Abstract
Rule acquisition is one of the main purposes in the analysis of formal decision contexts. Up to now, there have been several types of rules in formal decision contexts such as decision rules, decision implications, and granular rules, which can be viewed as rules since all of them have the following form: “if conditions and hold, then decisions hold.” In order to enrich the existing rule acquisition theory in formal decision contexts, this study puts forward two new types of rules which are called rules and  mixed rules based on formal, objectoriented, and propertyoriented concept lattices. Moreover, a comparison of rules,  mixed rules, and rules is made from the perspectives of inclusion and inference relationships. Finally, some real examples and numerical experiments are conducted to compare the proposed rule acquisition algorithms with the existing one in terms of the running efficiency.
1. Introduction
Formal concept analysis (FCA) is a field of applied mathematics based on the mathematization of formal concepts and conceptual hierarchy [1]. This theory starts with the notion of a formal context consisting of an object set , an attribute set , and an incidence relation between and [2]. Its key characteristic lies in the conceptual unfolding of data. Nowadays, FCA has been applied in many domains such as information retrieval [3], machine learning [4], knowledge discovery [5–15], and software engineering [16, 17]. What is more, it has shown a trend of multidisciplinary intersection.
In FCA, a basic way of describing dependencies between the attributes of a formal context is via implications [18, 19] or association rules [20]. Considering that directly deriving these types of rules from a formal context involves so many calculations and the number of rules is generally large, some researchers [21–23] discussed how to efficiently mine implications/association rules from a formal context and eliminate superfluous rules as many as possible. This issue was also investigated in generalized formal contexts [24].
In the real world, a formal context often contains target attributes for the purpose of making decision analysis. A formal context equipped with additional target attributes is called a formal decision context [25] (or a decision formal context sometimes) which is in fact a training context [26]. Note that rule acquisition is one of the main purposes in the analysis of formal decision contexts and in recent years this issue has attracted much attention from the Chinese FCA community. For instance, Zhang and Qiu [25] put forward the first type of rules, called decision rules, in formal decision contexts through combining conditional formal concepts with decision formal concepts. In order to eliminate superfluous decision rules, Li et al. [27] proposed the notion of a nonredundant decision rule and an approach to extract all nonredundant decision rules from a formal decision context, which was improved in [28] by integrating granular computing into decision rules for decreasing computation time. Moreover, the notion of a decision rule was extended into the cases of incomplete formal decision contexts [29] and real formal decision contexts [30, 31]. Besides, Qu et al. [32] presented the second type of rules, called decision implications, whose premises and conclusions are taken from conditional attributes and decision attributes, respectively. It should be pointed out that decision rules are special decision implications. Following the above discussion, Zhai et al. [33] discussed the semantical and syntactical aspects of decision implications. Additionally, based on granular computing, Wu et al. [34] defined the third type of rules, called granular rules, which are special decision rules, with their premises and conclusions being the intents of conditional formal concepts and decision formal concepts, respectively. A detailed investigation on relation between granular rules and decision rules can also be found in [28].
The aforementioned types of rules in formal decision contexts, which were investigated within the framework of classical FCA, can be viewed as rules since all of them have the following form: “if conditions , and hold, then decisions hold.” Considering that objectoriented concept lattice [35] and propertyoriented concept lattice [36] possess useful characteristics from both FCA and rough set theory [37] for data analysis and they have been proven to be beneficial to knowledge discovery [38, 39], this study puts forward two new types of rules, called rules and  mixed rules, using formal, objectoriented, and propertyoriented concept lattices. The current work can enrich the existing rule acquisition theory in formal decision contexts.
The rest of this paper is organized as follows. Section 2 reviews some basic notions and properties related to formal, objectoriented, and propertyoriented concept lattices. Section 3 discusses the issue of rule acquisition in formal decision contexts based on formal, objectoriented, and propertyoriented concept lattices. More specifically, the notions of a rule and a  mixed rule are proposed. Furthermore, we put forward approaches to derive all nonredundant rules and  mixed rules from a formal decision context. Section 4 makes a comparison of the inclusion and inference relationships among rules,  mixed rules, and rules. Section 5 conducts some numerical experiments to compare the performances of the proposed rule acquisition algorithms and the existing one.
2. Preliminaries
In what follows, we briefly recall some basic notions and properties related to formal, objectoriented, and propertyoriented concept lattices.
Definition 1 (see [1]). A formal context is a triple including an object set , an attribute set , and an incidence relation , in which indicates that the object has the attribute and means the opposite.
Definition 2 (see [25]). A formal context is said to be regular if, for any , the following conditions hold: (i)there exist such that and ;(ii)there exist such that and .
Without loss of generality, the formal contexts discussed hereinafter are all assumed to be regular.
In order to derive formal, objectoriented, and propertyoriented concept lattices, the following six operators are needed: for any and ,
Note that the pair of operators forms an antitone Galois connection, while the pairs of operators and form isotone Galois connections [39]. More properties about these operators can be found below.
Proposition 3 (see [2, 35, 36]). Let be a formal context. For and , the following properties hold:(i), , ;(ii), , ;(iii), ;(iv), ;(v), , ;(vi), , .
Definition 4 (see [1, 2, 35, 36]). Let be a formal context, , and . If and , then is called a formal concept; if and , then is called an objectoriented concept; if and , then is called a propertyoriented concept. For each of the cases, and are called the extent and intent of , respectively.
When the formal, objectoriented, and propertyoriented concepts of a formal context are, respectively, ordered by they form complete lattices which are called the formal, objectoriented, and propertyoriented concept lattices [1, 35, 36] of the formal context , respectively. Hereinafter, we denote formal concept lattice by , objectoriented concept lattice by , and propertyoriented concept lattice by .
In the concept lattices , , and , the infimum and supremum of two concepts and are, respectively, defined by
It should be pointed out that the relation among formal, objectoriented, and propertyoriented concept lattices was discussed in [38, 39], and knowledge reduction of objectoriented and/or propertyoriented concept lattices was investigated in [7, 40, 41].
3. Rule Acquisition in Formal Decision Contexts Based on Formal, ObjectOriented, and PropertyOriented Concept Lattices
Definition 5 (see [25, 26]). A formal decision context is a quintuple , where and with are two formal contexts. Here, and are called the conditional attribute set and the decision attribute set of , respectively.
Like the formal context, a formal decision context is also said to be regular [27] if both and are regular. Hereinafter, the concerned formal decision contexts are all assumed to be regular. Moreover, for convenience, , , and are, respectively, called conditional formal, objectoriented, and propertyoriented concept lattices of , and ,, and are, respectively, called decision formal, objectoriented, and propertyoriented concept lattices of .
To the best of our knowledge, the existing work of rule acquisition in formal decision contexts is based on (conditional and decision) formal concept lattices only, and the derived rules with and can be viewed as ones since they have the following form: “any object having all conditional attributes of also has all decision attributes of .” Up to now, such kinds of rules have successfully been applied to radar fault diagnosis under incomplete environment [42].
In order to widen the domain of application of rule acquisition, now we continue to put forward two new types of rules, called rules and  mixed rules, based on formal, objectoriented, and propertyoriented concept lattices.
3.1. Rule Acquisition Based on Formal and ObjectOriented Concept Lattices
In this subsection, we propose the notion of a rule in formal decision contexts based on formal and objectoriented concept lattices.
Definition 6. Let be a formal decision context, let be the objectoriented concept lattice of , and let be the formal concept lattice of . For any and , if , , and , then the expression is called a rule generated between the objectoriented concept and formal concept . Here, and are called the premise and conclusion of the rule , respectively. The set of all the rules generated between the objectoriented concepts in and the formal concepts in is denoted by .
Thus, for any , we conclude that each having at least one conditional attribute in has all the decision attributes in . More specifically, if and , then means that “if , then ,” where and denote logical disjunction and conjunction operators.
It should be pointed out that the rules have something to do with both the attribute implication rules and the association rules (see, e.g., [18, 19] for the detailed introduction of the attribute implication rules and, e.g., [20, 23] for that of the association rules). Concretely, a rule with and can be integrated by the following attribute implication rules (or association rules with their confidences being one): However, an attribute implication rule may not be a rule since its premise is not an expression of disjunction of conditional attributes except a singleton set. Yet, an association rule may not be a rule since its confidence is often less than one.
Definition 7. Let be a formal decision context. For , if and , one says that can be implied by . One denotes this implication relationship by . For any , if there exists such that , then is said to be redundant in ; otherwise, is said to be nonredundant in . We denote by the set of all of the nonredundant rules in .
It can be known from Definition 7 that, for a given formal decision context, it is more appealing to extract the nonredundant rules since they can imply others.
Let be a formal decision context. Denote For any , define
Theorem 8. For a formal decision context , , , and . Then, is redundant in if and only if or , or, equivalently, is nonredundant in if and only if and .
Proof. Necessity. If is redundant in , there exists such that , , and . By Definition 7, it follows that and , which implies that and . Noting that and is different from , we conclude that or . Consequently, or .
Sufficiency. If or , we can prove that is redundant in . In fact, when , there exists such that . Suppose . Then, . As a result, is redundant in . When , there exists such that . Suppose . Then, . Consequently, is redundant in .
Now, we are ready to put forward a method to derive the nonredundant rules from a formal decision context. The method can briefly be described as in Algorithm 1.

Note that the objectoriented concept lattice of can be derived from the formal concept lattice of the complementary formal context of [39]. Then, it is easy to prove that the time complexity of Algorithm 1 is where denotes the cardinality of the objectoriented concept lattice of and denotes that of the formal concept lattice of .
Example 9. Table 1 depicts a formal decision context , where , , and . The objectoriented concept lattice of is shown in Figure 1 and the formal concept lattice of is shown in Figure 2.
According to Algorithm 1, we can derive the following nonredundant rules from : : if , then and ; : if , then and ; : if , , , , or , then .
It should be pointed out that can be divided into the following attribute implication rules: : if , then ; : if , then ; : if , then ; : if , then ; : if , then .
3.2. Rule Acquisition Based on Formal and PropertyOriented Concept Lattices
In this subsection, we continue to put forward the notion of a  mixed rule in formal decision contexts based on formal and propertyoriented concept lattices.
Definition 10. Let be a formal decision context, let be the propertyoriented concept lattice of , and let be the formal concept lattice of . For any and , if , , and , then the expression is called a  mixed rule generated between the propertyoriented concept and the formal concept . Here, and are called the premise and conclusion of the  mixed rule , respectively. The set of all of the  mixed rules generated between the propertyoriented concepts in and the formal concepts in is denoted by .
Thus, for any , we conclude that each object having at least one conditional attribute in and no conditional attribute in has all the decision attributes in . More specifically, if , , and , then means the following: “if and , then ,” where , , and denote logical disjunction, conjunction, and negation operators, respectively.
It should be pointed out that the  mixed rules have something to do with both the attribute implication rules and the association rules. Concretely, a  mixed rule with , , and can be integrated by the following attribute implication rules (or association rules with their confidences being one): However, an attribute implication rule may not be a  mixed rule since its premise is generally not an expression of disjunction and conjunction of conditional attributes. Yet, an association rule may not be a  mixed rule since its confidence is often less than one.
Definition 11. Let be a formal decision context. For any , if and , one says that can be implied by . One denotes this implication relationship by . For any , if there exists such that , then is said to be redundant in ; otherwise, is said to be nonredundant in . One denotes by the set of all the nonredundant  mixed rules in .
It can be known from Definition 11 that, for a given formal decision context, it is more appealing to extract the nonredundant  mixed rules since they can imply the remainder.
Let be a formal decision context. Denote For any , one defines
Theorem 12. For a formal decision context , , , and . Then, is redundant in if and only if or , or, equivalently, is nonredundant in if and only if and .
Proof. It is similar to the proof of Theorem 8.
Now, we are ready to propose an approach to derive all the nonredundant  mixed rules from a formal decision context. The detailed steps are given in Algorithm 2.

Note that the propertyoriented concept lattice of can be derived from the formal concept lattice of the complementary formal context of [39]. Then, it is easy to prove that the time complexity of Algorithm 2 is where denotes the cardinality of the propertyoriented concept lattice of and denotes that of the formal concept lattice of .
Example 13. Let be the formal decision context in Table 1, where , , and . The propertyoriented concept lattice of is shown in Figure 3 and the formal concept lattice of can be found in Figure 2.
According to Algorithm 2, we can derive the following nonredundant  mixed rule from : : if , , , or and and , then and .
It should be pointed out that can be divided into the following attribute implication rules: : if , , and , then and ; : if , , and , then and ; : if , , and , then and ; : if , , and , then and .
4. A Comparison of Rules in Terms of Inclusion and Inference Relationships
In Section 3, we have compared the rules and  mixed rules with the attribute implication rules and association rules. In this section, we continue to make a comparison of rules,  mixed rules, and decision rules in terms of inclusion and inference relationships. Before embarking on this issue, we introduce the notion of a decision rule in formal decision contexts.
Definition 14 (see [27]). Let be a formal decision context, let be the formal concept lattice of , and let be the formal concept lattice of . For any and , if , , , and are all nonempty and , then the expression is called a decision rule generated between the formal concepts and . Here, and are called the premise and conclusion of the decision rule , respectively. The set of all the decision rules generated between the formal concepts in and those in is denoted by .
Definition 15 (see [27]). Let be a formal decision context. For , if and , one says that can be implied by . One denotes this implication relationship by . For any , if there exists such that , then is said to be redundant in ; otherwise, is said to be nonredundant in . One denotes by the set of all the nonredundant decision rules in .
Thus, for any , one concludes that each object having all the conditional attributes in also has all the decision attributes in . More specifically, if and , then means the following: “if , then ,” where denotes logical conjunction operator. Moreover, it is easy to observe that decision rules, rules, and  mixed rules are different from each other in terms of their logical reasoning methodologies.
The following example is used to show that there does not exist inclusion relationship among decision rules, rules, and  mixed rules. That is, we need to confirm three statements: a decision rule may not be a rule or  mixed rule; a rule may not be a decision rule or  mixed rule; a  mixed rule may not be a decision rule or rule.
Example 16. Let be the formal decision context in Table 1, where , , and . The formal concept lattice of is shown in Figure 4 and that of can be found in Figure 2.
According to the algorithm in [28] (interested readers can refer to [28] about how to efficiently derive all the nonredundant decision rules from a formal decision context), we can derive the following nonredundant decision rules from : : if , , and , then and ; : if , , , and , then and ; : if and , then .
Combining these decision rules with the results obtained in Examples 9 and 13, we conclude that there does not exist inclusion relationship among decision rules, rules, and  mixed rules.
Moreover, the following example is used to show that there does not exist inference relationship among decision rules, rules, and  mixed rules, where the inference rule is described as follows [32]: Note that the negation of a conditional attribute is treated as a new one and it is different from others.
Example 17. Let be the formal decision context in Table 1, where , , and . Then, according to the discussion in Examples 9, 13, and 16, we have the following:(i)the decision rule cannot be implied by the rules , , , , , , and based on the inference rule , neither can the  mixed rules , , , and ;(ii)each of the rules , , , , , , and cannot be implied by the decision rules , , and based on the inference rule , neither can the  mixed rules , , , and ;(iii)each of the  mixed rules , , and cannot be implied by the decision rules , , and based on the inference rule , neither can the rules , , , , , , and .
5. Experiments
Although, according to the discussion in Section 4, there does not exist inclusion or inference relationships among decision rules, rules, and  mixed rules, it is still necessary to conduct some experiments to compare the proposed rule acquisition algorithms with the existing one in [27] in terms of the running efficiency.
In the experiments, eight reallife databases, including Bacteria [43], Zoo [44], Breast Tissue [44], Acute Inflammations [44], Servo [44], Wine [44], Balance Scale [44], and Car Evaluation [44], are analyzed to achieve the task of comparing the running efficiency. The detailed information about the eight chosen reallife databases is shown in Table 2.
For each of the chosen databases, we took the classification attribute(s) as the decision attribute(s) and other attributes as the conditional attributes. Then, the scaling approach [2] was used to transform the eight chosen databases into formal decision contexts. More specifically, discrete (but not Boolean) or continuous attributes were converted into Boolean ones. The detailed information about the conversion is listed in Table 3, where “/” means “taking no action” and “trisection” means “classifying the values of each continuous attribute, from small to large, into three pairwise disjoint intervals whose lengths are the same.” We denote by data sets 1, 2, 3, 4, 5, 6, 7, and 8 the formal decision contexts which were obtained by applying the scaling approach (exactly, nominal scale and/or ordinal scale) to the eight chosen databases.
In the experiments, we still denote the proposed rule acquisition algorithms by Algorithms 1 and 2 (see Sections 3.1 and 3.2 for details) and the existing one in [27] by Algorithm 3. Then, Algorithms 1, 2, and 3 were applied to data sets 1, 2, 3, 4, 5, 6, 7, and 8. The corresponding running time is reported in Table 4 in which “Size” is the Cartesian product of the object set, conditional attribute set, and decision attribute set of the concerned formal decision context and , , and are the nonredundant rules,  mixed rules, and decision rules, respectively. It can be seen from Table 4 that, by the running time, Algorithms 1, 2, and 3 are all acceptable, and which one being more efficient than another seems to be dependent on the given databases including density and scaling approach.
6. Conclusion and Future Work
Rule acquisition is one of the main purposes in the analysis of formal decision contexts. Although there have been several types of rules in formal decision contexts such as decision rules, decision implications, and granular rules, these rules are all ones since they have the following form: “if conditions , and hold, then decisions hold.” In order to enrich the existing rule acquisition theory in formal decision contexts, we have proposed two new types of rules, called rules and  mixed rules, based on formal, objectoriented, and propertyoriented concept lattices. Moreover, a comparison of rules,  mixed rules, and rules has been made from the perspectives of inclusion and inference relationships. Finally, some numerical experiments have been conducted to compare the proposed rule acquisition algorithms with the existing one in terms of the running efficiency.
From the point of view of real applications, the results obtained in this paper need to be further extended to the cases of fuzzy formal decision contexts [45], incomplete formal decision contexts [29], and real formal decision contexts [30, 31] since in the real world the relationship between some objects and attributes of a formal decision context may be fuzzyvalued, intervalvalued, or even realvalued. This issue will be discussed in our future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 61305057, 61202018, 61203283, and 11371014) and the Natural Science Research Foundation of Kunming University of Science and Technology (no. 14118760).