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 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 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 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
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
formal contexts. In
Section 4, experiments and case studies are presented to show the effectiveness of the 3-valued
concept lattice model. Finally, conclusions are provided in the last section.
2. Preliminaries of Formal Concept Analysis
A formal context describes a set of related objects Q by a set of attributes T via a binary relation between Q and T. Concretely, we use (respectively, ) to express that the object q possesses (respectively, does not possess) the attribute t.
represents the attributes that are possessed by the object
and
denotes the set of objects which have the attribute
; these are defined as follows:
For
and
, we define two operations as follows:
gives the common attributes shared by every object in E and returns the set of objects that possess all the attributes in P. If and , then we call a formal concept of .
In addition,
and
can be defined as follows:
is the attributes that are not commonly shared by every object in
E and
returns the set of objects that do not possess all the attributes in
P. If
,
and
, then we call
an object-induced three-way concept of
[
15].
What is more, other types of operations,
,
and
, can also be defined to investigate the other aspects of granules [
25,
26,
27,
28,
29,
30]. Formally, for
, we have
, and .
Dually, for , we have:
, and .
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 and , then we call a property-oriented concept of . If and , then we call an object-oriented concept of . If and , then we call a dual formal concept of .
3. The Unique Properties of 2R Formal Contexts: A Formal Concept Analysis Viewpoint
In this section, we propose four types of concepts of formal contexts and describe the unique properties of 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., , , , , and . In the subsequent discussion, we collectively denote these six sets as . Put formally, we have .
Definition 1. A formal context for Russian Roulette is a quadruple , where Q and T are two non-empty finite sets and .
For simplicity, we call the above-defined formal context a Russian Roulette formal context, and a formal context for short. Here, the first R represents Russian and the second R indicates roulette.
A formal context is essentially a six-valued formal context and its complement is defined as follows.
Definition 2. Let . For and , the negation of is defined asMoreover, is called the complement context of K. It is easy to show that the complement context of a formal context is also a formal context.
Example 1. A formal context is shown in Table 1 and its complement is shown in Table 2. Qi et al. [
42] presented a 3-valued formal context. As an example,
Table 3 shows a 3-valued formal context, which can be seen as a special case of a
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 . For and , two associated operators and are defined aswhere . At first glance, ‘
’ 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.,
. In other words,
is not necessarily equal to ?. Alternatively,
containing the element ? is sufficient.
For example, considering the
formal context in
Table 1, we have
. However, if we follow the idea of 3-way concept analysis [
42], it turns out to be
as
.
Proposition 1. If E, , and P, , , then the following properties hold.
(i) ; .
(ii) ; .
(iii) ; .
(iv) .
(v) ; .
(vi) ; .
Definition 4. Let , and . If and ; then is called a concept.
Moreover, we use
to denote the set of
concepts and define a binary relation between
concepts as
Theorem 1. is a complete lattice and for , ; it follows that Hereinafter, we call a concept lattice of K. For the sake of a clear discussion, if ? of concept-forming operator is set to +, then we call the derived concepts positive concepts. Analogously, negative concepts and zero concepts can be obtained. As a consequence, for a formal context K, we have positive concept lattice, negative concept lattice and zero concept lattice and denote them as , and , respectively.
Example 2. The formal context K in Table 4 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 to indicate a concept .
Figure 1 shows the positive
concept lattice
, based on which we can find which gamblers prefer the choice ‘+’ in each trial. For instance, the concept
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
. For
and
, we define
Definition 5. Let , and . Two associated operators and are defined aswhere . Moreover, if and , we call an object-oriented concept, concept for short.
In addition, the set of
concepts is denoted as
and we call it the object-oriented
concept lattice of
K,
concept lattice for short, and the infimum and supremum in
are, respectively, defined as
Definition 6. Let , and . Two operators and are defined aswhere . Moreover, if and , we call a property-oriented concept, concept for short, and denote the set of concepts by .
In addition, we call
the property-oriented
concept lattice of
K,
concept lattice for short, and the infimum and supremum in
are, respectively, given by
Definition 7. Let , and . Two operators and are defined aswhere . Moreover, if and , we call a dual concept, concept for short, and denote the set of concepts by .
In addition, we call
a dual
concept lattice,
concept lattice for short, and the infimum and supremum in
are, respectively, defined as
Proposition 2. Let , and . The following properties hold.
(i) ; .
(ii) ; .
(iii) ; .
(iv) , ; , .
Proof. (iv) By (ii) and (iii), it follows that . Moreover, let . Then, , which leads to .
The rest can be proved by the duality principle. □
Theorem 2. Let . For , and , the following propositions are equivalent.
(i) is a concept of K.
(ii) is a dual concept of K.
(iii) is an object-oriented concept of .
(iv) is a property-oriented concept of .
Proof. (i) ⇔ (ii): by Proposition 2 (i), we obatin
(i) ⇔ (iii): by Proposition 2 (ii) and (iii), we obatin
(iii) ⇔ (iv): by Proposition 1 (iv), we obatin
□
The comparisons between
formal contexts and 3-valued formal contexts are summarized in
Table 5. 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 concepts and discuss their properties and the connections with the basic concepts.
Definition 8. Let . For and , two associated 3-valued operators and are defined as Let . Then, if and only if .
Proposition 3. If E, , and S, , , then the following properties hold.
(i) ; .
(ii) ; .
(iii) ; .
(iv) .
(v) ; .
(vi) ; .
Definition 9. Let , and . If and , we call a 3-valued concept.
The set of 3-valued concepts is denoted by and a partial order between any two 3-valued concepts in is defined as Theorem 3. is a complete lattice and for , , it follows that Hereinafter, we call the 3-valued concept lattice of K.
In what follows, Algorithm 1 describes the process of building the 3-valued concept lattice based on an incremental strategy. The main idea of algorithm 1 adopts an incremental manner to construct the 3-valued concept lattice. Concretely, we first initialize the 3-valued concept lattice to be empty and then for each object of the 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 in descending order in terms of the sizes of the extents of each 3-valued concept.
Step 2: For each 3-valued concept, perform necessary changes and generate new 3-valued concept as follows.
(i) If , then add into .
(ii) If
and there is no 3-valued
concept with an intent
, then add
into
.
Algorithm 1 Algorithm for building 3-valued concept lattice |
Require: . Ensure: .
- 1:
Initialized . - 2:
Fetch an object q from Q and put and into . - 3:
For each object q in Q - 4:
Sort in descending order in terms of the sizes of the extents of each 3-valued concept. - 5:
For each in - 6:
If - 7:
add into ; - 8:
Else - 9:
If there is no 3-valued concept with an intent , - 10:
add into . - 11:
End If - 12:
End If - 13:
End For - 14:
End For - 15:
Return and end the algorithm.
|
Example 3. The of the formal context K in Table 4 is shown in Figure 2. For the sake of discussion, given a formal context , we collectively call the positive concepts, negative concepts and zero concepts basic concepts of K.
Theorem 4. Let and be a 3-valued concept of K. Then, the following propositions hold.
(i) is a positive concept.
(ii) is a negative concept.
(iii) is a zero concept.
Proof. (i) As is a 3-valued concept, we have . Then, , which means that is a positive concept.
In addition, items (ii) and (iii) can be analogously proved. □
Definition 10. Let , , and . If , then denote this case as and denote the corresponding equivalence class by .
Theorem 5. is the least element of , if and only if , and .
Proof. “⇒”. As
,
, it follows that
. Similarly, we also have
and
. Therefore,
. In addition, by the properties of the operators, we have
,
and
. To sum up, we can conclude that
, which further implies that
Moreover, by the fact that
,
,
, we can derive
By the condition that is the least element of , we can conclude that , and .
“⇐”. For any , we have , by which we can derive . Similarly, we can show and . Then, it follows that , and ; i.e., is the least element of . □
Theorem 6. Let and Δ
be the collection of the least elements in . Then, Proof. Let .
On one hand, let
. We have
,
and
. Moreover, as
,
and
, by Theorem 3, we have
By the fact that , we can conclude , which implies .
On the other hand, let . By Theorem 3, we have , and . Then, by the definition of a 3-valued concept, is at hand, which implies .
Then, this theorem is proved. □
The above theorems actually indicate the strong connections between 3-valued concepts and basic concepts. That is, Theorem 4 implies that we can derive basic concepts from 3-valued concepts, while Theorem 6 states a 3-valued concept lattice can be derived by using its basic 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 concept lattices and the basic concept lattices.
In the experiments, Algorithm 2 derives the synthetic
formal contexts.
Algorithm 2 Generating a synthetic formal context |
Require: Q, T, two ratios and and a possibility p. Ensure: .
- 1:
For each pair of object and attribute - 2:
Initialize and a set . - 3:
Fetch one element from S with a possibility ratio . - 4:
. - 5:
Add into . - 6:
Fetch one element from S with a possibility ratio . - 7:
Add into with a possibility p. - 8:
End For - 9:
Return and end the algorithm.
|
(i) The influences caused by the number of objects
By setting
,
,
and
and changing the size of
Q from 20 to 40, we can derive a family of randomly generated
formal context, which have 25 attributes, but different numbers of objects. For each generated
formal context, we record the number of 3-valued
concepts and basic
concepts. Then, we show the experimental results in
Figure 3.
(ii) The influences caused by the number of attributes
Let
,
,
and
. By changing
from 20 to 40, we can derive a family of randomly generated
formal contexts, which has 35 objects but different numbers of attributes. By recording the number of 3-valued
concepts and basic
concepts, we show the experimental results in
Figure 4.
(iii) The influences caused by different fill ratios
By fixing
and setting five different fill ratios
,
,
,
and
for
, we randomly generate five different
formal contexts by Algorithm 2, namely
,
,
,
and
, where for each
formal context
and
. By recording the number of 3-valued
concepts and basic
concepts of the family of the randomly generated
formal contexts, we collectively list the experimental results in
Table 6, where
,
,
and
represent the numbers of concepts of
,
,
and
, respectively, and
denotes the ratio between the number of +, − and 0 contained in the
formal contexts.
The following can be analyzed from the experimental results.
There is a close correlation between () and the size of 3-valued concept lattices and the basic concept lattices. The larger the values of (), the larger the size of the 3-valued concept lattices and the basic concept lattices.
Given a formal context, the sizes of basic concept lattices are smaller than that of the 3-valued concept lattice, which implies 3-valued concept lattices embody some unique patterns compared with basic concept lattices.
When +, − and 0 are evenly filled, the formal contexts contain the fewest 3-valued concepts.
5.2. Case Study
In
Table 7, a survey on the usage of a type of Chinese herbal medicine is presented. In this table, there are 15 TCM doctors (denoted as
to
), who gave their suggested prescriptions independently to 10 patients (denoted as
to
). 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
has prescribed Aconiti Lateralis Radix Preparata for patients
and
and has prescribed Aconitum Carmichaeli and Tianxiong for patient
.
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 concepts from this dataset, and each 3-valued concept denotes a meaningful pattern, based on which many useful pieces of information can be obtained.
For instance, there are 79 3-valued concepts whose extents possess . Among these 79 3-valued concepts, the minimal one is , , which states that gives Aconiti Lateralis Radix Preparata for patients and , Aconitum Carmichaeli for patients and and Tianxiong for patients and and the maximal one is , which implies that all the 15 TCM physicians agree that Aconiti Lateralis Radix Preparata is necessary for patient and Tianxiong for patients and .
According to the prescriptions given by the TCM physicians, the consistency degree of an arbitrary group of TCM physicians can be obtained. Let , and . We define , where and there does not exist such that .
Then, the consistency degree .
By using the above formula, the consistency degrees of arbitrary cliques can be computed. For example, the consistency degrees of pairs that contain
are collectively listed as follows:
It can be concluded that and have the most similar prescriptions and and have the least similar prescriptions.