In this section, the definition of IF generalized consistent decision formal context is proposed and some important properties are discussed. Similar to the case in the classical decision formal context and the IF consistent decision formal context, attribute reduction in the IF generalized consistent decision formal context is still the key issue that needs to be investigated.
3.1. The Basic Definitions and Propositions
Definition 12. Letandbe two IF concept lattices,,. If, then we say thatimplies, denoted by.
Definition 13. Letandbe two IF concept lattices. If there exists an injection:, such that
- (1)
,,
- (2)
,,
then is called an implication mapping from to .
The set of all concepts’ extensions in the range of are defined to be and it is obviously that .
It is obvious that , and every element of is included in the corresponding element of under the implication mapping, denoted by
It is obvious that the following proposition holds.
Proposition 5. Letandbe two IF concept lattices. Subsequently, there exists an implication mapping:,.
Definition 14. Letandbe two IF concept lattices. If there exists an implication mapping:, we say thatis generalized coarser than, denoted by.
Obviously, the relation of coarser in Definition 8 is the special case and the following proposition holds.
Proposition 6. Letbe an IF formal context. If,, then there must exist the following relation.
Proposition 7. Letandbe two IF concept lattices.,
Proposition 8. Letandbe two IF concept lattices. Ifandhold, then.
Proof. Suppose that and , and then there exist two implication mappings :, :. Since are two injections, and are two finite sets, . It follows that are surjections, and so they are bijections. Next, we suffice . □
Firstly, we have , according to the Definition 13.
Secondly, assume that , , is the father concept of , , so we can obtain , . Suppose that , , it follows that , . Since is the father concept of , holds. Thus, . Likewise, is the father concept of . If studying the father concept of , we can obtain the similar conclusion. By analogy, we can conclude that .
Corollary 1. Let , then is a partial ordered set.
Definition 15. Letbe an IF decision formal context. Subsequently,is called generalized consistent, if.
It is obvious that IF decision consistent formal context proposed in paper [
53] is generalized consistently.
Proposition 9. Letbe an IF decision formal context, then the following three propositions are equivalent:
- (1)
generalized consistent.
- (2)
There exists an implication mapping : .
- (3)
.
Proof. It can be easily obtained from above discussions. □
3.2. Attribute Reductions in View of the Implication Mapping
Definition 16. Letbe an IF generalized consistent decision formal context,:be an implication mapping,. We say thatis a consistent set ofbased on, if. Furthermore, if for any,does not hold, and thenis called an attribute reduction ofbased on. The intersection set of all reductions is called core ofbased on.
Definition 17. Letbe an IF generalized consistent decision formal context,:is an implication mapping and suppose thatis an index set and all of the reductions denoted by. Afterwards, conditional attributes can be classified four sorts based onas follows:
- (1)
Absolutely necessary attribute (core attribute),
- (2)
Relatively necessary attribute,
- (3)
Absolutely unnecessary attribute,
- (4)
Unnecessary attribute.
Proposition 10. Letbe an IF generalized consistent decision formal contextare two implication mappings. The reduction based onis the same with that based on, if.
Proof. It is easy to be verified.
Obviously, we can obtain the following propositions by the above definitions. □
Proposition 11. Letbe an IF generalized consistent decision formal context and:is an implication mapping. Ifand, thenis a consistent set based on.
Proof. According to Proposition 3.15. and Definition 16, the conclusion can be easily obtained. □
Proposition 12. Letbe an IF generalized consistent decision formal context and:is an implication mapping, then there must exist a reduction ofbased on.
Proof. If for any , does not hold, and then is its reduction. If there exists an attribute such that , then we study . Further, if such that , and then is a reduction. Otherwise, we study . Repeating the above process, we can find one reduction at least because is a finite set. Thus, the reduction of must exist.
In general, the reduction of is not unique. □
Example 1. An example of an IF decision formal contextis depicted in Table 1. In this context,,and.
We can find all concepts of the
by the definition, which are
,
,
,
,
,
,
,
,
,
, and
respectively, and we denote objects set
by
, which is same to others, where
Furthermore, we can obtain IF concept lattice of
, as shown as
Figure 1.
Similarly, all the concepts of
can be obtained, which are
,
,
,
,
,
, and
, respectively, where
Similarly, the IF concept lattice of
can be obtained in the following
Figure 2.
It can be easily testified that
is generalized consistently in
Table 1, and we take an implication mapping
:
i.e.,
. If we take out
from the attributes set
, then we can obtain a new IF formal context
, where
. We can get all concepts of
, which are
,
,
,
,
,
,
,
,
, and
, respectively, where
In addition, concept lattice of I
can be obtained, as shown as
Figure 3.
From
Figure 1,
Figure 2 and
Figure 3, we can easily find that
. Accordingly,
is a consistent set of
. In fact, we can find
,
,
by calculating. Hence,
is a reduction of
.
Similarly, if we take out
from the attributes set
, then we can obtain a new IF formal context
, where
We can get all concepts of
, they are
,
,
,
,
,
,
,
,
, and
, respectively, where
In addition, we can obtain concept lattice of
, as shown as
Figure 4.
Furthermore, if taking the implication mapping : as and : as . It can be easily verified that i.e., the reduction based on is same with that based on .
Corollary 2. Letbe an IF generalized consistent decision formal context and:is an implication mapping. For: Proof. Obviously.
Assumed that the core is the reduction, and the reduction is not unique, that is, there are two reductions: at least. Hence, the core of the reductions . For is the reduction, the proper subset of it (where it is the core of the reductions) must not be the reduction. This clearly contradicts the known conditions. So, if the core is the reduction, the reduction is only one.
Obviously, the following corollaries can be obtained by the above definitions and propositions. □
Corollary 3. Letbe an IF generalized consistent decision formal context and:is an implication mapping. For: Corollary 4. Letbe an IF generalized consistent decision formal context, and:is an implication mapping. For:
Since the reduction of an IF generalized consistent decision formal context based on satisfies the following conditions: (1) a consistent set. (2) is not a consistent set. In order to get reductions, it is helpful to give the necessary and sufficient conditions of consistent sets in order to more easily obtain reductions.
Proposition 13. Letbe an IF generalized consistent decision formal context,and:is an implication mapping. Subsequently, for:
Proof. Assume that is a consistent set, and then we have , according to Definition 16. For any , it satisfies , which is to say that there exists , such that . Hence, .
Conversely, it is obvious. □
Corollary 5. Letbe an IF generalized consistent decision formal context,, and:is an implication mapping. Subsequently, for: In Definition 17, conditional attributes are classified four sorts, which are absolutely necessary attribute, relatively necessary attribute, absolutely unnecessary attribute, and unnecessary attribute based on the relation between conditional attributes and decision attributes. A different kind of attribute has a different effect in reduction. Next, some propositions of the attribute will be presented.
Proposition 14. Letbe an IF generalized consistent decision formal context, and:is an implication mapping. Subsequently, for:is a set of absolutely necessary attributes ifthere exists, such that, and for any.
Proof. We only need to prove that , because if contains more one element, then we can treat as one new attribute to deal with. Suppose that is an unnecessary attribute, then is a consistent set, i.e., . For any , it satisfies . So, for any , and , and .
However, i.e., and for any i.e., according to the above conditions, which comes a contradiction. Therefore, is an absolutely necessary attribute. □
Proposition 15. Letbe an IF generalized consistent decision formal context and:be an implication mapping. Subsequently, for:is an unnecessary attributes if the following conditions hold: For anyand, if, then there exists, such that. Moreover, if there existssuch that, then.
Proof. Suppose that , . It suffices to prove that is consistent set. By Corollary 5, it remains to prove that for any , there exists , such that . For any , there exists , such that . So, suppose that , where , , and .
If for any , , then let , so we can get .
Otherwise, assume that there are
, such that
, then there exist, according the condition,
such that
. Moreover, if there exists
such that
, then
. Let
, where
then it follows that
. Hence, we know that, if
, i.e.,
, then
. Subsequently,
and
. It follows that
and so
. If
, i.e., there exists
such that
. If
, i.e.,
, then
, i.e.,
and so
. If
, i.e., then
, i.e.,
, and so
. Hence,
.
Therefore, we conclude that for any , there exists , such that . □
Proposition 16. Letbe an IF generalized consistent decision formal context and:is an implication mapping. Afterwards, for:is an unnecessary attribute if there exists, such that: for anyand, ifimplies that. Moreover, ifis an absolutely necessary attribute, thenis an absolutely unnecessary attribute.
Proof. Suppose that , , . It suffices to prove that is a consistent set. By Corollary 5. it remains to prove that, for any , there exists such that . For any , there exists , such that . So, suppose that , where , , and . If for any , , then let , so we can get . □
Otherwise, there exists
, such that
, then
. Denote
to be the set whose elements satisfy the condition that
. Subsequently,
and thus
. Let
, where
then it follows that
. Accordingly, we know that, if
, i.e.,
, then
and
. Afterwards,
and
. It follows that
and so
. If
, then
or
. If
, then
. If
, then
, and then
, i.e.,
and so
i.e.,
. Hence,
. Thus,
.
Therefore, we conclude that for any , there exists , such that . In conclusion, is an unnecessary attribute.
Moreover, suppose that is an absolutely necessary attribute and is a consistent set that contains . Since is an absolutely necessary attribute, we have , thus is also a consistent set, i.e., is not a reduction. Therefore, is an absolutely unnecessary attribute.
Corollary 6. Letbe an IF generalized consistent decision formal context and:is an implication mapping. Subsequently, for:is an absolutely unnecessary attribute if for anyand,holds.
3.3. Approach to Attribute Reduction in View of the Implication Mapping
The discernibility matrix and discernibility function are useful tools in computing all reductions for information tables [
5], which we introduce to compute all reductions for an IF generalized consistent decision formal context that is based on the conclusions discussed above. Furthermore, we discuss the approach to reduction as well as the corresponding characteristics in the following.
Definition 18. Letbe an IF generalized consistent decision formal context,:is an implication mapping, and, we define.
Subsequently, is called discernibility attributes set between and based on . is referred as discernibility matrix of an IF formal context based on .
Proposition 17. Let be an IF generalized consistent decision formal context and : is an implication mapping. Afterwards, for , the following two propositions are equivalent.
- (1)
is a consistent set of based on .
- (2)
If , then , .
Proof. We assume that property (2) does not hold. i.e., , such that and . That is to say such that and , hence . In other words, . It is paradoxical that is a consistent set of based on .
If , then , such that or or . Hence, and , such that . Accordingly, there exists . Thus, is a consistent set of based on . □
Definition 19. Letbe an IF generalized consistent decision formal context,:is an implication mapping andis discernibility matrix of. We define Subsequently, is called discernibility function of an IF generalized consistent decision formal context based on .
Proposition 18. Letbe an IF generalized consistent decision formal context and:is an implication mapping. The minimal disjunctive normal form of discernibility function isDenote , then are all reductions of IF generalized consistent decision formal context , based on . Proof. It can be easily verified by the Proposition 19, Proposition 20, and the definition of minimal disjunctive normal of discernibility function.
In view of the implication , from the above discussion, we know that to get the attribute reductions in concept lattices based on IF generalized consistent decision formal context, is equal to find the minimum consistent set , which satisfies for any . □
Corollary 7. Letbe an IF generalized consistent decision formal context and:is an implication mapping.,is the core attribute, such that.
Example 2. (Renewal Example 1) All of the reductions can be computed by discernibility matrix and discernibility function in the Example 1.
By the definition of discernibility matrix, the results are presented in
Table 2.
Through calculation and analysis, there are two reductions, which are
,
for the IF formal context in
Table 1.
are relatively necessary attributes. There are no absolutely unnecessary attribute and absolutely unnecessary attributes in this IF formal context.