Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis

Knowledge-Based Systems xxx (xxxx) xxx

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Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis✩ Guangming Lang a,b , Junfang Luo b,c , Yiyu Yao b ,

∗

a

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan, 410114, PR China Department of Computer Science, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada c College of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 610031, PR China b

article

info

Article history: Received 8 August 2019 Received in revised form 18 January 2020 Accepted 22 January 2020 Available online xxxx Keywords: Conflict analysis Formal concept analysis Rough sets Three-valued situation table Three-way decision

a b s t r a c t The Pawlak model of conflict analysis uses three-valued ratings (i.e., positive, neutral, and negative) of a set of agents on a set of issues. Several extensions to the Pawlak model, namely, rough sets based qualitative and quantitative models, formal concept analysis based quantitative models, and three-way conflict analysis models, have been proposed in recent years. The main objective of this paper is to propose a more general model that unifies these existing models in an evaluation-based framework of three-way decision. The proposed model uses a pair of evaluations, one for support and the other for opposition, for trisecting the set of agents. By considering qualitative and quantitative evaluations, we derive a qualitative model and a quantitative model of three-way conflict analysis, respectively. The corresponding two models built based on rough sets and the corresponding two models built based on formal concept analysis are special cases. A unification of existing models provides insights into a common structure in formulating three-way conflict analysis with different choices of evaluations. We illustrate an application of the three-way conflict analysis model in making development plans for Gansu Province in China. © 2020 Elsevier B.V. All rights reserved.

1. Introduction In the early 1980s, Pawlak [1] introduced a simple model of conflict analysis by using a three-valued function φ that maps a pair of agents to values in {−1, 0, +1}. Two agents x and y are allied if φ (x, y) = +1; they are in conflict if φ (x, y) = −1; they are neutral if φ (x, y) = 0. According to the three values, he introduced alliance, conflict, and neutrality relations to describe three possible types of relationship between agents. The model is related to signed graphs [2] and can be studied with the help of graph theory. In his subsequent writings, Pawlak [3,4] suggested methods for constructing and interpreting function φ by using a three-valued situation table. The rows and the columns of the table represent, respectively, a set of agents and a set of issues, and each cell is a value from {−1, 0, +1} representing the rating of an agent on an issue. Based on the function φ of individual issues, for a subset of issues Pawlak built a conflict function ρ that ✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.knosys. 2020.105556. ∗ Corresponding author. E-mail addresses: [email protected] (G. Lang), [email protected] (J. Luo), [email protected] (Y. Yao).

maps a pair of agents to the unit interval [0, 1]. According to the values of the conflict function, the alliance, conflict, and neutrality relations are defined, respectively, as follows: two agents x and y are allied if ρ (x, y) < 0.5; they are in conflict if ρ (x, y) > 0.5; they are neutral if ρ (x, y) = 0.5. The Pawlak model of conflict analysis has inspired many studies [5–8]. Yao [9–13] proposed a theory of three-way decision for thinking, problem solving, and information processing in threes. In a set-theoretic setting, a model of three-way decision involves the division of a universal set into three pair-wise disjoint subsets whose union is the universal set [14–34]. The triplet of three subsets is called a trisection of the universal set. In the Pawlak model, it can be observed that the alliance, conflict, and neutrality relations are pair-wise disjoint and their union is the Cartesian product of the set of agents. That is, the three relations form a trisection of the set of all pairs of agents. The notion of threevalued situation table provides another way of thinking in threes in the sense that three-valued ratings lead naturally to a trisection of the set of agents and a trisection of the set of issues [35]. These observations suggest that there are close connections between Pawlak conflict analysis and the theory of three-way decision. In the light of three-way decision as thinking in threes, there are a total of four types of trisections: (i) trisections of the set of agents, (ii) trisections of the set of all pairs of agents, (iii) trisections of

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Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.

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the set of issues, and (iv) trisections of the set of all pairs of issues. The four types have been studied by many authors [35–42]. For trisecting the set of agents, several proposals have been made. Sun, Ma, and Zhao [41] provided a method for trisecting the set of agents based on rough set approximation operators over two universes, which gives a novel model of three-way conflict analysis for three-valued situation tables. But the conditions of rough sets-based conflict analysis models are so strict that limits its application in conflict analysis. Sun et al. [39] generalized the rough sets-based conflict analysis method by using probabilistic rough set approximations, which can trisect the set of agents with more tolerances. Fan, Qi, and Wei [36] proposed a different construction method by using a quantitative version of the derivation operators in formal concept analysis. In comparison with Sun, Ma, and Zhao’s models, they gave a new model of three-way conflict analysis from a different perspective. But they did not explicitly consider the qualitative derivation operators, as a special of quantitative derivation operators, for constructing the three-way conflict analysis model. So it is not easy to find the relationship between rough sets-based conflict analysis models and formal concept analysis-based conflict analysis models. Lang, Miao, and Fujita [38] employed the theory of group decision making to trisect the universe of agents by considering a situation table in which agents ratings on issues are expressed by Pythagorean fuzzy numbers, and provided a model of three-way conflict analysis based on ordering relations on Pythagorean fuzzy sets. The same method for trisecting the universe of agents can be applied to trisect the set of issues. For trisecting the set of all pairs of agents, Lang, Miao, and Cai [37] extended Pawlak’s three-way classification by using a pair of thresholds (α, β ) on the conflict function ρ to replace a single threshold 0.5. That is, two agents x and y are allied if ρ (x, y) ≤ β ; they are in conflict if ρ (x, y) ≥ α ; they are neutral if β < ρ (x, y) < α . With respective to a given agent, they trisected the set of agents into its allied, conflict, and neutral subsets of agents. They also considered conflict analysis in a dynamically changing situation table. These studies investigated separately trisecting agents and trisecting pairs of agents. They did not link together the two types of trisections in a common model. Yao [35] introduced a more general model of three-way conflict analysis by considering trisections of the set of agents and trisections of the set of all pairs of agents. A trisection of the set of issues and a trisection of the set of all pairs of issues can be easily obtained, respectively, by the same methods. Several important findings are reviewed here. First, for conflict analysis on a single issue, trisecting the set of agents and trisecting the set of all pairs of agents produce equivalent results. On the other hand, for conflict analysis on multiple issues, the corresponding results are different. Although a conflict function is useful for constructing a trisection of the set of all pairs of agents, it does not give a trisection of the set of agents. It is possible to build two different models of conflict analysis. Second, in the spirit of three-way decisions, there can be three levels of conflict, consisting of strong conflict, weak conflict, and non-conflict. The three levels of conflict can be interpreted in two ways. One is based on a trisection of the set of agents, and the other is based on the trisection of the set of all pairs of agents. Third, three-valued ratings is only a special case of many-valued ratings. The latter is more suitable for studying conflict analysis on multiple issues. By focusing on trisections of agents, the main objective of this paper is to propose a general model to unify models of conflict analysis based on rough set theory and formal concept analysis. Instead of using an evaluation-based three-way decision model with one evaluation, we use a model with two evaluations. This enables us to unify approaches based on, respectively, rough

set theory and formal concept analysis, in a common setting. To achieve this goal, the rest of the paper is organized as follows. Section 2 recalls some concepts of evaluation-based models of three-way decision. Section 3 provides a unified model of three-way conflict analysis for trisecting the universe of agents. Section 4 studies rough set theory-based three-way conflict analysis models. Section 5 investigates formal concept analysis-based three-way conflict analysis models. Section 6 gives an application of three-way conflict analysis models. 2. Evaluation-based models of three-way decision The philosophy of three-way decision is thinking in threes. The trisecting-acting-outcome (TAO) model of three-way decision consists of (1) dividing a whole into three separated and related parts, (2) devising strategies to act on the three parts, and (3) optimizing a desirable outcome [12]. For conflict analysis, it is sufficient to use evaluation-based models of three-way decision proposed by Yao [10] in a set-theoretic setting. That is, we trisect the universe of objects into the positive region, negative region, and boundary region by using evaluations on the set of objects. Instead of using a general model with poset-valued evaluations, we consider specials in which evaluation functions take values from the interval [−1, 1]. Intuitively, in the context of conflict analysis, +1 stands for full support, 0 stands for neutral, and −1 stands for full opposition. Definition 1. Let U be a finite nonempty set of objects and ν : U −→ [−1, +1] be an evaluation from U to [−1, +1]. Given a pair of thresholds (α, β ) with −1 ≤ β < 0 < α ≤ +1, the positive region POS(α,β ) (ν ), negative region NEG(α,β ) (ν ), and boundary region BND(α,β ) (ν ) induced by the evaluation ν are defined by: POS(α,β ) (ν ) = {x ∈ U | (x) ≥ α}, NEG(α,β ) (ν ) = {x ∈ U | (x) ≤ β}, BND(α,β ) (ν ) = {x ∈ U | β < (x) < α}. In Definition 1, the union of the three regions is the entire universe of objects, and the condition α > β ensures that the three regions are pair-wise disjoint. Since one or two of them may be the empty set, they do not necessarily form a partition of U. The triplet ⟨⟨POS(α,β ) (ν ), NEG(α,β ) (ν ), BND(α,β ) (ν )⟩⟩ is called a trisection of U [35]. A limitation of three-way decision with one evaluation function is the assumption of the existence of one function to represent both positive and negative opinions, which corresponds to univariate bipolarity [43]. In many situations, there is a need to consider bivariate unipolarity, that is, using two different independent evaluations, one for positive and the other for negative. Definition 2. Let U be a finite nonempty set of objects. A pair of functions νp : U −→ [0, +1] and νn : U −→ [−1, 0] are called a positive evaluation and a negative evaluation, respectively. Furthermore, νp (x) and νn (x) are referred to as the positive value and the negative value of x ∈ U, respectively. In formulating a pair of evaluations, we consider both qualitative and quantitative information. The signs of νp (x) and νn (x) reflect positive and negative opinions, respectively. The absolute values |νp (x)| = νp (x) and |νn (x)| = −νn (x) give the strength levels of opinions. Generally speaking, the positive evaluation and the negative evaluation are independent. For two objects x, y ∈ U, if νp (x) < νp (y), then x is less positive than y; if |νn (x)| < |νn (y)|, then x is less negative than y. With respect to a pair of evaluations, we can divide the set of objects into three regions [10].

Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.

G. Lang, J. Luo and Y. Yao / Knowledge-Based Systems xxx (xxxx) xxx Table 1 Combination of two two-way decision models. (P, P)

Table 2 A three-valued situation table.

(N, N)

A

x ∈ NEG(νn )

x ∈ NEG (νn )

x ∈ POS(νp )

x ∈ POS(νp ) ∩ NEG(νn ) (Over-commitment)

x ∈ POS(νp ) ∩ NEG (νn ) (Positive)

x ∈ POS (νp )

x ∈ POSc (νp ) ∩ NEG(νn ) (Negative)

x ∈ POSc (νp ) ∩ NEGc (νn ) (Under-commitment)

c

3

c

c

Definition 3. Let U be a finite nonempty set of objects. Given a pair of evaluations, namely, a positive evaluation νp : U −→ [0, +1] and a negative evaluation νn : U −→ [−1, 0], and a pair of thresholds (α, β ) with 0 < α ≤ +1 and −1 ≤ β < 0, we divide U into the positive region POS(α,β ) (νp , νn ), negative region NEG(α,β ) (νp , νn ), and boundary region BND(α,β ) (νp , νn ): POS(α,β ) (νp , νn ) = {x ∈ U | νp (x) ≥ α ∧ νn (x) > β}, NEG(α,β ) (νp , νn ) = {x ∈ U | νp (x) < α ∧ νn (x) ≤ β}, BND(α,β ) (νp , νn ) = (POS(α,β ) (νp , νn ) ∪ NEG(α,β ) (νp , νn ))c

= {x ∈ U | [νp (x) ≥ α ∧ νn (x) ≤ β] ∨ [νp (x) < α ∧ νn (x) > β]}. By Definition 3, the three regions are pair-wise disjoint and their union is the entire universe of objects U. The triplet ⟨⟨POS(α,β ) (νp , νn ), NEG(α,β ) (νp , νn ), BND(α,β ) (νp , νn )⟩⟩ is a trisection of U. To gain more insights into a trisection defined by a pair of evaluations, we may consider the combination of two models of two-way decision [16], that is, a positive model (P, P) and a negative model (N, N). In the positive model, given a threshold 0 < α ≤ +1, according to νp , we divide U into a pair of disjoint positive region POS(νp ) and non-positive region POSc (νp ): POS(νp ) = {x ∈ U | νp (x) ≥ α}, POSc (νp ) = {x ∈ U | νp (x) < α}. Similarly, in the negative model, given a threshold −1 ≤ β < 0, according to νn , we divide U into a pair of disjoint negative region NEG(νn ) and non-negative region NEGc (νn ): NEG(νn ) = {x ∈ U | νn (x) ≤ β}, NEG (νn ) = {x ∈ U | νn (x) > β}. c

The four regions of the two models can be combined according to Table 1. The trisection defined by two bisections is given as follows: POS(α,β ) (νp , νn ) = POS(νp ) ∩ NEGc (νn ), NEG(α,β ) (νp , νn ) = POSc (νp ) ∩ NEG(νn ), BND(α,β ) (νp , νn ) = (POS(νp ) ∩ NEG(νn )) ∪ (POSc (νp ) ∩ NEGc (νn )). We view both positive and negative decisions as commitments. It can be seen that the positive region and the negative region are defined by two agreed parts of the two models, that is, a positive decision agrees with a non-negative decision, and a negative decision agrees with a non-positive decision. The boundary region is the union of two parts, one accounting for over-commitment by the two models and the other accounting for under-commitment. 3. Trisection-based conflict analysis In this section, we propose a general model of three-way conflict analysis to unify several existing models [35,36,38,39,41] by using two evaluations.

I i1

i2

i3

i4

i5

a1

−1

+1

+1

+1

+1

a2

+1

0

−1

−1

−1

a3

+1

−1

−1

−1

0

a4

0

−1

+1

0

−1

a5

+1

−1

−1

−1

−1

a6

0

+1

−1

0

+1

3.1. Trisection of agents The Pawlak model of conflict analysis is based on a threevalued situation table. Definition 4. A three-valued situation table is a triplet S = (A, I , r), where A is a set of agents, I is a set of issues, the attitude of an agent x ∈ A to an issue i ∈ I is given by a function r : A × I −→ {−1, 0, +1}. If r(a, i) = +1, the agent a is positive about the issue i; if r(a, i) = −1, the agent a is negative about the issue i; if r(a, i) = 0, the agent a is neutral about the issue i. Example 1. We adopt an example from Pawlak’s seminal paper [3]. Table 2 is a three-valued situation table S = (A, I , r) about Middle East conflict, where A = {a1 , a2 , a3 , a4 , a5 , a6 }, I = {i1 , i2 , i3 , i4 , i5 }. The countries a1 , a2 , a3 , a4 , a5 , and a6 are Israel, Egypt, Palestine, Jordan, Syria, and Saudi Arabia, respectively; the issues i1 , i2 , i3 , i4 , and i5 are ‘‘Autonomous Palestinian state on the West Bank and Gaza’’, ‘‘Israeli military outpost along the Jordan River’’, ‘‘Israel retains East Jerusalem’’, ‘‘Israeli military outposts on the Golan Heights’’, ‘‘Arab countries grant citizenship to Palestinians who choose to remain within their borders’’, respectively. The rating r(a1 , i1 ) = −1 denotes that Israel opposes Autonomous Palestinian state on the West Bank and Gaza; the rating r(a2 , i1 ) = +1 denotes that Egypt supports Autonomous Palestinian state on the West Bank and Gaza; the rating r(a4 , i1 ) = 0 denotes that Jordan is neutral to Autonomous Palestinian state on the West Bank and Gaza. The three-valued ratings naturally trisect the universe of agents. We have three sets of positive agents, neutral agents, and negative agents towards an issue [35]. Definition 5. In a three-valued situation table S = (A, I , r), 0 for i ∈ I, the sets of positive agents A+ i , neutral agents Ai , and − negative agents Ai towards the issue i are given: A+ i = {a ∈ A | r(a, i) = +1}, A0i = {a ∈ A | r(a, i) = 0}, A− i = {a ∈ A | r(a, i) = −1}; According to Definition 5, A+ i consists of agents that support the issue i, A0i consists of agents that are neutral about the issue i, and A− i consists of agents that oppose the issue i. Therefore, we − 0 have a trisection ⟨⟨A+ i , Ai , Ai ⟩⟩ of the universe of agents. In order to trisect agents based on a subset of issues, we introduce the concepts of conditional support evaluation and conditional opposition evaluation. Definition 6. In a three-valued situation table S = (A, I , r), we introduce the notion of conditional evaluations. For J ⊆ I, a pair of functions es (·|J) : A −→ [0, +1] and eo (·|J) : A −→ [−1, 0] consists of a conditional support evaluation and a conditional opposition evaluation of agents given by the set of issues J.

Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.

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For simplicity, we use the same symbols es and eo to denote a conditional support evaluation and a conditional opposition evaluation of issues, and we may construct the conditional support evaluation by using the conditional opposition evaluation, and vice versa. In general, the conditional support evaluation and the conditional opposition evaluation are independent of each other. By applying the pair of evaluations in Definition 6 to Definition 3, we have a model of conflict analysis for trisecting the set of agents. Definition 7. In a three-valued situation table S = (A, I , r), for J ⊆ I, we have a pair of conditional evaluations (es (·|J), eo (·|J)). Given a pair of thresholds (αA , βA ) with 0 < αA ≤ +1 and −1 ≤ βA < 0, we define the sets of positive agents P(αA ,βA ) (J), negative agents N(αA ,βA ) (J), and neutral agents Z(αA ,βA ) (J) towards J:

P(αA ,βA ) (J) = {a ∈ A | es (a|J) ≥ αA ∧ eo (a|J) > βA }, N(αA ,βA ) (J) = {a ∈ A | es (a|J) < αA ∧ eo (a|J) ≤ βA }, Z(αA ,βA ) (J) = (P(αA ,βA ) (J) ∪ N(αA ,βA ) (J))c

= {a ∈ A | [es (a|J) ≥ αA ∧ eo (a|J) ≤ βA ] ∨ [es (a|J) < αA ∧ eo (a|J) > βA ]}. The subset of issues J ⊆ I is called an optimal feasible consensus strategy of conflict situation S = (A, I , r) if and only if |P(αA ,βA ) (J)|/|A| ≥ δA , where 1 ≥ δA ≥ 0. Furthermore, according to Definition 7, we have P(αA ,βA ) (J) ∪ N(αA ,βA ) (J) ∪ Z(αA ,βA ) (J) = A and P(αA ,βA ) (J) ∩ N(αA ,βA ) (J) = P(αA ,βA ) (J) ∩ Z(αA ,βA ) (J) = N(αA ,βA ) (J) ∩ Z(αA ,βA ) (J) = ∅. That is, the three sets are pair-wise disjoint, and their union is the entire set of agents. Therefore, we have a trisection ⟨⟨P(αA ,βA ) (J), N(αA ,βA ) (J), Z(αA ,βA ) (J)⟩⟩ of the universe of agents. We have the set of non-positive agents (P(αA ,βA ) (J))c , the set of non-negative agents (N(αA ,βA ) (J))c , and the set of non-neutral agents (Z(αA ,βA ) (J))c towards J: (P(αA ,βA ) (J))c = Z(αA ,βA ) (J) ∪ N(αA ,βA ) (J), (N(αA ,βA ) (J))c = Z(αA ,βA ) (J) ∪ P(αA ,βA ) (J), (Z(αA ,βA ) (J))c = N(αA ,βA ) (J) ∪ P(αA ,βA ) (J). Obviously, we have that (P(αA ,βA ) (J))c ∩ (N(αA ,βA ) (J))c = Z(αA ,βA ) (J), (P(αA ,βA ) (J))c ∩ (Z(αA ,βA ) (J))c = N(αA ,βA ) (J), (N(αA ,βA ) (J))c ∩ (Z(αA ,βA ) (J))c = P(αA ,βA ) (J), and (P(αA ,βA ) (J))c ∪ (N(αA ,βA ) (J))c ∪ (Z(αA ,βA ) (J))c = A. That is, the sets of non-positive agents, nonnegative agents, and non-neutral agents are not necessarily pairwise disjoint, and their union is the entire set of agents. Therefore, ⟨(P(αA ,βA ) (J))c , (N(αA ,βA ) (J))c , (Z(αA ,βA ) (J))c ⟩ is a weak tri-covering of the universe of agents [35]. Remark. In this work, the verb ‘‘trisect’’ means dividing the universe of agents into three parts, and the three parts are not necessary equal, and the union of the three parts is the entire set of agents, which is different from the problem of trisecting an angle into three equal parts. Furthermore, by simply switching the roles of agents and issues, the same method of trisecting the set of agents A towards a subset of issues can be easily adapted to trisect the set of issues I towards a subset of agents. 3.2. Three levels of conflict The trisection of agents enables us to construct coalitions of agents, and three levels of conflict [35]. Definition 8. In a three-valued situation table S = (A, I , r), for J ⊆ I,

(1) the three sets P(αA ,βA ) (J), N(αA ,βA ) (J), and Z(αA ,βA ) (J) are called strong coalitions of agents with respect to J; (2) the three sets (P(αA ,βA ) (J))c , (N(αA ,βA ) (J))c , and (Z(αA ,βA ) (J))c are called weak coalitions of agents with respect to J. The three coalitions P(αA ,βA ) (J), N(αA ,βA ) (J), and Z(αA ,βA ) (J) are the largest homogeneous groups with respect to the subset of issues J. That is, the conditional support evaluations and conditional opposition evaluations of agents in each group are bounded by the thresholds αA and βA . For a1 , a2 ∈ P(αA ,βA ) (J), the conditional support evaluations of a1 and a2 are not less than the threshold αA , and the conditional opposition evaluations of a1 and a2 are larger than the threshold βA ; for a1 , a2 ∈ N(αA ,βA ) (J), the conditional support evaluations of a1 and a2 are less than the threshold αA , and the conditional opposition evaluations of a1 and a2 are not larger than the threshold βA ; for a1 , a2 ∈ Z(αA ,βA ) (J), the conditional support evaluations of a1 and a2 are not less than the threshold αA , the conditional opposition evaluations of a1 and a2 are not larger than the threshold βA ; or the conditional support evaluations of a1 and a2 are less than the thresholds αA , the conditional opposition evaluations of a1 and a2 are larger than the threshold βA . Each group of ⟨(P(αA ,βA ) (J))c , (Z(αA ,βA ) (J))c , (N(αA ,βA ) (J))c ⟩, as the union of two homogeneous groups of ⟨⟨P(αA ,βA ) (J), Z(αA ,βA ) (J), N(αA ,βA ) (J)⟩⟩, is not a homogeneous group with respect to the subset of issues J. Furthermore, the agents in (P(αA ,βA ) (J))c , (Z(αA ,βA ) (J))c , and (N(αA ,βA ) (J))c are absolutely different from those in P(αA ,βA ) (J), Z(αA ,βA ) (J), and N(αA ,βA ) (J), respectively. For example, consider a1 , a2 ∈ (Z(αA ,βA ) (J))c = P(αA ,βA ) (J) ∪ N(αA ,βA ) (J). If a1 ∈ P(αA ,βA ) (J) and a2 ∈ N(αA ,βA ) (J), then es (a1 |J) ≥ αA , es (a2 |J) < αA , eo (a1 |J) > βA , and eo (a2 |J) ≤ βA . For a1 , a2 ∈ (Z(αA ,βA ) (J))c , and a3 ∈ Z(αA ,βA ) (J), the two agents a1 and a2 have a commitment (i.e., support or opposition) towards J, and the agent a3 has a noncommitment towards J. Therefore, according to the commitment and non-commitment, the agents a1 and a2 belong to a coalition relative to the agent a3 . We have three levels of conflict induced by trisecting the set of agents based on the conditional support and opposition evaluations. Definition 9. In a three-valued situation table S = (A, I , r), for J ⊆ I, the three levels of conflict among agents with respect to a set of issues J are defined as follows: (1) Strong Conflict : SC(P(αA ,βA ) (J), N(αA ,βA ) (J)), (2) Weak Conflict : WC(P(αA ,βA ) (J), Z(αA ,βA ) (J)), WC(N(αA ,βA ) (J), Z(αA ,βA ) (J)), (3) Non-Conflict : NC(P(αA ,βA ) (J), P(αA ,βA ) (J)), NC(N(αA ,βA ) (J), N(αA ,βA ) (J)), NC(Z(αA ,βA ) (J), Z(αA ,βA ) (J)). The disagreement between agents towards a subset of issues results in a conflict and the agreement leads to a non-conflict. The conditional support evaluations and the conditional opposition evaluations of agents in the coalition P(αA ,βA ) (J) are absolutely different from those of agents in the coalition N(αA ,βA ) (J); it results in a strong conflict. Only the conditional support evaluations or the conditional opposition evaluations of agents in the coalition P(αA ,βA ) (J) are in the same range as agents in the coalition Z(αA ,βA ) (J); it leads to a weak conflict. Only the conditional support evaluations or the conditional opposition evaluations of agents in the coalition N(αA ,βA ) (J) are in the same range as agents in the coalition Z(αA ,βA ) (J); it also leads to a weak conflict. The conditional support evaluations or the conditional opposition evaluations of agents in each coalition of ⟨⟨P(αA ,βA ) (J), N(αA ,βA ) (J), Z(αA ,βA ) (J)⟩⟩ are in the same range; it leads

Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.

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to a non-conflict. Therefore, we have three levels of conflict by using the conditional support evaluation and the conditional opposition evaluation of agents with respect to a subset of issues. Remark. By simply switching the roles of agents and issues, the same method of trisecting the set of all pairs of agents A × A towards a subset of issues can be easily adapted to trisect the set of all pairs of issues I × I towards a subset of agents. 4. Three-way conflict analysis models based on rough set theory In this section, we investigate qualitative and quantitative three-way conflict analysis models based on rough set approximations over two universes [44,45]. 4.1. A qualitative model Sun, Ma, and Zhao [41] proposed conflict analysis models based on rough set theory over two universes and provided a foundation for trisecting the set of agents by using approximation operators. We first consider a transformation of a three-valued situation table into a pair of two-valued situation tables. The latter is required for building rough set approximations over two universes, and can help to reflect the essence of the models of conflict analysis based on rough sets over two universes. Definition 10. Given a three-valued situation table S = (A, I , r), we construct a pair of two-valued situation tables or two approximations spaces (A, I , R+ ) and (A, I , R− ), where R+ and R− are defined by: R+ = {(a, i) ∈ A × I | r(a, i) = +1}, R− = {(a, i) ∈ A × I | r(a, i) = −1}. They represent, respectively, that an agent supports and opposes an issue. For a ∈ A, we have the set of issues supported by agent a and the set of issues opposed by agent a: Ia+ = {i ∈ I | (a, i) ∈ R+ } = {i ∈ I | r(a, i) = +1},

5

In Definition 11, we use additional conditions Ia− ̸ = ∅ and Ia ̸ = ∅ when defining the lower approximations of a subset of issues, which is slightly different from Sun, Ma, and Zhao’s definition [41]. For the purpose of conflict analysis, we interpret the lower and upper approximation of a subset of issues based on relation R+ . By Definition 11, an agent a ∈ apr + (J) supports a subset of issues of J and does not support any issues outside of J. In other words, an agent a ∈ apr + (J) only supports some + issues in J. An agent a ∈ apr (J) supports some issues of J and may support some issues outside of J. Similarly, we can interpret the lower and upper approximations based on relation R− . The pair of approximation spaces (A, I , R+ ) and (A, I , R− ) may be viewed as two two-way decisions. By combining the pair of lower approximations, Sun, Ma, and Zhao [41] proposed a method to trisect the set of agents into the sets of positive agents, negative agents, and neutral agents towards a subset of issues. −

Definition 12. In a three-valued situation table S = (A, I , r), for J ⊆ I, the sets of positive agents RP(J), negative agents RN(J), and neutral agents RZ(J) towards a set of issues J are defined by:

RP(J) = apr + (J) − apr − (J), RN(J) = apr − (J) − apr + (J), RZ(J) = (RP(J) ∪ RN(J))c . The sets of positive agents, negative agents, and neutral agents towards a subset of issues given by Definition 12 are pair-wise disjoint, and their union is the entire universe of agents. Therefore, we have a trisection ⟨⟨RP(J), RN(J), RZ(J)⟩⟩ of the universe of agents. We now show that Definition 12 is in fact a special case of Definition 7 introduced earlier. This is done by defining the conditional support evaluation and the conditional opposition evaluation. Definition 13. In a three-valued situation table S = (A, I , r), for J ⊆ I, we define a pair of functions, namely, a conditional support evaluation es (·|J) : A −→ {0, +1} and a conditional opposition evaluation eo (·|J) : A −→ {0, −1}, as follows:

{

+1, 0,

Ia+ ̸ = ∅ ∧ Ia+ ⊆ J , other w ise,

{

−1, 0,

Ia− ̸ = ∅ ∧ Ia− ⊆ J , other w ise.

es (a|J) =

Ia− = {i ∈ I | (a, i) ∈ R− } = {i ∈ I | r(a, i) = −1}.

eo (a|J) =

According to Definition 10, we have two relations R+ and R− over the universes of agents and issues in a three-valued situation table. For a ∈ A and i ∈ I, if the pair (a, i) ∈ R+ , it means that the agent a supports the issue i; if the pair (a, i) ∈ R− , then the agent a opposes the issue i. Furthermore, for a ∈ A, Ia+ is the set of issues supported by the agent a, and Ia− is the set of issues opposed by the agent a. That is, for i ∈ I, if the issue i ∈ Ia+ , it means that the issue i is supported by the agent a; if the issue i ∈ Ia− , it means that the issue i is opposed by the agent a. Based on the two approximation spaces, we can define lower and upper approximations of a subset of issues.

By inserting the conditional support and opposition evaluations of Definition 13 into Definition 7 and setting a pair of thresholds (αA , βA ) such that 0 < αA ≤ +1 and −1 ≤ βA < 0, we immediately obtain Sun, Ma, and Zhao’s definition, as shown in the next theorem.

Definition 11. In a three-valued situation table S = (A, I , r), the lower and upper approximations of a subset of issues J ⊆ I are defined by: apr + (J) = {a ∈ A | ∅ ̸ = Ia+ ⊆ J }, +

apr (J) = {a ∈ A | Ia+ ∩ J ̸ = ∅}; −

−

apr (J) = {a ∈ A | ∅ ̸ = Ia ⊆ J }, −

apr (J) = {a ∈ A | Ia− ∩ J ̸ = ∅}.

Theorem 1. In a three-valued situation table S = (A, I , r), consider a pair of conditional support and opposition evaluations of Definition 13 and a pair of thresholds (αA , βA ) with 0 < αA ≤ +1 and −1 ≤ βA < 0. We have:

RP(J) = P(αA ,βA ) (J), RN(J) = N(αA ,βA ) (J), RZ(J) = Z(αA ,βA ) (J). Proof. By Definitions 12 and 13, we have

RP(J) = apr + (J) − apr − (J)

= {a ∈ A | Ia+ ⊆ J } − {a ∈ A | Ia− ⊆ J } = {a ∈ A | es (a|J) = +1 ∧ eo (a|J) ̸= −1} = {a ∈ A | es (a|J) ≥ αA ∧ eo (a|J) > βA }

Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.

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= P(αA ,βA ) (J), RN(J) = apr − (J) − apr + (J) {a ∈ A | Ia− ⊆ J } − {a ∈ A | Ia+ ⊆ J } {a ∈ A | es (a|J) ̸= +1 ∧ eo (a|J) = −1} {a ∈ A | es (a|J) < αA ∧ eo (a|J) ≤ βA } N(αA ,βA ) (J), (RP(J) ∪ RN(J))c {a ∈ A | [Ia+ ⊆ J ∧ Ia− ⊆ J ] ∨ [Ia+ ⊈ J ∧ Ia− ⊈ J ]} {a ∈ A | [es (a|J) = +1 ∧ eo (a|J) = −1] ∨ [es (a|J) ̸= +1 ∧ eo (a|J) ̸= −1]} = {a ∈ A | [es (a|J) ≥ αA ∧ eo (a|J) ≤ βA ] ∨ [es (a|J) < αA ∧ eo (a|J) > βA ]} = Z(αA ,βA ) (J). □

= = = = RZ(J) = = =

By taking the conditional support and opposition evaluations of Definition 14 into Definition 7, we trisect the set of agents into the sets of positive agents, negative agents, and neutral agents towards a subset of issues. Definition 15. In a three-valued situation table S = (A, I , r). For J ⊆ I, we have a pair of evaluations (es (·|J), eo (·|J)). Given a pair of thresholds (αA , βA ) with 0 < αA ≤ +1 and −1 ≤ βA < 0, we define the sets of positive agents RP(αA ,βA ) (J), negative agents RN(αA ,βA ) (J), and neutral agents RZ(αA ,βA ) (J) towards J by:

RP(αA ,βA ) (J) = {a ∈ A | es (a|J) ≥ αA ∧ eo (a|J) > βA }, RN(αA ,βA ) (J) = {a ∈ A | es (a|J) < αA ∧ eo (a|J) ≤ βA }, RZ(αA ,βA ) (J) = (RP(αA ,βA ) (J) ∪ RN(αA ,βA ) (J))c

= {a ∈ A | [es (a|J) ≥ αA ∧ eo (a|J) ≤ βA ] ∨ [es (a|J) < αA ∧ eo (a|J) > βA ]}.

Theorem 1 illustrates that the positive agents, negative agents, and neutral agents towards a subset of issues given by Definition 12 are specials of those given by Definition 7. That is, the qualitative three-way conflict analysis model based on rough set theory given by Sun, Ma, and Zhao [41] is a special of the unified model given by Definition 7.

We employ the conditional support evaluation es (a|J) = +|Ia+ ∩ J |/|Ia+ | and the conditional opposition evaluation eo (a|J) = −|Ia− ∩ J |/|Ia− | to get a trisection ⟨⟨RP(αA ,βA ) (J), RN(αA ,βA ) (J), RZ(αA ,βA ) (J)⟩⟩ of the universe of agents towards J. According to Definition 15, we derive three rules for the agent a ∈ A towards

Example 2 (Continuation from Example 1). Taking J = {i1 , i2 , i3 }, first, by Definition 5, we have that Ia+1 = {i2 , i3 , i4 , i5 }, Ia+2 = {i1 }, Ia+3 = {i1 }, Ia+4 = {i3 }, Ia+5 = {i1 }, Ia+6 = {i2 , i5 }, Ia−1 = {i1 }, Ia−2 = {i3 , i4 , i5 }, Ia−3 = {i2 , i3 , i4 }, Ia−4 = {i2 , i5 }, Ia−5 = {i2 , i3 , i4 , i5 }, and Ia−6 = {i3 }. Second, by Definition 11, we get apr + (J) = {a2 , a3 , a4 , a5 } and apr − (J) = {a1 , a6 }. Third, by Definition 12, we have RP(J) = {a2 , a3 , a4 , a5 }, RN(J) = {a1 , a6 }, and RZ(J) = ∅. We also obtain (RP(J))c = {a1 , a6 }, (RN(J))c = {a2 , a3 , a4 , a5 }, and (RZ(J))c = {a1 , a2 , a3 , a4 , a5 , a6 }. Therefore, we obtain a trisection ⟨⟨RP(J), RN(J), RZ(J)⟩⟩ and a weak tricovering ⟨(RP(J))c , (RN(J))c , (RZ(J))c ⟩ of the universe of agents.

(1) If the conditional support evaluation es (a|J) = +|Ia+ ∩ J |/|Ia+ | ≥ αA and the conditional opposition evaluation eo (a|J) = −|Ia− ∩ J |/|Ia− | > βA for a ∈ A, then the agent a takes the support action towards J. (2) If the conditional opposition evaluation eo (a|J) = −|Ia− ∩ J | /|Ia− | ≤ βA and the conditional support evaluation es (a|J) = +|Ia+ ∩ J |/|Ia+ | < αA for a ∈ A, then the agent a takes the opposition action towards J. (3) If es (a|J) = +|Ia+ ∩ J |/|Ia+ | ≥ αA ∧ eo (a|J) = −|Ia− ∩ J |/|Ia− | ≤ βA , or es (a|J) = +|Ia+ ∩ J |/|Ia+ | < αA ∧ eo (a|J) = −|Ia− ∩ J | /|Ia− | > βA for a ∈ A, then the agent a takes the neutrality action towards J.

In Example 2, we have strong coalitions RP(J) = {a2 , a3 , a4 , a5 } and RN(J) = {a1 , a6 }, and weak coalitions (RP(J))c = {a1 , a6 }, (RN(J))c = {a2 , a3 , a4 , a5 }, and (RZ(J))c = {a1 , a2 , a3 , a4 , a5 , a6 }. We obtain a strong conflict SC({a2 , a3 , a4 , a5 }, {a1 , a6 }), and two non-conflicts NC({a1 , a6 }, {a1 , a6 }) and NC({a2 , a3 , a4 , a5 }, {a2 , a3 , a4 , a5 }). 4.2. A quantitative model Subsethood measures [46,47] are considered as a generalization of the set-inclusion relation for representing graded inclusion. For example, Yao and Deng [47] investigated quantitative rough set models based on subsethood measures, and provided effective methods for constructing the quantitative three-way conflict analysis model. We first propose a pair of conditional support evaluation and conditional opposition evaluation for trisecting the set of agents by using the subsethood measures. Definition 14. In a three-valued situation table S = (A, I , r), for J ⊆ I and a ∈ A, we define a pair of functions, namely, a conditional support evaluation es (·|J) : A −→ [0, +1] and a conditional opposition evaluation eo (·|J) : A −→ [−1, 0], as follows:

{ +

es (a|J) = +sh(Ia ⊑ J) =

{ eo (a|J) = −sh(Ia− ⊑ J) =

+

+ |I|aI +∩|J | , a 0, −

− |I|aI −∩|J | , a 0,

Ia+ ̸ = ∅, Ia+ = ∅, Ia− ̸ = ∅, −

Ia = ∅.

the subset of issues J:

By inserting the conditional support and opposition evaluations of Definition 14 into Definition 7 and setting a pair of thresholds (αA , βA ) such that 0 < αA ≤ +1 and −1 ≤ βA < 0, we immediately obtain Definition 15, as shown in the next theorem. Theorem 2. In a three-valued situation table S = (A, I , r), consider a pair of conditional support and opposition evaluations of Definition 14 and a pair of thresholds (αA , βA ) with 0 < αA ≤ +1 and −1 ≤ βA < 0. We have:

RP(αA ,βA ) (J) = P(αA ,βA ) (J), RN(αA ,βA ) (J) = N(αA ,βA ) (J), RZ(αA ,βA ) (J) = Z(αA ,βA ) (J). Proof. By Definitions 14 and 15, we have

RP(αA ,βA ) (J) = {a ∈ A | es (a|J) ≥ αA ∧ eo (a|J) > βA }

= RN (J) = = (αA ,βA ) RZ (J) = = (αA ,βA )

P(αA ,βA ) (J),

{a ∈ A | es (a|J) < αA ∧ eo (a|J) ≤ βA } N(αA ,βA ) (J), (RP(αA ,βA ) (J) ∪ RN(αA ,βA ) (J))c {a ∈ A | [es (a|J) ≥ αA ∧ eo (a|J) ≤ βA ] ∨ [es (a|J) < αA ∧ eo (a|J) > βA ]} (αA ,βA ) =Z (J). □

Theorem 2 illustrates that the positive agents, negative agents, and neutral agents towards a subset of issues given by Definition 15 are specials of those given by Definition 7. That is, the

Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.

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quantitative three-way conflict analysis model based on rough set theory given by Definition 15 is a special of the unified model given by Definition 7. Example 3 (Continuation from Example 2). Taking αA = + 54 and βA = − 35 , first, by Definition 14, we have es (a1 |J) = + 21 , es (a2 |J) = +1, es (a3 |J) = +1, es (a4 |J) = +1, es (a5 |J) = +1, es (a6 |J) = + 21 , eo (a1 |J) = −1, eo (a2 |J) = − 13 , eo (a3 |J) = − 32 , eo (a4 |J) = − 21 , eo (a5 |J) = − 12 , and eo (a6 |J) = −1. Second, by Definition 15, we get RP(αA ,βA ) (J) = {a2 , a4 , a5 }, RN(αA ,βA ) (J) = {a1 , a6 }, and RZ(αA ,βA ) (J) = {a3 }. Third, we obtain (RP(αA ,βA ) (J))c = {a1 , a3 , a6 }, (RN(αA ,βA ) (J))c = {a2 , a3 , a4 , a5 }, and (RZ(αA ,βA ) (J))c = {a1 , a2 , a4 , a5 , a6 }. Therefore, we have a trisection ⟨⟨RP(αA ,βA ) (J), RN(αA ,βA ) (J), RZ(αA ,βA ) (J)⟩⟩ and a weak tri-covering ⟨(RP(αA ,βA ) (J))c , (RN(αA ,βA ) (J))c , (RZ(αA ,βA ) (J))c ⟩ of the universe of agents. In Example 3, we have strong coalitions RP(αA ,βA ) (J) = {a2 , a4 , a5 }, RN(αA ,βA ) (J) = {a1 , a6 }, and RZ(αA ,βA ) (J) = {a3 }, and weak coalitions (RZ(αA ,βA ) (J))c = {a1 , a2 , a4 , a5 , a6 } (RN(αA ,βA ) (J))c = {a2 , a3 , a4 , a5 }, and (RP(αA ,βA ) (J))c = {a1 , a3 , a6 }. We get a strong conflict SC({a2 , a4 , a5 }, {a1 , a6 }), weak conflicts WC({a2 , a4 , a5 }, {a3 }) and WC({a1 , a6 }, {a3 }), non-conflicts NC({a2 , a4 , a5 }, {a2 , a4 , a5 }), NC({a1 , a6 }, {a1 , a6 }), and NC({a3 }, {a3 }). According to Examples 2 and 3, the qualitative and quantitative three-way conflict analysis models based on rough set theory have similar positive agents, negative agents, and neutral agents. In comparison with the qualitative model, more agents are classified into the neutral agents by using the quantitative model. By using more information, we can classify these agents into the positive agents or negative agents with less risks or losses. The qualitative model illustrates the original semantics of the quantitative model. 5. Three-way conflict analysis models based on formal concept analysis In this section, we study qualitative and quantitative threeway conflict analysis models based on formal concept analysis [48,49]. 5.1. A qualitative model Many researchers [21,49–56] studied formal concept analysis based on the theory of three-way decision. For example, Yao [49] investigated the relationship among three-way granular computing, rough sets, and formal concept analysis, which gives the foundation for trisecting the set of agents based on formal concept analysis. We first consider a transformation of a three-valued situation table into a pair of binary formal contexts. The latter is required for building derivation operators over two universes, and can help to reflect the essence of models of three-way conflict analysis based on formal concept analysis. Definition 16. Given a three-valued situation table S = (A, I , r), we have two two-valued situation tables or two binary formal contexts (A, I , R+ ) and (A, I , R− ), where R+ and R− are given by: R+ = {(a, i) ∈ A × I | r(a, i) = +1}, R− = {(a, i) ∈ A × I | r(a, i) = −1}, which represent, respectively, that an agent supports and opposes an issue. For a ∈ A, we have the set of issues supported by agent a and the set of issues opposed by agent a:

According to Definition 16, R+ and R− have the same meanings as those in Definition 10, and Ia+ and Ia− have the same meanings as those in Definition 10. Based on the two binary formal contexts, we associate a subset of issues with two subsets of agents by derivation operators. Definition 17. In a three-valued situation table S = (A, I , r), for J ⊆ I, we associate the subset of issues J with two subsets of agents: der + (J) = {a ∈ A | J ⊆ Ia+ }, der − (J) = {a ∈ A | J ⊆ Ia− }. In the framework of formal concept analysis, for the purpose of conflict analysis, we interpret the derivation operators of a subset of issues based on relation R+ . By Definition 17, an agent a ∈ der + (J) supports all issues of J. Similarly, we can interpret the derivation operators of a subset of issues based on relation R− . The pair of binary formal contexts (A, I , R+ ) and (A, I , R− ) may be viewed as two two-way decisions. By combining a pair of derivation operators given by Definition 17, we trisect the set of agents into the sets of positive agents, negative agents, and neutral agents towards a subset of issues. Definition 18. In a three-valued situation table S = (A, I , r), for J ⊆ I, the sets of positive agents FP(J), negative agents FN(J), and neutral agents FZ(J) towards J are defined by:

FP(J) = der + (J), FN(J) = der − (J), FZ(J) = (FP(J) ∪ FN(J))c . According to Definition 18, the sets of positive agents, negative agents, and neutral agents towards a subset of issues are pairwise disjoint, and their union is the entire universe of agents. Therefore, we have a trisection ⟨⟨FP(J), FN(J), FZ(J)⟩⟩ of the universe of agents. We now show that Definition 18 is in fact a special case of Definition 7 introduced earlier. This is done by defining the conditional support evaluation and the conditional opposition evaluation. Definition 19. In a three-valued situation table S = (A, I , r), for J ⊆ I and a ∈ A, we define a pair of functions, namely, a conditional support evaluation es (·|J) : A −→ {0, +1} and a conditional opposition evaluation eo (·|J) : A −→ {0, −1}, as follows:

{

+1, 0,

J ̸ = ∅ ∧ J ⊆ Ia+ , other w ise,

{

−1, 0,

J ̸ = ∅ ∧ J ⊆ Ia− , other w ise.

es (a|J) = eo (a|J) =

By inserting the conditional support and opposition evaluations of Definition 19 into Definition 7 and setting a pair of thresholds (αA , βA ) such that 0 < αA ≤ +1 and −1 ≤ βA < 0, we immediately obtain Definition 18, as shown in the next theorem. Theorem 3. In a three-valued situation table S = (A, I , r), consider a pair of conditional support and opposition evaluations of Definition 19 and a pair of thresholds (αA , βA ) with 0 < αA ≤ +1 and −1 ≤ βA < 0. We have:

FP(J) = P(αA ,βA ) (J),

Ia = {i ∈ I | (a, i) ∈ R } = {i ∈ I | r(a, i) = +1},

FN(J) = N(αA ,βA ) (J),

Ia = {i ∈ I | (a, i) ∈ R } = {i ∈ I | r(a, i) = −1}.

FZ(J) = Z(αA ,βA ) (J).

+

−

+

−

7

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Proof. By Definitions 18 and 19, we have

FP(J) = der + (J)

= = = = = FN(J) = = = = = = FZ(J) = = =

der + (J) − der − (J) +

−

{ a ∈ A | J ⊆ Ia } − { a ∈ A | J ⊆ Ia } {a ∈ A | es (a|J) = +1 ∧ eo (a|J) ̸= −1} {a ∈ A | es (a|J) ≥ αA ∧ eo (a|J) > βA } P(αA ,βA ) (J), der − (J) −

+

der (J) − der (J)

{a ∈ A | J ⊆ Ia− } − {a ∈ A | J ⊆ Ia+ } {a ∈ A | es (a|J) ̸= +1 ∧ eo (a|J) = −1} {a ∈ A | es (a|J) < αA ∧ eo (a|J) ≤ βA } N(αA ,βA ) (J), (FP(J) ∪ FN(J))c {a ∈ A | [J ⊆ Ia+ ∧ J ⊆ Ia− ] ∨ [J ⊈ Ia+ ∧ J ⊈ Ia− ]} {a ∈ A | [es (a|J) = +1 ∧ eo (a|J) = −1] ∨ [es (a|J) ̸= +1 ∧ eo (a|J) ̸= −1]} = {a ∈ A | [es (a|J) ≥ αA ∧ eo (a|J) ≤ βA ] ∨ [es (a|J) < αA ∧ eo (a|J) > βA ]} = Z(αA ,βA ) (J). □

Theorem 3 illustrates that the positive agents, negative agents, and neutral agents towards a subset of issues given by Definition 18 are specials of those given by Definition 7. That is, the qualitative three-way conflict analysis model based on formal concept analysis given by Definition 18 is a special of the unified model given by Definition 7. Example 4 (Continuation from Example 1). Taking J = {i1 , i2 , i3 }, by Definition 18, we have FP(J) = ∅, FN(J) = ∅, and FZ(J) = {a1 , a2 , a3 , a4 , a5 , a6 }. So we obtain a trisection ⟨⟨FP(J), FN(J), FZ(J)⟩⟩ of the universe of agents. Then, we have (FP(J))c = {a1 , a2 , a3 , a4 , a5 , a6 }, (FN(J))c = {a1 , a2 , a3 , a4 , a5 , a6 }, and (FZ(J))c = ∅. Therefore, we get a weak tri-covering ⟨(FP(J))c , (FN(J))c , (FZ(J))c ⟩ of the universe of agents. In Example 4, we have a strong coalition FZ(J) = {a1 , a2 , a3 , a4 , a5 , a6 } and a non-conflict NC({a1 , a2 , a3 , a4 , a5 , a6 }, {a1 , a2 , a3 , a4 , a5 , a6 }). Yao [35] proposed the concepts of positive agents, negative agents, and neutral agents of an issue and a subset of issues. In fact, they can be viewed as specials of Definition 7. Theorem 4. In a three-valued situation table S = (A, I , r), consider a pair of conditional support and opposition evaluations of Definition 19 and a pair of thresholds (αA , βA ) with 0 < αA ≤ +1 and −1 ≤ βA < 0. For i ∈ I and J ⊆ I, we have: (αA ,βA ) (1) A+ (i) = P(αA ,βA ) (i), i = FP (αA ,βA ) A− (i) = N(αA ,βA ) (i), i = FN

A0i = FZ(αA ,βA ) (i) = Z(αA ,βA ) (i); +

(2) AJ = FP

(αA ,βA )

(J) = P

(αA ,βA )

(J),

(αA ,βA ) A− (J) = N(αA ,βA ) (J), J = FN

A0J

= FZ

(αA ,βA )

(αA ,βA )

(J) = Z

(J).

Proof. (1) By Definitions 14 and 15, we have A+ i = {a ∈ A | r(a, i) = +1}

= {a ∈ A | {i} ⊆ Ia+ } − {a ∈ A | {i} ⊆ Ia− }

= = − Ai = = = = A0i = = = =

FP(αA ,βA ) (i) P(αA ,βA ) (i),

{a ∈ A | r(a, i) = −1} {a ∈ A | {i} ⊆ Ia− } − {a ∈ A | {i} ⊆ Ia+ } FN(αA ,βA ) (i) N(αA ,βA ) (i), − c (A+ i ∪ Ai )

{a ∈ A | r(a, i) = 0} FZ(αA ,βA ) (i) Z(αA ,βA ) (i).

(2) By Definitions 14 and 15, we have A+ J = {a ∈ A | ∀i ∈ J , r(a, i) = +1}

= = = A− J = = = = A0J = = = =

{a ∈ A | J ⊆ Ia+ } − {a ∈ A | J ⊆ Ia− } FP(αA ,βA ) (J) P(αA ,βA ) (J),

{a ∈ A | ∀i ∈ J , r(a, i) = −1} {a ∈ A | J ⊆ Ia− } − {a ∈ A | J ⊆ Ia+ } FN(αA ,βA ) (J) N(αA ,βA ) (J), − c (A+ J ∪ AJ )

{a ∈ A | ∃i ∈ J , r(a, i) = 0} FZ(αA ,βA ) (J) Z(αA ,βA ) (J). □

We employ some symbols given by Yao [35] to denote the positive agents, negative agents, and neutral agents towards an − issue and a subset of issues in Theorem 4. The three sets A+ i , Ai , 0 and Ai denote the positive agents, negative agents, and neutral − 0 agents, respectively, towards the issue i; A+ J , AJ , and AJ stand for the positive agents, negative agents, and neutral agents of the subset of issues J, respectively. 5.2. A quantitative model In this section, we investigate the quantitative three-way conflict model based on formal concept analysis, proposed by Fan, Qi, and Wei [36], for trisecting the universe of agents. We now show that Fan, Qi, and Wei’s three-way conflict model is in fact a special case of Definition 7 introduced earlier. This is done by defining a pair of conditional support and opposition evaluations. Definition 20. In a three-valued situation table S = (A, I , r), for J ⊆ I and a ∈ A, we define a pair of functions, namely, a conditional support function es (·|J) : A −→ [0, +1] and a conditional opposition function eo (·|J) : A −→ [−1, 0] towards J, as follows:

{

+ |Ia|J∩| J | , 0,

{

− |Ia|J∩| J | , 0,

+

es (a|J) = +sh(J ⊑ Ia ) = −

eo (a|J) = −sh(J ⊑ Ia ) =

+

J ̸ = ∅, J = ∅,

−

J ̸ = ∅, J = ∅.

By taking the conditional support and opposition evaluations of Definition 20 into Definition 7, we trisect the set of agents into the sets of positive agents, negative agents, and neutral agents towards a subset of issues.

Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.

G. Lang, J. Luo and Y. Yao / Knowledge-Based Systems xxx (xxxx) xxx

Definition 21. In a three-valued situation table S = (A, I , r). For J ⊆ I, we have a pair of evaluations (es (·|J), eo (·|J)). Given a pair of thresholds (αA , βA ) with 0 < αA ≤ +1 and −1 ≤ βA < 0, the sets of positive agents FP(αA ,βA ) (J), negative agents FN(αA ,βA ) (J), and neutral agents FZ(αA ,βA ) (J) towards J are defined by:

FP(αA ,βA ) (J) = {a ∈ A | es (a|J) ≥ αA ∧ eo (a|J) > βA }, FN(αA ,βA ) (J) = {a ∈ A | es (a|J) < αA ∧ eo (a|J) ≤ βA }, FZ(αA ,βA ) (J) = (FP(αA ,βA ) (J) ∪ FN(αA ,βA ) (J))c

= {a ∈ A | [es (a|J) ≥ αA ∧ eo (a|J) ≤ βA ] ∨ [es (a|J) < αA ∧ eo (a|J) > βA ]}. We employ the conditional support evaluation es (a|J) = +|Ia+ ∩ J |/|J | and the opposition evaluation eo (a|J) = −|Ia− ∩ J |/|J | to get a trisection ⟨⟨FP(αA ,βA ) (J), FN(αA ,βA ) (J), FZ(αA ,βA ) (J)⟩⟩ of the universe of agents towards J. According to Definition 21, we derive three rules for the agent a ∈ A towards the subset of issues J: (1) If the conditional support evaluation es (a|J) = +|Ia+ ∩ J | /|J | ≥ αA and the conditional opposition evaluation eo (a|J) = −|Ia− ∩ J |/|J | > βA for a ∈ A, then the agent a takes the support action towards J. (2) If the conditional opposition evaluation eo (a|J) = −|Ia− ∩ J | /|J | ≤ βA and the conditional support evaluation es (a|J) = +|Ia+ ∩ J |/|J | < αA for a ∈ A, then the agent a takes the opposition action towards J. (3) If es (a|J) = +|Ia+ ∩ J |/|J | ≥ αA ∧ eo (a|J) = −|Ia− ∩ J |/|J | ≤ βA , or es (a|J) = +|Ia+ ∩ J |/|J | < αA ∧ eo (a|J) = −|Ia− ∩ J |/|J | > βA for a ∈ A, then the agent a takes the neutrality action towards J. By inserting the conditional support and opposition evaluations of Definition 20 into Definition 7 and setting a pair of thresholds (αA , βA ) such that 0 < αA ≤ +1 and −1 ≤ βA < 0, we immediately obtain Definition 21, as shown in the next theorem. Theorem 5. In a three-valued situation table S = (A, I , r), consider a pair of conditional support and opposition evaluations of Definition 20 and a pair of thresholds (αA , βA ) with 0 < αA ≤ +1 and −1 ≤ βA < 0. We have:

FP(αA ,βA ) (J) = P(αA ,βA ) (J), FN(αA ,βA ) (J) = N(αA ,βA ) (J), FZ(αA ,βA ) (J) = Z(αA ,βA ) (J). Proof. By Definitions 20 and 21, we have

FP(αA ,βA ) (J) = {a ∈ A | es (a|J) ≥ αA ∧ eo (a|J) > βA }

= FN(αA ,βA ) (J) = = FZ(αA ,βA ) (J) = =

P(αA ,βA ) (J),

{a ∈ A | es (a|J) < αA ∧ eo (a|J) ≤ βA } N(αA ,βA ) (J), (FP(αA ,βA ) (J) ∪ FN(αA ,βA ) (J))c {a ∈ A | [es (a|J) ≥ αA ∧ eo (a|J) ≤ βA ] ∨ [es (a|J) < αA ∧ eo (a|J) > βA ]} (αA ,βA ) =Z (J). □

Theorem 5 illustrates that the positive agents, negative agents, and neutral agents towards a subset of issues given by Definition 21 are specials of those given by Definition 7. That is, the quantitative three-way conflict analysis model based on formal concept analysis given by Definition 18 is a special of the unified model given by Definition 7.

9

Example 5 (Continuation from Example 4). Taking αA = + 53 and βA = − 53 , by Definition 20, we have that es (a1 |J) = + 23 , es (a2 |J) = + 31 , es (a3 |J) = + 13 , es (a4 |J) = + 31 , es (a5 |J) = + 13 , es (a6 |J) = + 13 , eo (a1 |J) = − 31 , eo (a2 |J) = − 13 , eo (a3 |J) = − 32 , eo (a4 |J) = − 13 , eo (a5 |J) = − 32 , and eo (a6 |J) = − 31 . Then, by Definition 21, we get FP(αA ,βA ) (J) = {a1 }, FN(αA ,βA ) (J) = {a3 , a5 }, and FZ(αA ,βA ) (J) = {a2 , a4 , a6 }. Third, we obtain (FP(α1 ,β1 ) (J))c = {a2 , a3 , a4 , a5 , a6 }, (FN(αA ,βA ) (J))c = {a1 , a2 , a4 , a6 }, and (FZ(αA ,βA ) (J))c = {a1 , a3 , a5 }. Therefore, we have a trisection ⟨⟨FP(αA ,βA ) (J), FN(αA ,βA ) (J), FZ(αA ,βA ) (J)⟩⟩ and a weak tri-covering ⟨(FP(αA ,βA ) (J))c , (FN(αA ,βA ) (J))c , (FZ(αA ,βA ) (J))c ⟩ of the universe of agents. In Example 5, we have strong coalitions FP(αA ,βA ) (J) = {a1 }, FN (J) = {a3 , a5 }, and FZ(αA ,βA ) (J) = {a2 , a4 , a6 }. We have weak coalitions (FZ(αA ,βA ) (J))c = {a1 , a3 , a5 }, (FN(αA ,βA ) (J))c = {a1 , a2 , a4 , a6 }, and (FP(αA ,βA ) (J))c = {a2 , a3 , a4 , a5 , a6 }. We get a strong conflict SC({a1 }, {a3 , a5 }), weak conflicts WC({a1 }, {a2 , a4 , a6 }) and WC({a3 , a5 }, {a2 , a4 , a6 }), non-conflicts NC({a1 }, {a1 }), NC({a3 , a5 }, {a3 , a5 }), and NC({a2 , a4 , a6 }, {a2 , a4 , a6 }). According to Examples 4 and 5, the qualitative and quantitative three-way conflict analysis models based on formal concept analysis have different positive agents, negative agents, and neutral agents. In comparison with the quantitative model, more agents are classified into the neutral agents by using the qualitative model, and the condition of the qualitative model is so strict that cannot trisect the set of agents into three pair-wise disjoint parts for conflict analysis. (αA ,βA )

Remark. In Example 2, 3, 4, and 5, there are some contradicted results for the positive agents, neutral agents, and negative agents by using three-way conflict analysis models. The reasons are illustrated as follows. (1) The qualitative models of three-way conflict analysis based on rough sets and formal concept analysis are constructed from two different aspects. One is based on two conditions Ia+ ⊆ J and Ia− ⊆ J, and the other one is based on two conditions J ⊆ Ia+ and J ⊆ Ia− . It is obvious that the positive agents, negative agents, and neutral agents derived by using the two qualitative models have different semantics. (2) The quantitative models of three-way conflict analysis based on rough sets and formal concept analysis are constructed from two different aspects. One is based on two subsethood measures +sh(Ia+ ⊑ J) and −sh(Ia− ⊑ J), and the other one is based on two subsethood measures +sh(J ⊑ Ia+ ) and −sh(J ⊑ Ia− ). Therefore, the two quantitative models are generalizations of the two qualitative models, and the two qualitative models illustrate the classical semantics of the two quantitative models.

6. An application of three-way conflict analysis model In many practical situations, three-way conflict analysis may provide an effective tool for decision making. Sun et al. [39] have applied basic ideas of three-way conflict analysis to study development plans for Gansu Province in China. In this section, we use their example to demonstrate an application of the two models, i.e., rough set based model and formal concept analysis based model. We also compare the results from the two models. In China, the economic development and social stabilization are the most important assignments for the government, and there are many conflicts among cities involved in a new implemented policy. The government of Gansu Province intends to make a development plan for the next year. There are fourteen

Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.

10

G. Lang, J. Luo and Y. Yao / Knowledge-Based Systems xxx (xxxx) xxx

Table 3 The opinions of cities on all issues [39]. A

I i1

i2

i3

i4

i5

i6

i7

i8

i9

i10

a1

+

−

0

−

+

−

0

−

+

−

i11

+

a2

0

+

−

0

0

+

−

0

0

+

−

a3

−

0

−

−

−

+

+

−

−

0

0

a4

0

0

−

+

+

−

−

+

0

−

−

a5

−

+

−

0

−

+

0

0

−

+

+

a6

0

+

0

−

−

−

−

−

0

+

−

a7

+

+

0

+

0

+

0

+

+

+

0

a8

−

0

−

+

−

0

+

+

−

0

+

a9

+

+

0

−

+

+

−

−

+

+

−

a10

−

−

−

0

+

−

+

0

−

−

+

a11

−

0

−

−

−

−

−

−

−

0

−

a12

0

+

0

−

+

+

+

−

0

+

0

a13

−

0

−

+

0

0

0

+

−

0

+

a14

−

−

−

0

−

−

−

0

−

−

−

cities, namely, Lanzhou, Jinchang, Baiyin, Tianshui, Jiayuguan, Wuwei, Zhangye, Pingliang, Jiuquan, Qingyang, Dingxi, Longnan, Linxia, and Gannan in Gansu Province, which are denoted as a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 , a12 , a13 , and a14 . The development plan mainly involves eleven issues, namely, the construction of roads, factories, entertainment, educational institutions, total population of residence, ecology environment, the number of senior intellectuals, the traffic capacity, mineral resources, sustainable development capacity and water resources carrying capacity, which are denoted as i1 , i2 , i3 , i4 , i5 , i6 , i7 , i8 , i9 , i10 , and i11 . The opinions of each agent on all issues are depicted by Table 3 [39], where +1 stands for support on this issue, 0 stands for neutral on this issue, and −1 stands for against on this issue. Based on the model of three-way conflict analysis, Sun, Chen, Zhang, and Ma [39] provided the algorithm of computing the optimal feasible consensus strategy for the conflict situation. Algorithm 1 (The Algorithm based on Rough Set Theory). Step 1: Input a three-valued situation table S = (A, I , r); Step 2: Create all possible strategy set S = 2I \∅; Step 3: Construct the set of positive agents RP(αA ,βA ) (J) for all J ∈ S; Step 4: Compute the set RS = {J | |RP(αA ,βA ) (J)|/|A| ≥ δA } based on rough set theory; Step 5: Output the optimal feasible consensus strategy set RS. Meanwhile, we provide the algorithm of computing the optimal feasible consensus strategy based on formal concept analysis. Algorithm 2 (The Algorithm based on Formal Concept Analysis). Step 1: Input a three-valued situation table S = (A, I , r); Step 2: Create all possible strategy set S = 2I \∅; Step 3: Construct the set of positive agents FP(αA ,βA ) (J) for all J ∈ S; Step 4: Compute the set FS = {J | |FP(αA ,βA ) (J)|/|A| ≥ δA } based on formal concept analysis; Step 5: Output the optimal feasible consensus strategy set FS.

{i1 , i2 , i5 , i9 }, {i2 , i6 , i9 }, {i2 , i4 , i10 }, {i1 , i6 , i10 }, {i2 , i6 , i10 }, {i1 , i2 , i6 , i10 }, {i2 , i8 , i10 }, {i1 , i5 , i9 , i10 }, {i6 , i9 , i10 }, {i2 , i6 , i9 , i10 }, {i2 , i5 , i11 }, {i5 , i6 , i11 }, {i2 , i5 , i7 , i11 }, {i5 , i6 , i7 , i11 }, {i5 , i7 , i10 , i11 }}. (2) By Algorithm 2, taking αA = 0.5, βA = −0.3, and δA = 0.5, we have the optimal feasible consensus strategy set based on formal concept analysis as follows:

FS = {{i2 , i4 , i5 , i11 }, {i7 , i11 }, {i4 , i6 , i7 , i11 },

{i2 , i5 , i8 , i11 }, {i6 , i7 , i8 , i11 }}. (3) The government can make the development plan for Gansu Province by considering the above strategies. If there is a strategy in RS ∩ FS, then the government can choose this strategy. If we have RS ∩ FS = ∅, then the government can make a development plan by considering the strategies based on rough set theory and formal concept analysis. For example, since we have {i2 , i5 , i11 } ⊆ {i2 , i4 , i5 , i11 } for {i2 , i5 , i11 } ∈ RS and {i2 , i4 , i5 , i11 } ∈ FS, the government can choose the strategy {i2 , i5 , i11 } or {i2 , i4 , i5 , i11 }. Remark. In Example 6, we give the thresholds αA , βA , and δA . Actually, the thresholds αA , βA , and δA are very important for the model of three-way conflict analysis, and we will study how to provide appropriate thresholds for conflict analysis based on Bayesian decision theory in the future. 7. Conclusion In this paper, to integrate rough sets-based conflict analysis models and formal concept analysis-based conflict analysis models, we have provided a pair of evaluations, namely, conditional support evaluation and conditional opposition evaluation. Then, in the framework of the model of three-way decision, we have proposed a general three-way conflict analysis model, which unifies several models of conflict analysis for trisecting the universe of agents by using the two evaluations. We have illustrated the models of conflict analysis based on rough set theory and formal concept analysis are special cases of the general model. We have applied the two evaluations-based three-way conflict analysis model to make the development plan of Gansu Province in China. We have investigated conflict analysis with two evaluations in three-valued situation tables, and the same method can be applied to construct general three-way conflict analysis models in the future. Furthermore, we will study knowledge reduction of three-valued situation tables and provide effective algorithms for computing reducts of three-valued situation tables. CRediT authorship contribution statement Guangming Lang: Conceptualization, Investigation, Writing original draft, Software. Junfang Luo: Investigation, Writing review & editing. Yiyu Yao: Methodology, Supervision, Writing - review & editing, Validation. Acknowledgements

Example 6. Consider the opinions of cities on all issues given by Table 3. (1) By Algorithm 1, taking αA = 0.33, βA = −0.28, and δA = 0.50, we have the optimal feasible consensus strategy set based on rough set theory as follows:

RS = {{i1 , i2 , i6 }, {i2 , i4 , i6 }, {i2 , i6 , i8 },

We would like to thank the anonymous reviewers very much for their professional comments and valuable suggestions. This work was supported in part by the China Scholarship Council (No. 201808430120), the National Natural Science Foundation of China (No. 61603063), Hunan Provincial Natural Science Foundation, PR China (Nos. 2018JJ3518, 2018JJ2027), Hunan Provincial

Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.

G. Lang, J. Luo and Y. Yao / Knowledge-Based Systems xxx (xxxx) xxx

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Please cite this article as: G. Lang, J. Luo and Y. Yao, Three-way conflict analysis: A unification of models based on rough sets and formal concept analysis, Knowledge-Based Systems (2020) 105556, https://doi.org/10.1016/j.knosys.2020.105556.