two-evaluation based model of three-way decision. The second solution transforms the pair of membership and nonmembership functions into an evaluation and applies a single-evaluation based model of three-way decision. To address a fundamental issue of the determination of a pair of thresholds on evaluations, we present a general optimization based formulation and use Chebyshev distance to calculate the required thresholds. The results are useful for three-way decision with fuzzy sets, Atanassov intuitionistic fuzzy sets, and shadowed sets. [ABSTRACT FROM AUTHOR]

In the Pawlak model of conflict analysis, a rating of −1, 0, and +1 indicates that an agent is negative, neutral, and positive towards an issue. One defines measures of alliance and conflict of agents based on their three-valued ratings on a set of issues. Existing studies make some restrictive assumptions. One assumption is that a single distance function determines alliance, conflict, and neutrality. Another assumption is that all issues are equally important. The third assumption is that different pairs, such as (+ 1 , + 1) , (0 , 0) , and (− 1 , − 1) , contribute equally to alliance and conflict. Intuitively, different issues may have different importances and a (+ 1 , + 1) agreement may be stronger than a (0 , 0) agreement. To address these weaknesses, in this paper we use a pair of an alliance measure and a conflict measure, rather than a single distance measure. We propose and study a new class of additive measures of alliance and conflict by assigning different weights to different issues and assigning different values to different pairs of ratings. Existing measures are special cases of the new measures. By using the new measures, we construct a general model of three-way conflict analysis. Finally, we give an application of the new alliance and conflict measures in making the development plan of Gansu Province in China. [ABSTRACT FROM AUTHOR]

Three-way granular computing is a paradigm of thinking and information processing in three granules. We introduce a framework of three-way granular computing based on a hexagon induced by a trisection of a set. This enables us to unify rough-set concept analysis and formal concept analysis. For an object granule (i.e., a subset of a universal set of objects), we construct three attribute granules. Conversely, for an attribute granule (i.e., a subset of a universal set of attributes), we construct three object granules. Rough-set concept analysis uses one kind of trisections to study four types of disjunctive formal concepts and the associated concept lattices. Formal concept analysis uses another kind of trisections to study four types of conjunctive formal concepts and the associated concept lattices. The eight concept lattices can be grouped into two classes such that four lattices in each class are isomorphic. [ABSTRACT FROM AUTHOR]

Three-way decision is about thinking, problem-solving, and computing in threes or through triads. Its philosophy and practice of triadic thinking appear everywhere. In particular, there are a number of three-world or triworld models in different fields and disciplines, where a complex system, a complicated issue, or an intricate concept is explained and understood in terms of three interrelated worlds, with each world enclosing a group of elements or representing a particular view. This paper examines various triadic metaphors and structures in general and three-world metaphors and conceptions in particular for building and explaining basic concepts of three-way decision. The three major objectives are (a) to formulate a TAO (Triading-Acting-Optimizing) framework of three-way decision, (b) to discuss the Dao, the Way of three-way decision, and (c) to articulate the Dao, the Way of three-world thinking. Three-world conceptions offer more insights into three-way decision with new viewpoints, methods, and modes. An emphasis of the paper is on the philosophy and the power of three-way decision for building high-level conceptual models. By using three-part structures, by focusing on only three things, or by considering three basic ingredients, we may build a theory, a model, or a method that is simple-to-understand, easy-to-remember, and practical-to-use. [ABSTRACT FROM AUTHOR]

A Pawlak approximation space is a pair of a ground set/space and a quotient set/space, where the latter is induced by an equivalence relation on the former. With this two-space understanding, it is possible to lift any concepts and notions from the ground space to the quotient space. The results are granular versions that approximate the original concepts and notions. In this paper, we investigate the problem of lifting a fuzzy set in the ground space to granular fuzzy sets in the quotient space. By applying the principles of three-way decision, we introduce the idea of three-way granular approximations of fuzzy sets in terms of three granular fuzzy sets that represent the two extremes and one middle. The two extremes are given by granular rough fuzzy sets. We present several different ways to interpret and construct a middle. [ABSTRACT FROM AUTHOR]

Pedrycz introduces the notion of a shadowed set as a three-way approximation of a fuzzy set. Given a pair of thresholds (α, β), a shadowed set approximates a fuzzy set by three regions: an α -core of the fuzzy set, the difference between a β -support and the α -core, and the complement of the β -support. A fuzzy partition is a family of nonempty fuzzy sets satisfying certain conditions. When applying ideas of shadowed sets to a family of nonempty fuzzy sets, the result is a three-way approximation of the family, that is, a family of shadowed sets. This enables us to study a new type of three-way fuzzy partitions. We introduce and examine three pairs of properties for defining a three-way fuzzy partition: disjointness property, overlap property, and coverage property of α -cores and β -supports. We propose a general definition of a three-way fuzzy partition by using two sets of properties. By considering all possible non-equivalent subsets of properties, we obtain 21 classes of three-way fuzzy partition. We study three classes of three-way fuzzy partition and establish their connections to partitions, fuzzy δ -ɛ-partitions, and interval-set clusters. [ABSTRACT FROM AUTHOR]

Abstract Based on results from cognitive science, this paper examines the two fields of three-way decision and granular computing, as well as their interplay. The ideas from one field shed new light on the other field. The integration of the two gives rise to three-way granular computing, that is, thinking, problem solving, and information processing in threes. We discuss a wide sense of three-way decision and propose a trisecting–acting–outcome (TAO) model. We explain fundamental notions of granular computing based on the philosophy of three-way decision as thinking in threes. We discuss a model of three-way granular computing by making use of two particular types of granular structures represented, respectively, by three granules and three levels. We use examples across different disciplines to demonstrate the values of the two types. Our investigation suggests that, in many situations, the power of granular computing is indeed the power of three-way decision, i.e., thinking in threes. Highlights • Discuss a wide sense of three-way decision as thinking in threes. • Propose a TAO (trisecting–acting–outcome) model of three-way decision. • Explore the interplay of three-way decision and granular computing in the context of cognitive science. • Explain granular computing according to a philosophy of thinking in threes. • Illustrate the value of three-way decision by many examples. [ABSTRACT FROM AUTHOR]

In rough set theory, a decision class (i.e., a subset of objects) is approximated by three pair-wise disjoint positive, boundary, and negative regions. The concept of three-way decisions is introduced to provide a new interpretation of the three regions. We construct acceptance, non-commitment, and rejection rules, respectively, from the positive, boundary, and negative regions. The notion of class-specific attribute reducts concerns a minimal set of attributes used in constructing such rules. Existing studies on class-specific attribute reducts only consider the positive region and hence only the acceptance rules. In many situations such as medical diagnosis, we are also interested in negative rules or rule-out rules. This motivates the present study on three-way decision perspectives on class-specific attribute reducts. In addition to positive-region based attribute reducts, we study negative-region and positive-and-negative-region based attribute reducts. We investigate relationships among the three types of reducts. Although the three types of reducts are equivalent in consistent decision tables, they are not equivalent in inconsistent decision tables. By extending the framework, we study the three types of class-specific attribute reducts in probabilistic rough set models and their relationships. Finally, we give a general definition of class-specific attribute reducts. [ABSTRACT FROM AUTHOR]

The concept of attribute reducts plays a fundamental role in rough set analysis. There are at least two possibilities to define an attribute reduct. A classification-based or global attribute reduct is a minimal subset of condition attributes that preserves the positive region of the decision classification, namely, the positive regions of all decision classes, in a decision table. A class-specific, class-dependent, or local attribute reduct is a minimal subset of condition attributes that preserves the positive region of a particular decision class. While a classification-based reduct may not work equally well for every decision class, a class-specific attribute reduct is optimally tailored to a particular decision class. However, studies in rough set theory are dominated by classification-based reducts; there is very limited investigation on class-specific reducts. An objective of this paper is to draw attention to class-specific reducts. We systematically compare the two types of reducts and investigate their relationships with respect to both individual reducts and families of all reducts. It is possible to derive a class-specific reduct from a classification-based reduct and to derive a classification-based reduct from a family of class-specific reducts. The families of all class-specific reducts provide a pair of lower and upper bounds of the family of all classification-based reducts. Based on a three-way classification of attributes into the pair-wise disjoint sets of core, marginal, and nonuseful attributes, we examine relationships between the corresponding classes of classification-based and class-specific attributes. The union of the sets of class-specific core attributes is the set of classification-based core attributes. It is only possible to obtain an upper bound for the set of classification-based marginal attributes and a lower bound for the set of classification-based nonuseful attributes from the family of the class-specific corresponding sets of attributes. [ABSTRACT FROM AUTHOR]