Three-Valued Concept Analysis for 2R Formal Contexts

Abstract

Russian Roulette is a well-known cruel gambling game and its concepts and methods have been exploited in a lot of research fields for decades. However, abundant useful information contained in the process of Russian Roulette is seldom studied with a mathematical model with interpretability. To this end, we define the $2R$ formal context to model Russian Roulette and carry out 3-valued concept analysis for $2R$ formal contexts to mine useful information. At first, the uniqueness of $2R$ formal contexts is discussed from a formal concept analysis viewpoint. And then we propose 3-valued $2R$ concepts and discuss their properties and the connections with the basic $2R$ concepts. Experimental analysis demonstrates that 3-valued $2R$ concept lattices can show many more different details compared with basic $2R$ concept lattices. Finally, a case study about a Chinese herbal medicine is introduced to demonstrate the feasibility of the proposed model.

1. Introduction

Russian Roulette is a cruel gambling game. In Russian Roulette, players usually choose to put into a gun no bullets, or one or two bullets, rather than three or more. In other words, participants only have three options: no bullets, one bullet, or two bullets, i.e., three choices. The concepts and methods derived from Russian Roulette have been used in various types of scientific studies for decades. In the 1970s, the proposal of smoke-control legislation created a lot of crucial challenges for United States copper producers. White [1] reviewed the efforts of individual United States copper producers to reduce the emission of sulfur dioxide and found that copper smelters in the U.S. fell into a scenario of industrial Russian Roulette. In 1995, Murata et al. [2] conducted shielding analysis based on a three-dimensional Monte Carlo model. Two decades ago, Ghassoun and Jehouani [3] sampled energy parameters based on a Russian Roulette technique. Frackiewicz and Schmidt [4] generalized the concept of quantum Russian Roulette and gave a suitable quantum description for any finite number of players. In 2019, oriented for big data, Tokuyoshi and Harada [5] raised a hierarchical acceleration strategy for vertex connections.

It is obvious that there are abundant useful data embodied in Russian Roulette processes. However, those valuable data are seldom effectively investigated based on a mathematical model with interpretability. In this study, we deeply explore this issue with respect to the following aspects.

• Firstly, we define a mathematical model for Russian Roulette, the $2R$ formal context, which differs from the 3-valued formal context and the ordinary set-valued formal context in many aspects.
• In addition, we investigate the $2R$ formal context from multiple perspectives to show the uniqueness of 2R formal contexts.
• What is more, we define a novel type of concept for the $2R$ formal context, i.e., 3-valued 2R concepts and we manifest their effectiveness by systematic experiments and illustrative examples.

Nowadays, most of the studies based on machine learning exhibit a strict dichotomy between accuracy and interpretability. Conceptual knowledge-based data-mining techniques are powerful tools for explainable artificial intelligence. Formal concept analysis, which models a domain with a cognitive viewpoint, has become a wonderful theory for knowledge-based applications [6].

To cater for different cases, many scholars have proposed various types of concepts as well as concept-learning methods. Li et al. [7] described a way to acquire formal concepts from a pseudo-concept by using the granular computing technique. Xu and Li [8] built a two-way concept-acquisition system from several types of information granules. Zhi and Li [9] proposed updating strategies of concept lattices and implication rules with the arrival of new objects. Lately, Lang and Yao [10] provided a new perspective on conflict analysis by modeling and analyzing cliques with a set of formal concepts. Bazin et al. [11] described a method to discover causal relations based on an adaptation of classical formal concept analysis. Belohlavek and Mikula [12] formally explained typicality as an important psychological property of human concepts. Jǎkel and Schmidt [13] designed an abstract framework based on formal concepts to solve optimization problems.

However, formal concepts do not explicitly exhibit the not-possessed features. As a successful solution, Qi et al. [14,15] proposed three-way concept analysis (3WCA) by thinking in threes [16,17]. Since the proposal of 3WCA, many researchers have devoted their efforts to this novel concept model [18,19,20,21,22,23,24]. Zhi et al. [24] defined three-way dual formal concepts by introducing the operations of possibility analysis. Hu et al. [18] adopted interval sets to extend three-way concepts for a specific case in medical diagnosis. Mouliswaran et al. [19] proposed an effective access control method by analyzing the roles of visitors by 3WCA. Yan and Li [22] constructed three-way network concepts to mine topological information in dynamic social networks.

From a possibility theoretic view, more meaningful types of concepts, such as object-oriented and property-oriented concepts, can be defined [25,26,27,28,29,30]. There have been several theoretical studies in this field. For instance, Ma et al. [31] proposed a hierarchical concept-mining approach to acquire object-oriented concepts. In addition, Ma et al. [32] carried out attribute reduction of lattices aided by the topological structures of attributes. She et al. [33] surveyed the properties of multi-scale contexts based on necessary and possible attributes. Wei and Wan [34] discussed the transformation between three types of concepts based on equivalence relations. Recently, Zhi et al. [35] provided new semantics to property-oriented concepts, based on which they further proposed a feasible strategy to discover close contacts in disease-transmission networks. Analogously, object-oriented concepts, property-oriented concepts and dual formal concepts can be combined with three-way decisions. As a consequence, we can accordingly obtain three kinds of three-way concepts [24]. Qian et al. [36] explored the fast computation of three-way concepts by dividing and merging formal contexts.

In addition, all the above-mentioned types of concepts can be adjusted for fuzzy contexts. For example, He et al. [37] defined three-way L-fuzzy concepts to improve the precision of uncertainty reasoning. Chen et al. [38] combined fuzzy concept lattices and three-way decisions to mine useful patterns in unlabelled texts. Zhi et al. [39] adopted the fuzzy concept lattice to realize multi-level conflict analysis and proposed a way to obtain the most appropriate analysis level to make fast decisions. Xu et al. [40] described a two-way concept-acquisition method, which can derive fuzzy concepts from a given clue.

Motivated by the above-mentioned problems, i.e., how to model the obtained record of Russian Roulette, how to define new concepts to mine useful patterns from Russian Roulette and how to obtain these concepts, we put forward 3-valued concept analysis for $2R$ formal contexts. In the rest of the paper, Section 2 briefly reviews some basic notions related to this study. Section 3 proposes 3-valued concept analysis for $2R$ formal contexts. In Section 4, experiments and case studies are presented to show the effectiveness of the 3-valued $2R$ concept lattice model. Finally, conclusions are provided in the last section.

2. Preliminaries of Formal Concept Analysis

A formal context $\Pi =(Q,T,\rho )$ describes a set of related objects Q by a set of attributes T via a binary relation $\rho$ between Q and T. Concretely, we use $\rho (q,t)=1$ (respectively, $\rho (q,t)=0$) to express that the object q possesses (respectively, does not possess) the attribute t.

$R\left(q\right)$ represents the attributes that are possessed by the object $q\in Q$ and $R\left(t\right)$ denotes the set of objects which have the attribute $t\in T$; these are defined as follows:

$$R\left(q\right)=\{t\in T\mid \rho (o,p)=1\} \mathrm{and} R\left(t\right)=\{q\in Q\mid \rho (q,t)=1\}.$$

For $E\subseteq Q$ and $P\subseteq T$, we define two operations as follows:

$$E^{\ast }=\{t\in T\mid \forall q\in E,\rho (q,t)=1\} \mathrm{and} P^{\ast }=\{q\in Q\mid \forall t\in T,\rho (q,t)=1\}.$$

$E^{\ast }$ gives the common attributes shared by every object in E and $P^{\ast }$ returns the set of objects that possess all the attributes in P. If $E^{\ast }=P$ and $P^{\ast }=E$, then we call $(E,P)$ a formal concept of $\Pi$.

In addition, $E^{\overline{\ast }}$ and $P^{\overline{\ast }}$ can be defined as follows:

$$E^{\overline{\ast }}=\{t\in T\mid \forall q\in E,\rho (q,t)=0\} \mathrm{and} P^{\overline{\ast }}=\{q\in Q\mid \forall t\in T,\rho (q,t)=0\}.$$

$E^{\overline{\ast }}$ is the attributes that are not commonly shared by every object in E and $P^{\overline{\ast }}$ returns the set of objects that do not possess all the attributes in P. If $E^{\ast }=P_1$, $E^{\overline{\ast }}=P_2$ and $P_1^{\ast }\cap P_2^{\overline{\ast }}=E$, then we call $(E,(P_1,P_2))$ an object-induced three-way concept of $\Pi$ [15].

What is more, other types of operations, ${(·)}^{\diamond }$, ${(·)}^{\square }$ and ${(·)}^{\#}$, can also be defined to investigate the other aspects of granules [25,26,27,28,29,30]. Formally, for $E\subseteq Q$, we have

$E^{\diamond }=\{t\in T\mid E\cap R\left(t\right)\ne Ø\}$, $E^{\square }=\{t\in T\mid R\left(t\right)\subseteq E\}$ and $E^{\#}=\{t\in T\mid E\cup R\left(t\right)\ne Q\}$.

Dually, for $P\subseteq T$, we have:

$P^{\diamond }=\{q\in Q\mid P\cap R\left(q\right)\ne Ø\}$, $P^{\square }=\{q\in Q\mid R\left(q\right)\subseteq P\}$ and $P^{\#}=\{q\in Q\mid P\cup R\left(q\right)\ne T\}$.

Then, we can define property-oriented concepts, object-oriented concepts and dual formal concepts by using the above-defined operations [27,28,29,30,41].

Concretely, if $E^{\diamond }=P$ and $P^{\square }=E$, then we call $(E,P)$ a property-oriented concept of $\Pi$. If $E^{\square }=P$ and $P^{\diamond }=E$, then we call $(E,P)$ an object-oriented concept of $\Pi$. If $E^{\#}=P$ and $P^{\#}=E$, then we call $(E,P)$ a dual formal concept of $\Pi$.

3. The Unique Properties of 2R Formal Contexts: A Formal Concept Analysis Viewpoint

In this section, we propose four types of concepts of $2R$ formal contexts and describe the unique properties of $2R$ formal contexts.

3.1. Formal Concepts of 2R Formal Contexts

In Russian Roulette, a gambler will trigger one or two bullets. In other words, a gambler will definitely trigger one bullet and definitely does not trigger all three bullets. To conform with the existing studies, we use ‘+’, ‘−’ and ‘0’ to denote three bullets, i.e., three kinds of choices.

It is clear that there are six kinds of possible actions taken by the gamblers of a Russian Roulette game, i.e., $\{+\}$, $\{-\}$, $\left\{0\right\}$, $\{+,0\}$, $\{-,0\}$ and $\{+,-\}$. In the subsequent discussion, we collectively denote these six sets as $\Omega$. Put formally, we have $\Omega =2^{\{+,-,0\}}-\{+,-,0\}-Ø$.

Definition 1.

A formal context for Russian Roulette is a quadruple $K=(Q,T,\Omega ,I)$, where Q and T are two non-empty finite sets and $I:Q\times T\to \Omega$.

For simplicity, we call the above-defined formal context a Russian Roulette formal context, and a $2R$ formal context for short. Here, the first R represents Russian and the second R indicates roulette.

A $2R$ formal context is essentially a six-valued formal context and its complement is defined as follows.

Definition 2.

Let $K=(Q,T,\Omega ,I)$. For $q\in Q$ and $t\in T$, the negation of $I(q,t)$ is defined as

$$I^c(q,t)=\{+,-,0\}-I(q,t).$$

Moreover, $K^c=(Q,T,\Omega ,I^c)$ is called the complement context of K.

It is easy to show that the complement context of a $2R$ formal context is also a $2R$ formal context.

Example 1.

A $2R$ formal context is shown in

Table 1. The $2R$ formal context K of Example 1.

	a	b	c	d
1	$\{+\}$	$\{+\}$	$\{-,0\}$	$\{+,0\}$
2	$\{-,0\}$	$\{+\}$	$\{+,-\}$	$\{+,-\}$
3	$\{+,-\}$	$\{-\}$	$\{+\}$	$\{-\}$

and its complement is shown in

Table 2. The complement context $K^c$ of Example 1.

	a	b	c	d
1	$\{-,0\}$	$\{-,0\}$	$\{+\}$	$\{-\}$
2	$\{+\}$	$\{-,0\}$	$\left\{0\right\}$	$\left\{0\right\}$
3	$\left\{0\right\}$	$\{+,0\}$	$\{0,-\}$	$\{0,+\}$

.

Qi et al. [42] presented a 3-valued formal context. As an example,

Table 3. A 3-valued formal context.

	a	b	c	d	e
1	-	+	+	+	+
2	+	0	-	-	-
3	+	-	-	-	0
4	0	-	-	0	-
5	+	-	-	-	-
6	0	+	-	0	+

shows a 3-valued formal context, which can be seen as a special case of a $2R$ formal context in essence. However, it is clear that the complement cannot be derived in the settings of 3-valued formal contexts.

Definition 3.

Let $K=(Q,T,\Omega ,I)$. For $E\in 2^Q$ and $P\in 2^T$, two associated operators $\{\ast ,?\}:2^Q\to 2^T$ and $\{\ast ,?\}:2^T\to 2^Q$ are defined as

$$E^{\{\ast ,?\}}=\{t\in T\mid \forall q\in E,\exists ?\in I(q,t)\},$$

$$P^{\{\ast ,?\}}=\{q\in Q\mid \forall t\in P,\exists ?\in I(q,t)\},$$

where $?\in \{+,-,0\}$.

At first glance, ‘$\{\ast ,?\}$’ has been defined by Qi [42] in another format denoted by ‘#’. Actually, they are totally different from each other. Semantically, if an object q possesses an attribute t with a value ?, then the attribute t may be involved in the common features of E, i.e., $E^{\{\ast ,?\}}$. In other words, $I(q,t)$ is not necessarily equal to ?. Alternatively, $I(q,t)$ containing the element ? is sufficient.

For example, considering the $2R$ formal context in Table 1, we have ${\{2,3\}}^{\{\ast ,+\}}=\left\{c\right\}$. However, if we follow the idea of 3-way concept analysis [42], it turns out to be ${\{2,3\}}^{+}=Ø$ as $R(2,c)\ne R(3,c)$.

Proposition 1.

If E, $E_1$, $E_2\subseteq Q$ and P, $P_1$, $P_2\subseteq T$, then the following properties hold.

(i) $E\subseteq E^{\{\ast ,?\}\{\ast ,?\}}$; $P\subseteq P^{\{\ast ,?\}\{\ast ,?\}}$.

(ii) $E_1\subseteq E_2\Rightarrow E_2^{\{\ast ,?\}}\subseteq E_1^{\{\ast ,?\}}$; $P_1\subseteq P_2\Rightarrow P_2^{\{\ast ,?\}}\subseteq P_1^{\{\ast ,?\}}$.

(iii) $E^{\{\ast ,?\}}=E^{\{\ast ,?\}\{\ast ,?\}\{\ast ,?\}}$; $P^{\{\ast ,?\}}=P^{\{\ast ,?\}\{\ast ,?\}\{\ast ,?\}}$.

(iv) $E\subseteq P^{\{\ast ,?\}}\iff P\subseteq E^{\{\ast ,?\}}$.

(v) ${(E_1\cup E_2)}^{\{\ast ,?\}}=E_1^{\{\ast ,?\}}\cap E_2^{\{\ast ,?\}}$; ${(P_1\cup P_2)}^{\{\ast ,?\}}=P_1^{\{\ast ,?\}}\cap P_2^{\{\ast ,?\}}$.

(vi) ${(E_1\cap E_2)}^{\{\ast ,?\}}\supseteq E_1^{\{\ast ,?\}}\cup E_2^{\{\ast ,?\}}$; ${(P_1\cap P_2)}^{\{\ast ,?\}}\supseteq P_1^{\{\ast ,?\}}\cup P_2^{\{\ast ,?\}}$.

Definition 4.

Let $K=(Q,T,\Omega ,I)$, $E\in 2^Q$ and $P\in 2^T$. If $E^{\{\ast ,?\}}=P$ and $P^{\{\ast ,?\}}=E$; then $(E,P)$ is called a $2R$ concept.

Moreover, we use $L2R\left(K\right)$ to denote the set of $2R$ concepts and define a binary relation between $2R$ concepts as

$$(E_1,P_1)\le (E_2,P_2)\iff E_1\subseteq E_2\iff P_1\supseteq P_2.$$

Theorem 1.

$L2R\left(K\right)$ is a complete lattice and for $(E_1,P_1)$, $(E_2,P_2)\in L2R\left(K\right)$; it follows that

$$(E_1,P_1)\wedge (E_2,P_2)=(E_1\cap E_2,{(P_1\cup P_2)}^{\{\ast ,?\}\{\ast ,?\}}),$$

$$(E_1,P_1)\vee (E_2,P_2)=({(E_1\cup E_2)}^{\{\ast ,?\}\{\ast ,?\}},P_1\cap P_2).$$

Hereinafter, we call $L2R\left(K\right)$ a $2R$ concept lattice of K. For the sake of a clear discussion, if ? of concept-forming operator $\{\ast ,?\}$ is set to +, then we call the derived $2R$ concepts positive $2R$ concepts. Analogously, negative $2R$ concepts and zero $2R$ concepts can be obtained. As a consequence, for a $2R$ formal context K, we have positive $2R$ concept lattice, negative $2R$ concept lattice and zero $2R$ concept lattice and denote them as $PL2R\left(K\right)$, $NL2R\left(K\right)$ and $ZL2R\left(K\right)$, respectively.

Example 2.

The $2R$ formal context K in

Table 4. The $2R$ formal context K of Example 2.

	a	b	c	d	e
1	$\{+\}$	$\{+\}$	$\{-,0\}$	$\{+,0\}$	$\{+,-\}$
2	$\{-,0\}$	$\{+\}$	$\{+,-\}$	$\{+,-\}$	$\{-,0\}$
3	$\{+,-\}$	$\{-,0\}$	$\{+\}$	$\{-\}$	$\{+\}$
4	$\{+\}$	$\{+,-\}$	$\{+,0\}$	$\{-,0\}$	$\{-\}$
5	$\{+,-\}$	$\{+\}$	$\{-,0\}$	$\{+\}$	$\{+,-\}$

describes five gamblers in a Russian Roulette game. Each gambler has tried five times, represented, respectively, by a, b, c, d and e.

Hereinafter, for simplicity, we omit the brackes and commas when denoting a concept in a figure. For example, we adopt $(1245,b)$ to indicate a $2R$ concept $\left(\right\{1,2,4,5\},\{b\left\}\right)$.

Figure 1 shows the positive $2R$ concept lattice $PL2R\left(K\right)$, based on which we can find which gamblers prefer the choice ‘+’ in each trial. For instance, the concept $\left(\right\{1,2,4,5\},\{b\left\}\right)$ manifests that in the second trial, i.e., trial b, players 1, 2, 4 and 5 have chosen ‘+’.

3.2. 2R Concepts with a Possibility Theoretic View

Let $K=(Q,T,\Omega ,I)$. For $q\in Q$ and $t\in T$, we define

$$f(q,?)=\{t\in T\mid \exists ?\in I(q,t\left)\right\},$$

$$f(t,?)=\{q\in Q\mid \exists ?\in I(q,t\left)\right\}.$$

Definition 5.

Let $K=(Q,T,\Omega ,I)$, $E\in 2^Q$ and $P\in 2^T$. Two associated operators $\{\square ,?\}:2^Q\to 2^T$ and $\{\diamond ,?\}:2^T\to 2^Q$ are defined as

$$E^{\{\square ,?\}}=\{t\in T\mid f(t,?)\subseteq E\},$$

$$P^{\{\diamond ,?\}}=\{q\in Q\mid f(q,?)\cap P\ne Ø\},$$

where $?\in \{+,-,0\}$.

Moreover, if $E^{\{\square ,?\}}=P$ and $P^{\{\diamond ,?\}}=E$, we call $(E,P)$ an object-oriented $2R$ concept, $O2R$ concept for short.

In addition, the set of $O2R$ concepts is denoted as $OL2R\left(K\right)$ and we call it the object-oriented $2R$ concept lattice of K, $O2R$ concept lattice for short, and the infimum and supremum in $OL2R\left(K\right)$ are, respectively, defined as

$$(E_1,P_1)\wedge (E_2,P_2)=({(E_1\cap E_2)}^{\{\square ,?\}\{\diamond ,?\}},P_1\cap P_2),$$

$$(E_1,P_1)\vee (E_2,P_2)=(E_1\cup E_2,{(P_1\cup P_2)}^{\{\diamond ,?\}\{\square ,?\}}).$$

Definition 6.

Let $K=(Q,T,\Omega ,I)$, $E\in 2^Q$ and $P\in 2^T$. Two operators $\{\diamond ,?\}:2^Q\to 2^T$ and $\{\square ,?\}:2^T\to 2^Q$ are defined as

$$E^{\{\diamond ,?\}}=\{t\in T\mid f(t,?)\cap E\ne Ø\},$$

$$P^{\{\square ,?\}}=\{q\in Q\mid f(q,?)\subseteq P\},$$

where $?\in \{+,-,0\}$.

Moreover, if $E^{\{\diamond ,?\}}=P$ and $P^{\{\square ,?\}}=E$, we call $(E,P)$ a property-oriented $2R$ concept, $P2R$ concept for short, and denote the set of $P2R$ concepts by $PL2R\left(K\right)$.

In addition, we call $PL2R\left(K\right)$ the property-oriented $2R$ concept lattice of K, $P2R$ concept lattice for short, and the infimum and supremum in $PL2R\left(K\right)$ are, respectively, given by

$$(E_1,P_1)\wedge (E_2,P_2)=(E_1\cap E_2,{(P_1\cap P_2)}^{\{\diamond ,?\}\{\square ,?\}}),$$

$$(E_1,P_1)\vee (E_2,P_2)=({(E_1\cup E_2)}^{\{\square ,?\}\{\diamond ,?\}},P_1\cup P_2).$$

Definition 7.

Let $K=(Q,T,\Omega ,I)$, $E\in 2^Q$ and $P\in 2^T$. Two operators $\{\#,?\}:2^Q\to 2^T$ and $\{\#,?\}:2^T\to 2^Q$ are defined as

$$E^{\{\#,?\}}=\{t\in T\mid \exists q\in E^c,?\notin I(q,t)\},$$

$$P^{\{\#,?\}}=\{q\in Q\mid \exists t\in P^c,?\notin I(q,t)\},$$

where $?\in \{+,-,0\}$.

Moreover, if $E^{\{\#,?\}}=P$ and $P^{\{\#,?\}}=E$, we call $(E,P)$ a dual $2R$ concept, $D2R$ concept for short, and denote the set of $D2R$ concepts by $DL2R\left(K\right)$.

In addition, we call $DL2R\left(K\right)$ a dual $2R$ concept lattice, $D2R$ concept lattice for short, and the infimum and supremum in $DL2R\left(K\right)$ are, respectively, defined as

$$(E_1,P_1)\wedge (E_2,P_2)=({(E_1\cap E_2)}^{\{\#,?\}\{\#,?\}},P_1\cup P_2),$$

$$(E_1,P_1)\vee (E_2,P_2)=(E_1\cup E_2,{(P_1\cap P_2)}^{\{\#,?\}\{\#,?\}}).$$

Proposition 2.

Let $E\subseteq Q$, $P\subseteq T$ and $?\in \{+,-,0\}$. The following properties hold.

(i) $E^{\{\ast ,?\}}=E^{c\{\#,?\}c}$; $T^{\{\ast ,?\}}=T^{c\{\#,?\}c}$.

(ii) $E_R^{\{\ast ,?\}}=E_{R^c}^{\{\diamond ,?\}c}$; $T_R^{\{\ast ,?\}}=T_{R^c}^{\{\diamond ,?\}c}$.

(iii) $E_R^{\{\ast ,?\}}=E_{R^c}^{c\{\square ,?\}}$; $T_R^{\{\ast ,?\}}=T_{R^c}^{c\{\square ,?\}}$.

(iv) $E^{\{\diamond ,?\}c}=E^{c\{\square ,?\}}$, $E^{\{\square ,?\}c}=E^{c\{\diamond ,?\}}$; $T^{\{\diamond ,?\}c}=T^{c\{\square ,?\}}$, $T^{\{\square ,?\}c}=T^{c\{\diamond ,?\}}$.

Proof. 

(i)

$$\begin{array}{c}{E}^{c\{\#,?\}c}\hfill \\ ={\{t\in T\mid \exists q\in E,?\notin I(q,t)\}}^c\hfill \\ =\{t\in T\mid \forall q\in E,?\in I(q,t\left)\right\}\hfill \\ =E^{\{\ast ,?\}}.\hfill \end{array}$$

(ii)

$$\begin{array}{c}{E}_{R^c}^{\{\diamond ,?\}c}\hfill \\ ={\{t\in T\mid f_{I^c}(t,?)\cap E\ne Ø\}}^c\hfill \\ =\{t\in T\mid f_{I^c}(t,?)\cap E=Ø\}\hfill \\ =\{t\in T\mid \forall q\in E,?\in I(q,t\left)\right\}\hfill \\ =E_I^{\{\ast ,?\}}.\hfill \end{array}$$

(iii)

$$\begin{array}{c}{E}_{I^c}^{c\{\square ,?\}}\hfill \\ =\{t\in T\mid f_{I^c}(t,?)\subseteq E^c\}\hfill \\ =\{t\in T\mid E\subseteq {\left(f_{I^c}(t,?)\right)}^c\}\hfill \\ =\{t\in T\mid E\subseteq f(t,?\left)\right\}\hfill \\ =\{t\in T\mid \forall q\in E,?\in I(q,t\left)\right\}\hfill \\ =E_I^{\{\ast ,?\}}.\hfill \end{array}$$

(iv) By (ii) and (iii), it follows that $E^{\{\diamond ,?\}c}=E^{c\{\square ,?\}}=E_{I^c}^{c\{\ast ,?\}}$. Moreover, let $F=E^c$. Then, $F^{c\{\diamond ,?\}c}=F^{\{\square ,?\}}$, which leads to $F^{c\{\diamond ,?\}}=F^{\{\square ,?\}c}$.

The rest can be proved by the duality principle.    □

Theorem 2.

Let $K=(Q,T,\Omega ,I)$. For $?\in \{+,-,0\}$, $E\in 2^Q$ and $P\in 2^T$, the following propositions are equivalent.

(i) $(E,P)$ is a $2R$ concept of K.

(ii) $(E^c,P^c)$ is a dual $2R$ concept of K.

(iii) $(E^c,P)$ is an object-oriented $2R$ concept of $K^c$.

(iv) $(E,P^c)$ is a property-oriented $2R$ concept of $K^c$.

Proof. 

(i) ⇔ (ii): by Proposition 2 (i), we obatin

$$\begin{array}{c}(E,P) \mathrm{is} \mathrm{a} 2R \mathrm{concept} \mathrm{of} K\hfill \\ \iff E^{c\{\#,?\}c}=P \mathrm{and} P^{c\{\#,?\}c}=E\hfill \\ \iff E^{c\{\#,?\}}=P^c \mathrm{and} P^{c\{\#,?\}}=E^c\hfill \\ \iff (E^c,P^c) \mathrm{is} \mathrm{a} \mathrm{dual} 2R \mathrm{concept} \mathrm{of} K.\hfill \end{array}$$

(i) ⇔ (iii): by Proposition 2 (ii) and (iii), we obatin

$$\begin{array}{c}(E,P) \mathrm{is} \mathrm{a} 2R \mathrm{concept} \mathrm{of}K\hfill \\ \iff E_I^{\{\ast ,?\}}=P \mathrm{and} P_I^{\{\ast ,?\}}=E\hfill \\ \iff E_{I^c}^{c\{\square ,?\}}=P \mathrm{and} P_{I^c}^{\{\diamond ,?\}c}=E\hfill \\ \iff E_{I^c}^{c\{\square ,?\}}=P \mathrm{and} P_{I^c}^{\{\diamond ,?\}}=E^c\hfill \\ \iff (E^c,P) \mathrm{is} \mathrm{an} \text{object-oriented} 2R \mathrm{concept} \mathrm{of} K.\hfill \end{array}$$

(iii) ⇔ (iv): by Proposition 1 (iv), we obatin

$$\begin{array}{c}(E^c,P) \mathrm{is} \mathrm{an} \text{object-oriented} 2R \mathrm{concept} \mathrm{of} K^c\hfill \\ \iff E^{c\{\square ,?\}}=P \mathrm{and} P^{\{\diamond ,?\}}=E^c\hfill \\ \iff E^{\{\diamond ,?\}c}=P \mathrm{and} P^{\{\diamond ,?\}c}=E\hfill \\ \iff E^{\{\diamond ,?\}}=P^c \mathrm{and} P^{c\{\square ,?\}}=E\hfill \\ \iff (E,P^c) \mathrm{is} \mathrm{a} \text{property-oriented} 2R \mathrm{concept} \mathrm{of} K^c.\hfill \end{array}$$

   □

The comparisons between $2R$ formal contexts and 3-valued formal contexts are summarized in

Table 5. The uniqueness of $2R$ formal contexts compared with 3-valued formal contexts.

	$2\mathit{R}$ Formal Contexts	3-Valued Formal Contexts
Can we define complement contexts?	Yes	No
Can we define classical concepts?	Yes	Yes
Can we define object-oriented concepts?	Yes	Yes
Can we define property-oriented concepts?	Yes	Yes
Can we define dual concepts?	Yes	Yes
Are there connections among the defined concepts?	Yes	No

. In fact, the main reason for these differences is that the complement of a 3-valued formal context cannot be defined. As a consequence, it is impossible to relate formal concepts with object-oriented concepts, property-oriented concepts and dual concepts via its complement.

4. Concept Analysis for 2R Formal Contexts

In this section, we propose 3-valued $2R$ concepts and discuss their properties and the connections with the basic $2R$ concepts.

Definition 8.

Let $K=(Q,T,\Omega ,I)$. For $E\in 2^Q$ and $(M,N,O)\in 2^T\times 2^T\times 2^T$, two associated 3-valued operators $\alpha :2^Q\to 2^T\times 2^T\times 2^T$ and $\beta :2^T\times 2^T\times 2^T\to 2^Q$ are defined as

$$E^{\alpha }=(E^{\{\ast ,+\}},E^{\{\ast ,-\}},E^{\{\ast ,0\}}),$$

$${(M,N,O)}^{\beta }=M^{\{\ast ,+\}}\cap N^{\{\ast ,-\}}\cap O^{\{\ast ,0\}}.$$

Let $(M_1,N_1,O_1),(M_2,N_2,O_2)\subseteq T\times T\times T$. Then, $(M_1,N_1,O_1)\subseteq (M_2,N_2,O_2)$ if and only if $M_1\subseteq M_2,N_1\subseteq N_2,O_1\subseteq O_2$.

Proposition 3.

If E, $E_1$, $E_2\subseteq Q$ and S, ${\mathbf{S}}_1$, ${\mathbf{S}}_2\subseteq T\times T\times T$, then the following properties hold.

(i) $E\subseteq E^{\alpha \beta }$; $\mathbf{S}\subseteq {\mathbf{S}}^{\beta \alpha }$.

(ii) $E_1\subseteq E_2\Rightarrow E_2^{\alpha }\subseteq E_1^{\alpha }$; ${\mathbf{S}}_1\subseteq {\mathbf{S}}_2\Rightarrow {\mathbf{S}}_2^{\beta }\subseteq {\mathbf{S}}_1^{\beta }$.

(iii) $E^{\alpha }=E^{\alpha \beta \alpha }$; ${\mathbf{S}}^{\beta }={\mathbf{S}}^{\beta \alpha \beta }$.

(iv) $E\subseteq {\mathbf{S}}^{\beta }\iff \mathbf{S}\subseteq E^{\alpha }$.

(v) ${(E_1\cup E_2)}^{\alpha }=E_1^{\alpha }\cap E_2^{\alpha }$; ${({\mathbf{S}}_1\cup {\mathbf{S}}_2)}^{\beta }={\mathbf{S}}_1^{\beta }\cap {\mathbf{S}}_2^{\beta }$.

(vi) ${(E_1\cap E_2)}^{\alpha }\supseteq E_1^{\alpha }\cup E_2^{\alpha }$; ${({\mathbf{S}}_1\cap {\mathbf{S}}_2)}^{\beta }\supseteq {\mathbf{S}}_1^{\beta }\cup {\mathbf{S}}_2^{\beta }$.

Definition 9.

Let $K=(Q,T,\Omega ,I)$, $E\in 2^Q$ and $(M,N,O)\in 2^T\times 2^T\times 2^T$. If $E^{\alpha }=(M,N,O)$ and ${(M,N,O)}^{\beta }=E$, we call $(E,(M,N,O\left)\right)$ a 3-valued $2R$ concept.

The set of 3-valued $2R$ concepts is denoted by $3VL2R\left(K\right)$ and a partial order between any two 3-valued $2R$ concepts in $3VL2R\left(K\right)$ is defined as

$$(E_1,(M_1,N_1,O_1))\le (E_2,(M_2,N_2,O_2))\iff E_1\subseteq E_2\iff (M_1,N_1,O_1)\supseteq (M_2,N_2,O_2).$$

Theorem 3.

$3VL2R\left(K\right)$ is a complete lattice and for $(E_1,(M_1,N_1,O_1))$, $(E_2,(M_2,N_2,O_2))\in 3VL2R\left(K\right)$, it follows that

$$(E_1,(M_1,N_1,O_1))\wedge (E_2,(M_2,N_2,O_2))=(E_1\cap E_2,{((M_1,N_1,O_1)\cup (M_2,N_2,O_2))}^{\beta \alpha }),$$

$$(E_1,(M_1,N_1,O_1))\vee (E_2,(M_2,N_2,O_2))=({(E_1\cup E_2)}^{\alpha \beta },(M_1,N_1,O_1)\cap (M_2,N_2,O_2)).$$

Hereinafter, we call $3VL2R\left(K\right)$ the 3-valued $2R$ concept lattice of K.

In what follows, Algorithm 1 describes the process of building the 3-valued $2R$ concept lattice based on an incremental strategy. The main idea of algorithm 1 adopts an incremental manner to construct the 3-valued $2R$ concept lattice. Concretely, we first initialize the 3-valued $2R$ concept lattice to be empty and then for each object of the $2R$ formal context, we dynamically adjust the lattice until all the objects are considered. Lines 5 to 13 describe the procedure for updating the current lattice when dealing with a new object, which can be divided into the following two steps.

Step 1: Sort $3VL2R\left(K\right)$ in descending order in terms of the sizes of the extents of each 3-valued $2R$ concept.

Step 2: For each 3-valued $2R$ concept, perform necessary changes and generate new 3-valued $2R$ concept as follows.

(i) If $(M,N,O)\cap {\left\{q\right\}}^{\alpha }=(M,N,O)$, then add $(E\cup \{q\},(M,N,O\left)\right)$ into $3VL2R\left(K\right)$.

(ii) If $(M,N,O)\cap {\left\{q\right\}}^{\alpha }\ne (M,N,O)$ and there is no 3-valued $2R$ concept with an intent $(M,N,O)\cap {\left\{q\right\}}^{\alpha }$, then add $(E\cup \left\{q\right\},(M,N,O)\cap {\left\{q\right\}}^{\alpha })$ into $3VL2R\left(K\right)$.

Algorithm 1 Algorithm for building 3-valued $2R$ concept lattice

Require: $K=(Q,T,\Omega ,I)$. Ensure: $3VL2R\left(K\right)$.   1: Initialized $3VL2R\left(K\right)=Ø$.   2: Fetch an object q from Q and put $(\left\{q\right\},{\left\{q\right\}}^{\alpha })$ and $(Ø,(T,T,T\left)\right)$ into $3VL2R\left(K\right)$.   3: For each object q in Q   4:      Sort $3VL2R\left(K\right)$ in descending order in terms of the sizes of the extents of each 3-valued $2R$ concept.   5:      For each $(E,(M,N,O\left)\right)$ in $3VL2R\left(K\right)$   6:           If $(M,N,O)\cap {\left\{q\right\}}^{\alpha }=(M,N,O)$   7:                add $(E\cup \{q\},(M,N,O\left)\right)$ into $3VL2R\left(K\right)$;   8:           Else   9:                If there is no 3-valued $2R$ concept with an intent $(M,N,O)\cap {\left\{q\right\}}^{\alpha }$, 10:                     add $(E\cup \left\{q\right\},(M,N,O)\cap {\left\{q\right\}}^{\alpha })$ into $3VL2R\left(K\right)$. 11:                End If 12:           End If 13:      End For 14: End For 15: Return $3VL2R\left(K\right)$ and end the algorithm.

Example 3.

The $3VL2R\left(K\right)$ of the $2R$ formal context K in Table 4 is shown in Figure 2.

For the sake of discussion, given a $2R$ formal context $K=(U,V,\Omega ,R)$, we collectively call the positive $2R$ concepts, negative $2R$ concepts and zero $2R$ concepts basic $2R$ concepts of K.

Theorem 4.

Let $K=(Q,T,\Omega ,I)$ and $(E,(M,N,O\left)\right)$ be a 3-valued $2R$ concept of K. Then, the following propositions hold.

(i) $(M^{\{\ast ,+\}},M)$ is a positive $2R$ concept.

(ii) $(N^{\{\ast ,-\}},N)$ is a negative $2R$ concept.

(iii) $(O^{\{\ast ,0\}},O)$ is a zero $2R$ concept.

Proof. 

(i) As $(E,(M,N,O\left)\right)$ is a 3-valued $2R$ concept, we have $E\ast \{\ast ,+\}=M$. Then, $(M^{\{\ast ,+\}},M)=(E^{\{\ast ,+\}\{\ast ,+\}},E^{\{\ast ,+\}})$, which means that $(M^{\{\ast ,+\}},M)$ is a positive $2R$ concept.

In addition, items (ii) and (iii) can be analogously proved.    □

Definition 10.

Let $K=(Q,T,\Omega ,I)$, $(E,M),(E^{\prime },M^{\prime })\in PL2R\left(K\right)$, $(F,N),(F^{\prime },N^{\prime })\in NL2R\left(K\right)$ and $(G,O),(G^{\prime },O^{\prime })\in ZL2R\left(K\right)$. If $E\cap F\cap G=E^{\prime }\cap F^{\prime }\cap G^{\prime }=H$, then denote this case as $((E,M),(F,N),(G,O))\sim ((E^{\prime },M^{\prime }),(F^{\prime },N^{\prime }),(G^{\prime },O^{\prime }))$ and denote the corresponding equivalence class by ${\left[H\right]}_{\sim }$.

Theorem 5.

$\left(\right(E,M),(F,N),(G,O\left)\right)$ is the least element of ${\left[H\right]}_{\sim }$, if and only if ${(E\cap F\cap G)}^{\{\ast ,+\}}=M$, ${(E\cap F\cap G)}^{\{\ast ,-\}}=N$ and ${(E\cap F\cap G)}^{\{\ast ,0\}}=O$.

Proof. 

“⇒”. As $H\subseteq E$, $E^{\{\ast ,+\}\{\ast ,+\}}=E$, it follows that $H^{\{\ast ,+\}\{\ast ,+\}}\subseteq E$. Similarly, we also have $H^{\{\ast ,-\}\{\ast ,-\}}\subseteq F$ and $H^{\{\ast ,0\}\{\ast ,0\}}\subseteq G$. Therefore, $H^{\{\ast ,+\}\{\ast ,+\}}\cap H^{\{\ast ,-\}\{\ast ,-\}}\cap H^{\{\ast ,0\}\{\ast ,0\}}\subseteq E\cap F\cap G=H$. In addition, by the properties of the operators, we have $H^{\{\ast ,+\}\{\ast ,+\}}\supseteq H$, $H^{\{\ast ,-\}\{\ast ,-\}}\supseteq H$ and $H^{\{\ast ,0\}\{\ast ,0\}}\supseteq H$. To sum up, we can conclude that $H^{\{\ast ,+\}\{\ast ,+\}}\cap H^{\{\ast ,-\}\{\ast ,-\}}\cap H^{\{\ast ,0\}\{\ast ,0\}}=H$, which further implies that

$$((H^{\{\ast ,+\}\{\ast ,+\}},H^{\{\ast ,+\}}),(H^{\{\ast ,-\}\{\ast ,-\}},H^{\{\ast ,0\}}),(H^{\{\ast ,0\}\{\ast ,0\}},H^{\{\ast ,0\}}))\in {\left[H\right]}_{\sim }.$$

Moreover, by the fact that $H^{\{\ast ,+\}\{\ast ,+\}}\subseteq E$, $H^{\{\ast ,-\}\{\ast ,-\}}\subseteq F$, $H^{\{\ast ,0\}\{\ast ,0\}}\subseteq G$, we can derive

$$((H^{\{\ast ,+\}\{\ast ,+\}},H^{\{\ast ,+\}}),(H^{\{\ast ,-\}\{\ast ,-\}},H^{\{\ast ,0\}}),(H^{\{\ast ,0\}\{\ast ,0\}},H^{\{\ast ,0\}}))\le ((E,M),(F,M),(G,O)).$$

By the condition that $\left(\right(E,M),(F,M),(G,O\left)\right)$ is the least element of ${\left[H\right]}_{\sim }$, we can conclude that $H^{\{\ast ,+\}}=M$, $H^{\{\ast ,-\}}=N$ and $H^{\{\ast ,0\}}=O$.

“⇐”. For any $((E^{\prime },M^{\prime }),(F^{\prime },N^{\prime }),(G^{\prime },O^{\prime }))\in {\left[H\right]}_{\sim }$, we have $E^{\prime }\cap F^{\prime }\cap G^{\prime }=H$, by which we can derive $M^{\prime }=E^{\prime \{\ast ,+\}}\subseteq {(E^{\prime }\cap F^{\prime }\cap G^{\prime })}^{\{\ast ,+\}}=H^{\{\ast ,+\}}=M$. Similarly, we can show $N^{\prime }\subseteq N$ and $O^{\prime }\subseteq O$. Then, it follows that $(E^{\prime },M^{\prime })\ge (E,M)$, $(F^{\prime },M^{\prime })\ge (F,M)$ and $(G^{\prime },O^{\prime })\ge (G,O)$; i.e., $\left(\right(E,M),(F,N),(G,O\left)\right)$ is the least element of ${\left[H\right]}_{\sim }$.    □

Theorem 6.

Let $K=(Q,T,\Omega ,I)$ and Δ be the collection of the least elements in $PL2R\left(K\right)\times NL2R\left(K\right)\times ZL2R\left(K\right)/\sim$. Then,

$$3VL2R\left(K\right)=\left\{\right(E\cap F\cap G,(M,N,O))\mid ((E,M),(F,N),(G,O))\in \Delta \}.$$

Proof. 

Let $\left\{\right(E\cap F\cap G,(M,N,O))\mid ((E,M),(F,N),(G,O))\in \Delta \}=\Pi$.

On one hand, let $(E,(M,N,O\left)\right)\in 3VL2R\left(K\right)$. We have $(M^{\{\ast ,+\}},M)\in PL2R\left(K\right)$, $(N^{\{\ast ,-\}},N)\in NL2R\left(K\right)$ and $(O^{\{\ast ,0\}},O)\in ZL2R\left(K\right)$. Moreover, as ${(M^{\{\ast ,+\}}\cap N^{\{\ast ,-\}}\cap O^{\{\ast ,0\}})}^{\{\ast ,+\}}=E^{\{\ast ,+\}}=M$, ${(M^{\{\ast ,+\}}\cap N^{\{\ast ,-\}}\cap O^{\{\ast ,0\}})}^{\{\ast ,-\}}=E^{\{\ast ,-\}}=N$ and ${(M^{\{\ast ,+\}}\cap N^{\{\ast ,-\}}\cap O^{\{\ast ,0\}})}^{\{\ast ,0\}}=E^{\{\ast ,0\}}=O$, by Theorem 3, we have

$$((M^{\{\ast ,+\}},M),(N^{\{\ast ,-\}},N),(O^{\{\ast ,0\}},O))\in \Pi .$$

By the fact that $(E,(M,N,O))=(M^{\{\ast ,+\}}\cap N^{\{\ast ,-\}}\cap O^{\{\ast ,0\}},(M,N,O))$, we can conclude $(E,(M,N,O\left)\right)\in \Pi$, which implies $3VL2R\left(K\right)\subseteq \Pi$.

On the other hand, let $\left(\right(E,M),(F,N),(G,O\left)\right)\in \Delta$. By Theorem 3, we have ${(E\cap F\cap G)}^{\{\ast ,+\}}=M$, ${(E\cap F\cap G)}^{\{\ast ,-\}}=N$ and ${(E\cap F\cap G)}^{\{\ast ,0\}}=O$. Then, by the definition of a 3-valued $2R$ concept, $(E\cap F\cap G,(M,N,O\left)\right)\in 3VL2R\left(K\right)$ is at hand, which implies $\Pi \subseteq 3VL2R\left(K\right)$.

Then, this theorem is proved.    □

The above theorems actually indicate the strong connections between 3-valued $2R$ concepts and basic $2R$ concepts. That is, Theorem 4 implies that we can derive basic $2R$ concepts from 3-valued $2R$ concepts, while Theorem 6 states a 3-valued $2R$ concept lattice can be derived by using its basic $2R$ concept lattices.

5. Experimental Analysis

In this section, some experiments will be conducted to derive several important conclusions.

5.1. Testing on Synthetic Data Sets

This subsection discusses the influences of the number of objects, the number of attributes and the fill ratios on the sizes of 3-valued $2R$ concept lattices and the basic $2R$ concept lattices.

In the experiments, Algorithm 2 derives the synthetic $2R$ formal contexts.

Algorithm 2 Generating a synthetic $2R$ formal context

Require: Q, T, two ratios $r_1:r_2:r_3$ and $r_1^{\prime }:r_2^{\prime }$ and a possibility p. Ensure: $K=(Q,T,\Omega ,I)$.   1: For each pair of object $q\in Q$ and attribute $t\in T$   2:      Initialize $I(q,t)=Ø$ and a set $S=\{+,-,0\}$.   3:      Fetch one element $s_1$ from S with a possibility ratio $r_1:r_2:r_3$.   4:      $S\leftarrow S-\left\{s_1\right\}$.   5:      Add $s_1$ into $I(q,t)$.   6:      Fetch one element $s_2$ from S with a possibility ratio $r_1^{\prime }:r_2^{\prime }$.   7:      Add $s_2$ into $I(q,t)$ with a possibility p.   8: End For   9: Return $K=(Q,T,\Omega ,I)$ and end the algorithm.

(i) The influences caused by the number of objects

By setting $\left|T\right|=25$, $r_1:r_2:r_3=1:1:1$, $r_1^{\prime }:r_2^{\prime }=1:1$ and $p=0.5$ and changing the size of Q from 20 to 40, we can derive a family of randomly generated $2R$ formal context, which have 25 attributes, but different numbers of objects. For each generated $2R$ formal context, we record the number of 3-valued $2R$ concepts and basic $2R$ concepts. Then, we show the experimental results in Figure 3.

(ii) The influences caused by the number of attributes

Let $\left|Q\right|=35$, $r_1:r_2:r_3=1:1:1$, $r_1^{\prime }:r_2^{\prime }=1:1$ and $p=0.5$. By changing $\left|T\right|$ from 20 to 40, we can derive a family of randomly generated $2R$ formal contexts, which has 35 objects but different numbers of attributes. By recording the number of 3-valued $2R$ concepts and basic $2R$ concepts, we show the experimental results in Figure 4.

(iii) The influences caused by different fill ratios

By fixing $\left|U\right|=\left|V\right|=30$ and setting five different fill ratios $1:1:1$, $1:1.5:2.25$, $1:2:4$, $1:2.5:6.25$ and $1:3:9$ for $r_1:r_2:r_3$, we randomly generate five different $2R$ formal contexts by Algorithm 2, namely $K_1$, $K_2$, $K_3$, $K_4$ and $K_5$, where for each $2R$ formal context $r_1^{\prime }:r_2^{\prime }=1:1$ and $p=0.5$. By recording the number of 3-valued $2R$ concepts and basic $2R$ concepts of the family of the randomly generated $2R$ formal contexts, we collectively list the experimental results in

Table 6. The experimental results with different fill ratios.

$2\mathit{R}$ Formal Context	${\mathit{K}}_1$	${\mathit{K}}_2$	${\mathit{K}}_3$	${\mathit{K}}_4$	${\mathit{K}}_5$
$\left|3VL2R\right(K\left)\right|$	65,374	404,874	121,637	122,478	173,146
$\left|PL2R\right(K\left)\right|$	6074	1146	838	651	591
$\left|NL2R\right(K\left)\right|$	3977	4284	2592	2434	2288
$\left|ZL2R\right(K\left)\right|$	3664	16,478	44,718	55,126	83,608
$|+|:|-|:\left|0\right|$	1:0.93:0.92	1:1.28:1.57	1:1.29:1.96	1:1.39:2.19	1:1.42:2.32

, where $\left|3VL2R\right(K\left)\right|$, $\left|PL2R\right(K\left)\right|$, $\left|NL2R\right(K\left)\right|$ and $\left|ZL2R\right(K\left)\right|$ represent the numbers of concepts of $3VL2R\left(K\right)$, $PL2R\left(K\right)$, $NL2R\left(K\right)$ and $ZL2R\left(K\right)$, respectively, and $|+|:|-|:\left|0\right|$ denotes the ratio between the number of +, − and 0 contained in the $2R$ formal contexts.

The following can be analyzed from the experimental results.

• There is a close correlation between $\left|Q\right|$ ($\left|T\right|$) and the size of 3-valued $2R$ concept lattices and the basic $2R$ concept lattices. The larger the values of $\left|Q\right|$ ($\left|T\right|$), the larger the size of the 3-valued $2R$ concept lattices and the basic $2R$ concept lattices.
• Given a $2R$ formal context, the sizes of basic $2R$ concept lattices are smaller than that of the 3-valued $2R$ concept lattice, which implies 3-valued $2R$ concept lattices embody some unique patterns compared with basic $2R$ concept lattices.
• When +, − and 0 are evenly filled, the $2R$ formal contexts contain the fewest 3-valued $2R$ concepts.

5.2. Case Study

In

Table 7. A survey on the clinical usage of three kinds of Chinese herbal medicines.

	${\mathit{p}}_1$	${\mathit{p}}_2$	${\mathit{p}}_3$	${\mathit{p}}_4$	${\mathit{p}}_5$	${\mathit{p}}_6$	${\mathit{p}}_7$	${\mathit{p}}_8$	${\mathit{p}}_9$	${\mathit{p}}_{10}$
$d_1$	$\{+\}$	$\{+\}$	$\{-,0\}$	$\{+\}$	$\{+,0\}$	$\{+\}$	$\{+\}$	$\{+,0\}$	$\{+,0\}$	$\{+,-\}$
$d_2$	$\{-,0\}$	$\{+\}$	$\{-,0\}$	$\{+,0\}$	$\{+,0\}$	$\{+\}$	$\{+\}$	$\{-,0\}$	$\left\{0\right\}$	$\{+,-\}$
$d_3$	$\{+\}$	$\{+\}$	$\{+,0\}$	$\{+,0\}$	$\{+,0\}$	$\{+,0\}$	$\{+,-\}$	$\{-,0\}$	$\left\{0\right\}$	$\{-\}$
$d_4$	$\{+,-\}$	$\{-,0\}$	$\{+,0\}$	$\{+,0\}$	$\{+,0\}$	$\{-,0\}$	$\{+,-\}$	$\{+,0\}$	$\left\{0\right\}$	$\{-,0\}$
$d_5$	$\{-,0\}$	$\{+,0\}$	$\{+,0\}$	$\{+,0\}$	$\{+,0\}$	$\{-,0\}$	$\{+,-\}$	$\{-,0\}$	$\{+,0\}$	$\{-\}$
$d_6$	$\{-,0\}$	$\{+,0\}$	$\{+,-\}$	$\{+,0\}$	$\{+,0\}$	$\{+,0\}$	$\{+,-\}$	$\{+,0\}$	$\{+,0\}$	$\{+,-\}$
$d_7$	$\{-\}$	$\{+,0\}$	$\{-\}$	$\{+\}$	$\{+,0\}$	$\{+,0\}$	$\{+,-\}$	$\{-,0\}$	$\{-,0\}$	$\{-\}$
$d_8$	$\{+,-\}$	$\{+\}$	$\{+\}$	$\{+\}$	$\{-,0\}$	$\{+,0\}$	$\{+,-\}$	$\{+,0\}$	$\{-,0\}$	$\{-,0\}$
$d_9$	$\{+,-\}$	$\{+\}$	$\{+,0\}$	$\{+,-\}$	$\left\{0\right\}$	$\{+,0\}$	$\{+,-\}$	$\{+,0\}$	$\left\{0\right\}$	$\{-,0\}$
$d_{10}$	$\{+\}$	$\{+,-\}$	$\{+,-\}$	$\{+,-\}$	$\{+,0\}$	$\{+,0\}$	$\{+,0\}$	$\{+,0\}$	$\left\{0\right\}$	$\{-,0\}$
$d_{11}$	$\{+,0\}$	$\{+,-\}$	$\{+,0\}$	$\{+,-\}$	$\{-,0\}$	$\{+,0\}$	$\{+,0\}$	$\{+,0\}$	$\left\{0\right\}$	$\{-,0\}$
$d_{12}$	$\{+,0\}$	$\{+,-\}$	$\{+,0\}$	$\{+,0\}$	$\{-,0\}$	$\{+,0\}$	$\{+,0\}$	$\left\{0\right\}$	$\{+,0\}$	$\{-,0\}$
$d_{13}$	$\{+\}$	$\{+,0\}$	$\{+,-\}$	$\{+,0\}$	$\{-,0\}$	$\{+,0\}$	$\{+,0\}$	$\left\{0\right\}$	$\{+,0\}$	$\left\{0\right\}$
$d_{14}$	$\{+,0\}$	$\{+,0\}$	$\{+\}$	$\{+,0\}$	$\{-\}$	$\{+\}$	$\{+,0\}$	$\left\{0\right\}$	$\{+,0\}$	$\left\{0\right\}$
$d_{15}$	$\{+,0\}$	$\{+,0\}$	$\{+\}$	$\{+,0\}$	$\{+,-\}$	$\{+\}$	$\{-,0\}$	$\{-,0\}$	$\{+,0\}$	$\{+,0\}$

, a survey on the usage of a type of Chinese herbal medicine is presented. In this table, there are 15 TCM doctors (denoted as $d_1$ to $d_{1}5$), who gave their suggested prescriptions independently to 10 patients (denoted as $p_1$ to $p_{1}0$). Concretely, +, − and 0, respectively, denote Aconiti Lateralis Radix Preparata, Aconitum Carmichaeli and Tianxiong. Any one or two of them can cure a certain disease, but all three are toxic to the body. For instance, from this table, it can be observed that doctor $d_1$ has prescribed Aconiti Lateralis Radix Preparata for patients $p_1$ and $p_2$ and has prescribed Aconitum Carmichaeli and Tianxiong for patient $p_3$.

In this survey, Aconiti Lateralis Radix Preparata appears 106 times, Aconitum Carmichaeli 55 times and Tianxiong 100 times. In addition, we can derive 406 3-valued $2R$ concepts from this dataset, and each 3-valued $2R$ concept denotes a meaningful pattern, based on which many useful pieces of information can be obtained.

For instance, there are 79 3-valued $2R$ concepts whose extents possess $u_1$. Among these 79 3-valued $2R$ concepts, the minimal one is $(\left\{d_1\right\},(\{p_1,p_2,p_4,p_5,p_6,p_7,p_8,p_9,p_{10}\},\{p_3,p_{10}\}$, $\{p_3,p_5,p_8,a_9\}\left)\right)$, which states that $d_1$ gives Aconiti Lateralis Radix Preparata for patients $p_1,p_2,p_4,p_5,p_6,p_7,p_8,p_9$ and $a_{10}$, Aconitum Carmichaeli for patients $p_3$ and $p_{10}$ and Tianxiong for patients $p_3,p_5,p_8$ and $p_9$ and the maximal one is $(\{u_1,\cdots ,u_{15}\},(\left\{p_4\right\},Ø,\{p_8,p_9\}))$, which implies that all the 15 TCM physicians agree that Aconiti Lateralis Radix Preparata is necessary for patient $p_4$ and Tianxiong for patients $p_8$ and $p_9$.

According to the prescriptions given by the TCM physicians, the consistency degree of an arbitrary group of TCM physicians can be obtained. Let $E\subseteq Q$, $\Delta =\{E\mid (E,E^{\alpha })\in 3VL2R\left(K\right)\}$ and $E^{\prime }\notin \Delta$. We define $g\left(E\right)=E^{\prime }$, where $E^{\prime }\in \Delta$ and there does not exist $F\in \Delta$ such that $E\subset F\subset E^{\prime }$.

Moreover, we define

$$f\left(E\right)=\{\begin{array}{cc}E,\hfill & \mathrm{if} E\in \Delta ,\hfill \\ g\left(E\right),\hfill & \mathrm{otherwise}.\hfill \end{array}$$

Then, the consistency degree $\eta \left(E\right)={\displaystyle \frac{{|f\left(E\right)}^{\{\ast ,+\}}{|+|f\left(E\right)}^{\{\ast ,-\}}{|+|f\left(E\right)}^{\{\ast ,0\}}|}{\underset{e\in f{\left(E\right)}^{\alpha }}{min}{\left(\right|\left\{e\right\}}^{\{\ast ,+\}}{|+|\left\{e\right\}}^{\{\ast ,-\}}{|+|\left\{e\right\}}^{\{\ast ,0\}}\left|\right)}}$.

By using the above formula, the consistency degrees of arbitrary cliques can be computed. For example, the consistency degrees of pairs that contain $u_1$ are collectively listed as follows:

$$\begin{array}{cc}\eta \left(\{d_1,d_2\}\right)={\displaystyle \frac{12}{15}},\hfill & \eta \left(\{d_1,d_3\}\right)={\displaystyle \frac{11}{15}},\hfill \\ \eta \left(\{d_1,d_4\}\right)={\displaystyle \frac{10}{15}},\hfill & \eta \left(\{d_1,d_5\}\right)={\displaystyle \frac{10}{15}},\hfill \\ \eta \left(\{d_1,d_6\}\right)={\displaystyle \frac{13}{15}},\hfill & \eta \left(\{d_1,d_7\}\right)={\displaystyle \frac{10}{15}},\hfill \\ \eta \left(\{d_1,d_8\}\right)={\displaystyle \frac{10}{15}},\hfill & \eta \left(\{d_1,d_9\}\right)={\displaystyle \frac{11}{15}},\hfill \\ \eta \left(\{d_1,d_{10}\}\right)={\displaystyle \frac{12}{15}},\hfill & \eta \left(\{d_1,d_{11}\}\right)={\displaystyle \frac{11}{15}},\hfill \\ \eta \left(\{d_1,d_{12}\}\right)={\displaystyle \frac{11}{15}},\hfill & \eta \left(\{d_1,d_{13}\}\right)={\displaystyle \frac{10}{15}},\hfill \\ \eta \left(\{d_1,d_{14}\}\right)={\displaystyle \frac{8}{15}},\hfill & \eta \left(\{d_1,d_{15}\}\right)={\displaystyle \frac{9}{15}}.\hfill \end{array}$$

It can be concluded that $d_1$ and $d_6$ have the most similar prescriptions and $d_1$ and $d_{14}$ have the least similar prescriptions.

6. Conclusions

In this study, motivated by the well-known gambling game Russian Roulette, we defined the $2R$ formal context. Then, we undertook a systematic study on formal concept analysis for $2R$ formal contexts. It can be shown that the $2R$ formal context is a generalized 3-valued formal context, but it has more different properties. In addition, we explored 3-valued concept analysis for 2R formal contexts and presented a novel concept lattice model, i.e., 3-valued $2R$ concept lattice. Finally, we obtained several important conclusions by experimental analysis and raised a case in Chinese medicine to show the effectiveness of 3-valued $2R$ concept lattices in knowledge discovery.

Further research needs to consider more complicated cases. For instance, missing data are encountered from time to time due to various reasons in our daily life [43,44]. In addition, sometimes the data are obtained in a distributed and progressive manner [45] and the data may be inaccurate and contain some sort of vagueness [46]. How to model these problems and build the formal concept-analysis model is interesting and deserves our effort. What is more, the inputs of trainers in machine learning are of vital importance [47]. Thus, it is also appealing to model the inputs of trainers in the form of concepts, which may provide some interpretability to machine learning. Last but not least, how to reduce time consumption especially for big data when applying a concept lattice model is an important issue.