Rule reductions of decision formal context based on mixed information

Formal concept analysis is an effective tool in knowledge discovery. Nonetheless, it ignores the negative information, especially in rule extraction. And some applications require to mix positive and negative information for management and representation explicitly. This paper discusses an emerging method for rule reduction in formal contexts with mixed information by presenting a mixed decision rule based on a mixed concept lattice. Based on the mixed concept model, we discuss the relationships between mixed concept, formal concept and three-way concept, respectively. In this paper, to fully consider the mixed information, we construct a mixed decision rule. Moreover, to reduce the redundance of rule, a novel approach for weak-basis from the viewpoint of granular computing is designed to selection the core mixed decision rules and necessary mixed decision rules. Finally, the comparison of mixed decision rules and three-way decision rules is perfectly discussed, and case study is presented for the difference.


Introduction
Formal concept analysis (FCA) has been a useful mathematical method to knowledge representation and knowledge discovery. And it is a successful tool to analyze data aroused scholars' wide concern [2, 4-6, 9, 10, 14, 20, 25, 29].
As an important infrastructure for FCA, formal contexts are generally only took care of the positive information. Namely, the presence of an attribute in an object, but barely considered the hidden negations. However, the negations from formal contexts also contain a lot of useful information. In this case, Qi et al. [12] first proposed three-way concept analysis (3WCA) incorporating with three-way decisions and FCA, in which not only the positive information but also the negative information play important roles. Then, Wei et al. [23] further developed the necessitypossibility semi-three way concept. Later, Qi et al. [13] Yidong Lin yidong lin@yeah.net 1 introduced the relationship between 3WCA and FCA systematically. Zhi et al. [31] common-possible concept analysis based on a granule description viewpoint. To make the concepts of 3WCA easier to understand, Ren et al. [18] put forward an approach of attribute reduction subsequently. In addition, the conflict analysis model was furthermore formulated based on three-way analysis by Zhi et al. [30]. And a new method of constructing attribute-induced three-way and object-induced three-way concept lattices is proposed by Yang et al. [26]. Then Zhao et al. [32] investigated the relationship between different kinds of three-way concept lattices. And the relationship between various L3W concept lattices was further explored by Zhao and Miao [33].
Rule acquisition and association are definite two main patterns of knowledge discovering. Therefore, from logical viewpoint, Zhai et al. [27,28] paid much attention to decision implication canonical basis, inference rule and the semantic interpretation for decision formal context. In addition, some scholars [7,8,19] have ulteriorly served to derive all the non-redundant ones methodologically. Notably, most of the findings were ill-advised to tackle the negative information of contexts actually. And position and negation in 3WCA are involved separately. Meanwhile, a few works [11,15] have recently been served to rule acquisition or implications by integrating with negative information. Although Wei et al. [22] explored rules acquisition of decision formal context based on three-way concept lattices, from which the rules are usually the forms of "If A, then C" and "If not B, then not D".Nonetheless, "If A then not B" and "If not A then B" fail to be acquired in 3WCA. By inference, the information of formal context is not fully mined by 3WCA. For instance, a rule that "cyclist with short and sharp accelerations are not great climbers" [14,15] will not be gained typically from both classical formal concept lattice and three-way formal concept lattice. For this reason, we will address rule acquisition from decision formal context by making a mixed consideration of position and negation.
In reality, driven by requirements from practical applications, except for 3WCA, people had made different attempts to extend FCA with negative information. As early as 2000, Wille [24] had put forward diverse types of negation. Subsequently, Missaoui et al. [11] developed a method for generating implications with negation from formal context by a minimal generator. Rodriguez et al. [15] formulated a novel approach to gain mixed implications from formal context. Furthermore, Rodriguez et al. [16] studied in depth mixed concept lattices (MCA) and proposed a characterization theorem. To devote to breast cancer, data mining algorithms for mixed concepts were moreover developed [17]. Bartl et al. [1] introduced formal fuzzy concept analysis by using positive and negative attributes in the same year. Nevertheless, few works have recently been paid attention to the mixed concepts and the rule acquisition according to mixed information with respect to decision formal contexts. Although there is rule acquisition based on 3WCA [22], the positive and negative decision rules were managed respectively without stirring together. In order to further address this deficiency, in this article we focus on implementing mixed decision rules by using mixed concept lattices.
In this paper, we first recall several basic notions in Section 2. Subsequently, two kinds of mixed concepts are investigated. And then the relationships between three types of concept lattices are explored in Section 3. Moreover, in Section 4, two types of mixed decision rules acquisition with respect to decision formal context are discussed, respectively. With the framework of non-redundant mixed decision rules, a mechanism of rule classification is proposed. In Section 5, the connection between mixed decision rules and three-way rules is studied in depth. And the summary and future work are presented in Section 7.

Basic notations in FCA
A formal context is recorded as F = (G, M, I ), in which G is the set of considered objects, M is the collection of attributes and I ⊆ G × M.
Take X ⊆ G and B ⊆ M. Maps * : P(G) → P(M) and * : P(M) → P(G) are respectively specified by: The tuple (X, B) is called a formal concept derived from F whenever X * = B and B * = X, in which the former and the latter are called extent and intent, respectively. The relation"≤" between any two concepts (Y, C) and (X, B) is: And the operators "∧" and "∨" are defined as follows: In this case, the set of all the concepts derived from F specifies the concept lattice and is denoted as L(G, M, I ).
Simultaneously, we record L G (G, M, I ) as the collection of the extensions and L M (G, M, I ) the family of the intensions of all concepts. To save confusion, the two maps * are called positive operators [12], and the corresponding induced concepts is called a P-concept. In the similar fashion, Qi et al. [12] presented the negative operators in the following.
Maps * : P(G) → P(M) and * : P(M) → P(G) for any X ⊆ G and B ⊆ M w.r.t. F called negative operators are respectively defined by And if X

Basic concepts of 3WCA
3WCA is a newly proposed mechanism based three-way decisions. Combining positions and negations, Qi et al. [12] proposed the following three-way operators. Take X ⊆ G and B, C ⊆ M from a formal context F . The definitions of object-induced three-way maps : P(G) → DP(M) and : DP(M) → P(G) are presented as follows:

Negative attributes
It is widely known knowledge discovery, association and implication rules from a formal context are usually constructed via positive information. Formally, for B 1 , B 2 ⊆ M from a formal context F , B 1 → B 2 is true if each object which has all attributes from B 1 also has all attributes from B 2 [27]. In this case, B 1 → B 2 is called an implication of F , and the set B 1 is the premise and B 2 is the consequence. Furthermore, if two implications B 1 → B 2 and C 1 → C 2 satisfy B 1 ⊆ C 1 and C 2 ⊆ B 2 , then we say B 1 → B 2 implies C 1 → C 2 , and the rule C 1 → C 2 is redundant. All the non-redundances constitute the implication basis of F , by which any implication in F can be induced. Nevertheless, this is incomplete in some requirements from practical situations.
For example, from the formal context listed in Table 1, one can easily obtain the following implication basis = {m 5 ↔ m 2 m 3 , m 4 → m 3 , m 1 → m 2 }. In this case, any implication rule holds in F if and only if it can be derived from . Thus, both m 2 → m 3 and m 2 → m 4 do not hold. However, this two implications have different reasons for failure. For the former, any object has the attribute m 2 does not possess the attribute m 3 . In contrast, the latter implies that such objects shared by m 2 can be identified by the presence or absence of m 4 . Both them are not available without negative information. Therefore, it is more meaningful to combine with positive and negative attributes in data mining, knowledge discovery and rule extraction.
To overcome this problem, Rodriguez-Jimenez et al. [15][16][17] developed mixed concept lattice with mixing the position and negation. First, we begin with the introduction of an extended notation that allows us to consider the negation of attributes as follows.  Table 1, m 2 → m 4 holds by combining with positive and negative information, where m 4 means the absence of m 4 (negative attribute). Simultaneously, neither m 2 → m 3 nor m 2 → m 3 holds.
Then, the extended definitions and formal concepts are presented in the following. Definition 1 [15][16][17] Let (F |F ) be the mixed formal context with respect to objects derived from formal context F = (G, M, I ). The mixed derivation maps ⇑: P(G) → P(M ∪ M) and ⇓: The pair (X, B) is named a mixed concept of F whenever X ⇑ = B and B ⇓ = X.
Especially, ∅ ⇑ = M ∪ M and (M ∪ M) ⇓ = ∅. At the same time, ∅ ⇓ = G. For the sake of simplicity, we record {x} ⇑ as x ⇑ and {a} ⇓ as a ⇓ for x ∈ G and a ∈ M ∪ M. In addition, the partial order between any two mixed concepts (X, B) and (Y, C) is expressed as: Example 1 From Table 2, the derived mixed formal context with respect to objects is shown on Table 3. Take

Object induced mixed concepts and attribute induced mixed concepts
The target of this subsection is to enrich the theory of MCA. First of all, we discuss the properties of mixed maps in the following.
From Definition 1, we denote OML(G, M, I ) as the collection of all mixed concepts of formal context F . That is, And take OML G (G, M, I ) and OML M (G, M, I ) be the set of extensions and the family of intensions of all mixed concepts in OML(G, M, I ), respectively. The "∧" and "∨" between any two mixed concepts (X, B) and (Y, C) are presented as follows: With the partial order "≤" from Eq.(13) and operations "∧" and "∨", OML(G, M, I ) exactly forms a complete lattice and is called the mixed concept lattice of F .
Since such mixed concepts focus on positive and negative attributes, we call them object-induced mixed concepts (O-mixed concepts shortly) for convenience.
In the ordinary way, an element a ∈ L a lattice is said to be join-irreducible if x = 0 (in case L has zero) and a = x∈X x implies a ∈ X. A meet-irreducible element is dually defined (see [3] for detail). In essence, every element in L can be interpreted as a join of some join-irreducible elements. It is easy to see that for (X, B) ∈ OML(G, M, I ). Thus, any join-irreducible element has the form of (g ⇑⇓ , g ⇑ ). Then we have the following conclusion.
On the other side, we also have P os(g From this lemma, we can see that the there are identical rows in the raw formal context for different objects if existing object g ∈ G such that |g ⇑⇓ | ≥ 2. Furthermore, we have the following theorem based on Lemma 1.

Theorem 2 For any
Proof On the contrary, suppose (g ⇑⇓ , g ⇑ ) is join reducible, then there exists an index set T such that for i ∈ T , and thus (g ⇑⇓ , g ⇑ ) may not be interpreted as a join of some join-irreducible elements in OML(G, M, I ). Therefore, (g ⇑⇓ , g ⇑ ) is a join irreducible in OML(G, M, I ).
Likewise, we can also consider the negative objects, the opposite of objects. These situation also prevails in real life. For instance, in personnel management, the employers usually hire staffs according to job vacancy. In this case, it is also meaningful to search the negative information with respect to attributes.
Then (X, B) is called an attribute-induced mixed formal concept of F (A-mixed concept for short) whenever X = B and B = X, in which X is the extension and B is the intention.
Worthy of note is that the symbol F |F is different from (F |F ). The latter has one more bracket than the former. The "≤" between any A-mixed concepts (X, B) and (Y, C) is similarly defined by The collection of all the A-mixed concepts of F forms a concept lattice. And we record it as AML(G, M, I ). In addition, the following properties hold for the mixed operators with respect to attributes. Proof Similar to Theorem 2. Tables 3 and 4

The connections between FCA, 3WCA and MCA
This subsection will focus on the connections between MCA, FCA and 3WCA. First of all, the link between O-mixed concept lattice and concept lattice is discussed as follows.
Theorem 4 Let F = (G, M, I ) be a formal context. The following conclusions hold. Table 4 The mixed formal context F |F of Table 2 I  Proof It is only needed to formulate items (1) and (2).
Then (X, B) forms a formal concept of (F |F ) whenever f (X) = B and g(B) = X.  Example 3 Table 5 depicts the corresponding mixed formal context of a decision formal context S. Figures 3  and 4 are the Hasse diagrams of OML (G, M, I ) and OML (G, N, J ), respectively. Then the mixed decision rules are shown in Table 6.

Mixed decision rules in the O-consistent decision formal context
Classically, decision rules acquired from formal context are often described as "if B, then D". Even if from a negative formal context, the decision rules are "if not B, then not D". The commons are positive attributes and negative attributes separate from each other. In other terms, the premise of negative attributes does not have positive conclusion, and positive premise would never infer negative conclusion. For mixed decision rules, as illustrated by Example 3, both the premises and the conclusions may be composed by positive and negative attributes. Notably, there also exists redundance in R(S). The details are as follows.
or else, B → D is known as nonredundant in R(S). We denote R * (S) as the collection of all the non-redundances in R(S). Table 7. Comparing to Table 6, there are only 9 nonredundant mixed decision rules. This is more conducive to understanding knowledge.

Example 4 From Example 2, we search for all the nonredundant mixed decision rules in R(S) and show them in
If S is O-consistent, we then have the following results.
Thereby, the following result is available:

Theorem 9 With respect to an O-consistent decision formal context S, mixed decision rule B → D ∈ R(S), B and D are non-empty, is non-redundant iff
It is generally known that i∈T g  . 3 The OML(G, M, I ) of Table 5 Fig . 4 The OML(G, N, J ) of Table 5   Table 6 Mixed decision rules

Definition 9
With respect to an O-consistent decision formal context S, we call g ⇑ M → g ⇑ N the core mixed decision rule for any g ∈ G. All the core mixed decision rules in S is recorded as C(S). Furthermore, a mixed decision rule B → D satisfying B = ∅, D = ∅ and B → D = i∈T (g is called a relatively necessary mixed decision rule. And we denote N (S) as the collection of all such mixed decision rules in R(S). In addition, we denote wR(S) as the join of C(S) and N (S).
For instance, from Table 6, it is easy to see that the core mixed decision rules are   Table 6. In this case, m 5 m 6 → n 3 / ∈ R * (S). Then, the conclusion is available below.

Theorem 10 Let S be O-consistent. Then R * (S) ⊆ wR(S). If all rules in
We then call wR(S) the weak-basis of R(S). wR(S) may be smaller than R(S) in a general way. In applications, it is unnecessary to obtain all the mixed decision rules because of a large number of redundant mixed rules. However, it is a difficult and time-consuming task to mine all the non-redundant mixed decision rules. By Theorem 10, we can only explore the weak-basis via C(S) and N (S) directly so that the computational complexity will be reduced.

Theorem 11 Let S be O-consistent with the weak-basis wR(S). For any B → D ∈ wR(S), there does not exist
Proof Necessity. We only need to proof that there exists no B → D 0 such that D ⊂ D 0 . Actually, such B → D 0 does not exist if B → D ∈ C(S). Besides, B → D can be described as a meet of some core mixed decision rules when B → D ∈ N (S). That is to say, there is an index i . In such case, for any g 0 ∈ T − T 0 , we have i∈T 0 g with the definition of mixed decision rules. On the other hand, g Sufficiency. It is conceivable that there does not exist

Corollary 2 Let S be O-consistent. For any
Proof It is immediate from Theorem 11. Thus, Theorem 11 provides a suggestion for all the nonredundant mixed decision rules from the weak-basis from the weak-basis.

Mixed decision rules in the decision formal context
Generally, a decision formal context may not always Oconsistent. As previously stated, B → D ∈ R(S) is non-redundant if B ⇓ M = D ⇓ N . In this subsection, we shall discuss the unequal situations. In addition, it is worth pointing out B → X ⇑ N ∈ R(S) for (X, B) ∈ OML(G, M, I ) with none of X and B is empty.
It is conceivable that if there are not two identical rows in (G, M, I ), that is, g

Corollary 3 There exists no
Proof It is immediate from the above analysis.
That is to say, g ⇑ M → D 0 / ∈ R(S). Furthermore, we shall have the following results. Table 8 shows a decision formal context. The lattice OML(G, M, I ) is presented in Fig. 3. And the lattice OML(G, N, J ) is as follows:
Comparing OML (G, M, I ) to OML(G, N, J ), we can see that S is not O-consistent. For g 1 , we have g  = {n 1 , n 2 , n 3 }. In this case, g 1 satisfies the conclusion of Corollary 3, and the others also meet it evidently. Furthermore, all the mixed decision rules of S is depicted in Table 9.
Sufficiency. Assume that B → X ⇑ N is a redundant mixed decision rule in R(S). By Lemma 3, we only need to Thus, we further have the following conclusions. M, I ) with both X 0 and B 0 are non-empty satisfying the following two conditions:

which implies there exists
by Lemma 1. That is to say, g Proof It easily follows from the Corollary 4.
For example, from Table 9 That is to say B → X ⇑ N ∈ R * (S) by Theorem 13.

Corollary 7 For (X, B) ∈ OML(G, M, I ) with none of X and B is empty, if
To trim some proofs, we shall present the following Lemma.

Lemma 4
For any X ∈ OML G (G, M, I ) from S, if X = ∅, the following results are available: Consequently, each mixed decision rule B → X ⇑ N is the mergence of the rules g ⇑ M → g ⇑ M ⇓ M ⇑ N (g ∈ X), that is, Then the following theorem is available.

Theorem 14
For any (X, B) ∈ OML(G, M, I ) from S with none of X and B are empty, g Proof Hypothetically, suppose an object g 0 ∈ X such that g Subsequently, such g 1 , g 2 ∈ X are available for In this case, assume that g 3 ∈ {g 1 , g 2 } ⇑ M ⇓ M while g 3 / ∈ X ⇑ N ⇓ N . Then g 3 / ∈ g∈X g ⇑ M ⇓ M ⇑ N ⇓ N so that g 3 / ∈  Proof Straightforward.
The above results show that OE −PR and OE −N R are available by decomposing all mixed decision rules in R(S)    Example 8 Comparing with Tables 6 and 11, it easy to see that each OE decision rule may be derived from a mixed decision rule. Nevertheless, m 3 m 4 m 6 → n 3 fails to be a mergence of an OE-P decision rule and an OE-N decision rule.
However, there are not certain relationships between non-redundant OE decision rules and non-redundant mixed decision rules, as well as redundant OE decision rules and redundant mixed decision rules.

Example 9
The mixed decision rule m 3 m 4 m 6 → n 3 in Table 6 is non-redundant while the OE-P decision rule m 3 m 6 → n 3 derived from it is redundant in OER in Table 11. Moreover, a mixed decision rule m 1 m 2 m 6 → n 3 in Table 6 is redundant. However, m 6 → n 3 induced from it is non-redundant.
In addition, OE decision rules in OER fail to reflect the relationship between positive condition attributes and negative decision attributes, and reveal the link among negative condition attributes and positive decision rules. And as discussed above, we can see that mixed decision rules overcome such weak point. It shows, R(S) are more complete than OER.

Case study
In this section, we apply the proposed method to a real data to make a comparison analysis to check the effectiveness. It is needed to point out the approach of non-redundant decision rules extraction is usually presented as follows, which can be seen as a general mechanism.
From above Algorithm 1, we can see that it can not neglect to construct the conditional concept lattice and decision concept lattice. This indicates Algorithm 1 is exponential time complexity. Furthermore, considering the mixed information, the algorithm for all non-redundant mixed decision rules can be obtained based on Algorithm 1. Obviously, the modified algorithm is also time consuming with exponential. In contrast, our method in fact is in light Algorithm 1 An algorithm for calculating all non redundant decision rules of formal decision context. of granular mixed decision rules, which is presented in the following.
Therefore, Algorithm 2 searches for the weak basis without the need of constructing the mixed decision concept lattice. On the other hand, the range of the iterations is |OML (G, M, I )| while the modified Algorithm 1 is |OML(G, M, I ) × OML(G, N, J )|. Not only that, a complex judge process is required in the modified Algorithm 1 from steps 5 to 9. And for the Rules acquisition of formal decision contexts based on three-way concept lattices [22], the approach is similar to Algorithm 1. Summarily, our approach has an advantage in time complexity.
Next, a data "Zoo" from UCI is used as an example to illustrate the rule count. There are 16 conditional attributes and 7 classes in Zoo. It is worth pointed out the values of conditional attributes are binary except for the 13th conditional attribute. Thereby, to construct a decision formal context, we first delete the 13th conditional attribute. On the other hand, we change the 7 classes into a multi-decision table, in which any pair columns are disjoint. Then this table is added complementarily to replace last column of Zoo. In Algorithm 2 An algorithm for the weak basis of formal decision context based on mixed information. The general decision formal context is performed by Lattice Miner, and the base of mixed decision rules and generic base of decision rules are mined respectively, from which the minimum support and confidence are set by 0 and 100%. The rule count of the former is 589, and the latter is unlikely 0, which indicates that rule with positive and negation implies more information. The reason maybe the deleted attribute. And the support distribution of the base of mixed decision rules of Zoo is presented in Fig. 5, in which we can find that different mixed decision rules have different support. Figure 6 depicts the number of rules whose support are not less than 10%, 20%, 30%, 40%, 50%, Fig. 6 The statistical information of support distribution 60%, 70%, 80% and 90% are 145, 91, 56, 35, 14,11, 6, 3 and 0, respectively.
Last, without the multi-decision table, the formal context via the conditional attributes of Zoo is considered. When the minimum support and confidence are set by 0 and 100% respectively, 330 association rule base can be obtained. And if the minimum support and confidence are set by 50% and 100% respectively, we only acquire 14 association rules. However, we can acquire 2479 association rule base with negative and positive information. In this case, if the negation is considered, the more information will be acquire.

Discussion and conclusion
In most decision making analysis, decision rules are typically built using positive information only. Nevertheless, driven by demands of practical situations, negative information also needs to be exactly represented and managed. The main results of this work was that classifying mixed decision rules into diverse parts, and the weak-basis was proposed to replace the basis of mixed decision rules. We have first discussed two types of mixed concept lattices and explored the connections between mixed concept lattice, classical concept lattice and three-way lattice, respectively. Subsequently, non-redundant mixed decision rules have been proposed and the weak-basis has been investigated to approximate the basis of mixed decision rules. In the last part, we have compared the difference between mixed decision and three-way rules. And some theoretical examples case study have been given to show the main results of our work. Notably, an improved definition of core mixed decision rules for a decision formal context is needed further to be studied to to ensure both core mixed decision rules are non-redundant. And how to proof the performance of weak-basis by comparing the classical rule reduction is also included in the future plans. Furthermore, the approach of attribute reduction is also included in our future studies.