5.1. The Defects of -Approximate Descriptions Based on FCA
Following the discussions in the previous section, we analyze the defects of -approximate descriptions based on FCA.
Proposition 8. Let be a formal context and be a granule. If for any object , we have , then X is -indefinable.
Proof. According to the assumption, if there exists an object and , then there is no attribute to distinguish y from X. Hence, we obtain the conclusion. □
Proposition 9. Down-set complete sets can be -definable or -indefinable.
We use the following example to confirm our statement in Proposition 9.
Example 6. Consider the formal context in Table 3. Let . Then, , , . By Proposition 8, it can be concluded that is (∧, ¬)-indefinable. Next, we discuss two down-set complete sets and . Since , we conclude that is -definable. Note that we cannot find a formula containing the connectives ∧ and ¬ only to describe . So, is -indefinable.
Proposition 10. Up-set complete sets can be -definable or -indefinable.
We use the following example to confirm our statement in Proposition 10.
Example 7. Consider the formal context in Table 3. Discuss two up-set complete sets and . Since , we conclude that is -definable. Note that we cannot find a formula containing the connectives ∧ and ¬ only to describe . So, is -indefinable.
Following the above discussions, we can obtain the following two propositions immediately.
Proposition 11. Let X be a granule. Then, both and can be -definable or -indefinable.
Proposition 12. Let X be a granule, and Then, both and can be -definable or -indefinable.
In summary, we know that both and cannot serve as the upper -approximate granule, and both and cannot act as the lower -approximate granule. Then, we conclude that FCA is not an effective tool for the -approximate descriptions of granules.
5.2. -Approximate Descriptions of Indefinable Granules Based on the Three-Way Concept Lattice
In this section, we present a method for -approximate descriptions of indefinable granules based on the three-way concept lattice. Let be a formal context and . If X is an extent of a three-way concept of K, then we call X a basic granule in the settings of three-way concept analysis.
In order to obtain simpler descriptions of granules, we first extend the notion of a minimal generator from a formal concept [
39,
47] to a three-way concept.
Definition 20. Let be a formal context and . If there are a subset E of A and a subset F of B such that and for any , , then we call a minimal generator of . Furthermore, if , we call an intent reduct of .
Proposition 13. Let be a formal context and . Then, both and are three-way concepts.
Proof. Let and . On the one hand, we have . On the other hand, as , and , we can obtain . Then, is a three-way concept. Moreover, the rest of the proposition can be proved likewise. □
According to Proposition 13, we can define two types of attribute-induced three-way concepts.
Definition 21. Let be a formal context. If a three-way concept can be rewritten as or , then this concept is referred to as an attribute-induced three-way concept. Moreover, we say that is a type-I attribute-induced three-way concept and is a type-II attribute-induced three-way concept.
Theorem 5 ([
38])
. Let and be two three-way concepts of a formal context . Then, the infimum and supremum are given by: Theorem 6. Let be a formal context and be the three-way concept lattice of K. If a three-way concept of has only one upper neighbor, then it is an attribute-induced three-way concept.
Proof. As has only one upper neighbor, it follows that is a lower bound irreducible element. In other words, , which implies that . Let . Then, according to Theorem 5, we have . Therefore, we have , which means or . If , then there must exist an attribute and such that can be rewritten as . If , then there must exist an attribute and such that can be rewritten as . To sum up, is an attribute-induced three-way concept. □
Theorem 7. Let be a formal context and a three-way concept of has at least two upper neighbors , where I is an index set and . If , then is an attribute-induced three-way concept.
Proof. Since , there exists or . If there is , then this concept can be rewritten as . If there is , then this concept can be rewritten as . To sum up, the required conclusion is at hand. □
Theorem 8. Let be a formal context and be the three-way concept lattice of K. is a minimal generator of , and is a minimal generator of .
Proof. According to the notions of an attribute-induced three-way concept and minimal generator of a three-way concept, the proof can be completed easily. □
Definition 22. Let be a pair of two attribute sets. The simplification function carried on is defined as: Definition 23. Let be a pair of two attribute sets. The simplification function carried on the first element and second element of are, respectively, defined as: Definition 24. Let and be two pairs of attribute sets. The simplification-union function carried on and is defined as: Lemma 2. Let be a pair of two attribute sets and . Then, the following properties hold:
(i) , ;
(ii) .
Proof. (i) Without loss of generality, we assume that there exist two attributes and with . Then, , so a simple manipulation leads to the equation . Moreover, can be proved likewise.
(ii) It follows immediately from Definition 24. □
Lemma 3. In a three-way concept lattice, if a concept is not an attribute-induced three-way concept, then this concept has at least two upper neighbors.
Proof. It follows immediately from Theorem 6. □
Lemma 4. In a three-way concept lattice, if a concept is not an attribute-induced three-way concept, then one of its intent reducts is the union of any two upper neighbors’ minimal generators.
Proof. Assume that is not an attribute-induced three-way concept. Lemma 3 indicates that has at least two upper neighbors and we denote them by , where I is an index set. Moreover, we denote the minimal generator of by . Furthermore, . In addition, we take and into consideration, and then we have . Thence, the proof is completed. □
Theorem 9. In a three-way concept lattice, if a concept is not an attribute-induced three-way concept, then one of its minimal generators is the simplification union of any two upper neighbors’ minimal generators.
Proof. This theorem follows immediately from Lemmas 2–4. □
To sum up, according to Theorems 6 and 7, we can identify attribute-induced three-way concepts. Based on Theorems 8 and 9, we can obtain the minimal generators of each three-way concept. Algorithm 4 computes minimal generators of each three-way concept of a given three-way concept lattice.
Algorithm 4 Computing the minimal generator of each three-way concept. |
Require:
Three-way concept lattice . Ensure: The minimal generator of each three-way concept. 1: Consider the three-way concept lattice as an undirected graph and traverse it downwards from the maximal three-way concept by using breadth-first search; 2: For the current visited concept 3: If it fulfills Theorem 5 or Theorem 6 4: Then, it is an attribute-induced three-way concept and obtains its minimal generator by using Theorem 7; 5: Else compute its minimal generator by using Theorem 8; 6: End If 7: End For 8: Return the minimal generators of all three-way concepts.
|
Let N be the number of concepts of . In Algorithm 4, it is obvious that each three-way concept is visited just once, so the time complexity of this algorithm is .
Example 8. Consider the formal context in Table 3. The three-way concept lattice is shown in Figure 4. Actually, each three-way concept corresponds to a -definable granule. By using Algorithm 4, we obtain the minimal generators of all three-way concepts except and . Then, we obtain their descriptions shown in Table 4. Definition 25. Let be a formal context and be a granule. If there exist two three-way concepts and such that , we say that X is approximately -definable.
Theorem 10. All granules are approximately -definable.
Proof. This theorem can be proved similarly to that of Theorem 3. □
Definition 26. Let be a formal context and be a granule. is called the upper -approximate three-way concept of X if and there does not exist such that . Correspondingly, is called the lower -approximate three-way concept of X if and there does not exist such that .
Similar to the case in Algorithm 1, here both the upper and lower -approximate three-way concepts of X may not be unique. According to Definitions 2 and 26, there exists an upper -approximate three-way concept of X such that
and reaches the biggest value. Similarly, there exists a lower -approximate three-way concept of X such that
and
reaches the biggest value. Actually,
is the upper
-approximate granule of
X and
is the lower
-approximate granule of
X. Then, based on Definitions 5 and 6, we can use the upper and lower
-approximate three-way concepts of
X to construct the
-prior approximate optimal description, the
-prior approximate optimal description, and the
-approximate description
of
X. The detailed procedure is shown in Algorithm 5.
Algorithm 5 Computing the -approximate description of a granule. |
Require:
Three-way concept lattice and a granule . Ensure: If X is -definable, then this algorithm outputs the description of X; otherwise, this algorithm outputs the -approximate description of X. 1: If there exists a three-way concept 2: Then Return “X is -definable” and , where is the minimal generator of ; 3: Else find the upper approximate three-way concept and lower approximate three-way concept of X; 4: Based on , we derive , where is the minimal generator of ; 5: Based on , we derive , where is the minimal generator of ; 6: Return . 7: End Else 8: End If
|
Note that the minimal generator of a concept is often more concise to describe a granule than the intent of the concept [
39]. So, Algorithm 5 adopts such a technique as usual.
The time complexity of Algorithm 5 depends on the cost of finding the upper and lower approximate three-way concepts and the calculation of minimal generators. Apparently, in order to find the upper and lower -approximate three-way concepts, the number of visited concepts is less than N. Moreover, when computing minimal generators, the number of visited concepts is still less than N. So, the time complexity of Algorithm 5 is .
Example 9. Consider the formal context in Table 3. Let . It is easy to check that X is a -indefinable granule. Then, by using Algorithm 5, we can compute its -approximate description in three steps:
(i) we obtain the upper -approximate three-way concept of X and the result is ;
(ii) we obtain the lower -approximate thee-way concept of X and the result is or ;
(iii) we derive the approximate description of X and the result is or or with .
5.3. -Approximate Descriptions of Indefinable Granules Based on a Three-Way Concept Lattice
In this section, we propose a method for -approximate descriptions of indefinable granules based on a three-way concept lattice.
Proposition 14. Let be a formal context and be a granule. If there exist and such that , then X is -indefinable; otherwise, X is -definable.
Proof. If there exist and such that , then we cannot distinguish x from y by using both positive and negative attributes. By the definition of an -indefinable granule, it follows that X is -indefinable. Otherwise, we always have a description . Hence, X is -definable. □
Definition 27. Let be a formal context. For any , a pair of operators are defined, respectively, as: Based on Proposition 14 and Definition 27, we immediately obtain the following Corollaries 4, 5, and 6.
Corollary 4. Let X be a granule. Then, both and are -definable.
Corollary 5. Let X be a granule. If or , then X is -definable.
Corollary 6. Let X be a granule. is the lower -approximate granule of X and is the upper -approximate granule of X.
Definition 28. Let be a formal context and be a granule. If there exist two -definable granules and such that , we say that X is approximately -definable.
Theorem 11. All granules are approximately -definable.
Proof. Let be a formal context and be a granule. First, we can always derive and . Moreover, Corollary 4 states that both and are -definable. Then, it follows that X is approximately -definable. □
For a given -definable granule X, we have . However, it needs time to obtain the simplest form of by performing logical operations. To solve this problem, we resort to a three-way concept lattice.
By using Algorithm 6, we can calculate the descriptions of
-definable granules.
Algorithm 6 Computing the description of a -definable granule. |
Require:
Three-way concept lattice and a -definable granule . Ensure: This algorithm outputs the description of X. 1: Represent X by its elements , , ⋯, ; 2: Let ; 3: Consider the three-way concept lattice as an undirected graph and traverse it upwards from the minimal three-way concept by using breadth-first search; 4: If there exist subsets and a three-way concept such that 5: Then delete from S and add Y into S; 6: Denote the updated result of S as ; 7: , where is the -description of ; 8: Return .
|
Example 10. Consider the formal context in Table 3. Let . By using Algorithm 6, we obtain the description of X by combining the -descriptions of the granules , and . That is, . Similar to the case in Algorithm 3, there exists an upper
-approximate granule
of
X such that
and
reaches the biggest value. Similarly, there exists a lower
-approximate granule
of
X such that
and
reaches the biggest value. Then, based on Definitions 5 and 6, we can use the upper and lower
-approximate granules of
X to construct the
-prior approximate optimal description, the
-prior approximate optimal description, and the
-approximate description
of
X. The detailed procedure is shown in Algorithm 7. In addition, the time complexity of Algorithm 7 is
.
Algorithm 7 Computing the -approximate description of a granule. |
Require:
Three-way concept lattice and a granule . Ensure: If X is -definable, then this algorithm outputs the description of X; otherwise, this algorithm outputs the -approximate description of X. 1: Compute and ; 2: If 3: Then X is -definable and compute its description by calling Algorithm 6; 4: Return ; 5: Else compute the descriptions of and by calling Algorithm 6; 6: Return . 7: End If
|
Example 11. Consider the formal context in Table 3. Let . If we add a new object 8 with the same description as that of object 3 into X, then X changes to be a -indefinable granule. According to Algorithm 7, we obtain the upper and lower -approximate granules and , respectively. And then, we have the -approximate descriptionwith