Abstract
The idea of stability has been used in many applications. However, computing stability is still a challenge and the best algorithms known so far have algorithmic complexity quadratic to the size of the lattice. To improve the effectiveness, a critical term is introduced in this paper, that is, minimal generator, which serves as the minimal set that makes a concept stable when deleting some objects from the extent. Moreover, by irreducible elements, minimal generator is derived. Finally, based on inclusion-exclusion principle and minimal generator, formulas for the calculation of concept stability are proposed.
1. Introduction
When making scientific hypotheses about the cause of a natural phenomenon, some data should be gathered randomly to certain extent. A best hypothesis should be independent of this randomness, which means the hypothesis is not determined by any particular piece of data. In Kuznetsov’s research, this sort of independence was called stability [1, 2], and it was used in many occasions such as succinct representation of lattice based taxonomies [3–5], jackknife estimation towards sample functions [6], and the work of Carnap on inductive logic [7].
However, computing stability of a concept was proved to be a #P-complete problem and the best algorithms known so far have algorithmic complexity quadratic to the size of the lattice [8]. In this paper we will reconsider this problem and give a new method to improve the computational efficiency.
2. Basic Definitions in FCA
Formal concept analysis (FCA) is a generally appropriate framework for building categories defined as object sets sharing some attributes, irrespectively of a particular domain of application [9–13]. This presents a convincing formal model of the philosophical notion of a concept characterized extensionally by the set of entities and intensionally by the set of attributes they have in common.
Before proceeding, we briefly recall the FCA terminology and properties [12, 13]. Given a formal context , where is called a set of objects, is called a set of attributes, and the binary relation specifies which objects have which attributes, the derivation functions and are defined for and as follows:
is the set of attributes common to all objects of and is the set of objects sharing all attributes of , respectively.
A formal concept of the extent is a pair , where , , , and . The set is called the extent and is called the intent of the concept .
A concept is subconcept of if (equivalently, ). In this case, is called a superconcept of . We write and define the relations , , and as usual. If and there is no such that , then is a lower neighbor of and is an upper neighbor of ; notation: and .
The set of all concepts ordered by relation ≤ forms a lattice, which is denoted by and called the concept lattice of the context . The relation defines the covering graph of .
Let be a formal context and , , , , , . Then the following propositions hold:(1), ,(2), ,(3), ,(4).
3. Stability Calculation Based on Minimal Generator
Before proceeding, let us give two symbols which will be used frequently in the following discussion. For a given set ,(1) indicates the cardinality of ;(2) indicates the power set of .
Definition 1 (see [1, 2, 8]). Let be a formal context and a formal concept of . The stability index of is defined as follows:
Proposition 2. Let be a formal context and be a formal concept of . If there is a set with , then where .
Proof. Given , then . Since is a concept and , then is valid.
Proposition 3. Let , be two sets and . Then .
Proof. Noting is the subset of with where , it follows that .
Definition 4 (see [14]). Let be a formal context and a formal concept of . If there is a subset of which makes , and for any , , then we call the minimal generator of concept .
Remark 5. It is worthwhile to note that for a given concept, it may have more than one minimal generator. A minimal generator of a formal concept is a minimal set that makes the intent of a concept stable.
Theorem 6. Let be a formal context and a formal concept of . If has only one minimal generator , then
Proof. Since is the minimal generator of , by Propositions 2 and 3, we obtain . Then by Definition 1, we have , and a simple manipulation leads to the conclusion .
Theorem 7. Let be a formal context and a formal concept of . If has a family of minimal generators , then
Proof. Since is the family of the minimal generators of , by Propositions 2 and 3, we have .
According to inclusion-exclusion principle, we can show that = − + + . By Definition 1, it follows that = = .
Corollary 8. Let be a formal context and a formal concept of . If has only two minimal generators and , then
Followed by the above discussion, the only thing left to get the stability index of a formal concept is to find its minimal generator, which will be discussed in Section 4.
4. Minimal Generator Computation
Definition 9 (see [15]). Let be a formal context. Object is called a full attributes object if . Dually, attribute is called the largest common attribute if .
Definition 10 (see [16]). Let be a formal context. Object is called a reducible object if there exists a series of objects that makes where is the index set. Dually, attribute is called a reducible attribute if there exists a series of attributes that makes where is the index set.
Definition 11. Let be a formal context. is called a purified formal context, provided that there exists no full attributes object, no largest common attribute, no reducible object, and no reducible attribute.
Definition 12. Let be a formal context. A concept is called an object concept if it has the form , , and is called its object label. Dually, a concept is called an attribute concept if it has the form , , and is called its attribute label.
Theorem 13. Let be a purified formal context. Then any object concept of must be an upper bound irreducible element and vice versa.
Proof. Assume that an object concept is not an upper bound irreducible element; then has at least two lower neighbors, and denoting them by , is the index set. Since is an object concept, there exists an object such that . By basic theorem on concept lattice, we have . By the fact that , it follows that , which means is a irreducible object, and this is contradict to the definition of purified formal context. The reverse implication is proved in much the same way. Hence, this theorem holds.
Theorem 14. Let be a purified formal context. If a concept is an object concept, then its minimal generator consists of its object label.
Proof. The theorem is immediate from Definitions 4 and 12.
Definition 15. Let ( is the index set) be a set of objects. Function simplification on set is defined as
Lemma 16. Let ( is the index set) be a set of objects. Then .
Proof. Without loss of generality, we assume that there exist two objects and with . Then , so a simple manipulation leads to the equation .
Lemma 17. Let be a purified formal context and suppose is not an object concept. Then minimal generator of is contained in the union of the minimal generators of its any two lower neighbors.
Proof. As is not an object concept, there exist at least two lower neighbors, denoting them by with their minimal generators , respectively. By the fact that , it follows that . Moreover, for any , we can see
Suppose of the above expression holds. Then there exists a concept with , such that , for any , and this is contradiction to the assumption that are the lower neighbors of . Hence, the theorem is proved.
Theorem 18. Let be a purified formal context. If is not an object concept, then its minimal generator is the simplification of the union of its any two lower neighbors’ minimal generators.
Proof. By Lemmas 16 and 17 and in light of Definitions 4 and 15, the proof is trivial.
Based on Theorems 14 and 18, we get the following recursive formula on the calculation of the minimal generator of a given concept .
If is an object concept, then consists of its object label.
Otherwise, is the simplification of union of its any two lower neighbors’ minimal generators.
5. Example and Analysis
Example 1. Consider descriptions of several objects in Table 1, which is the well known formal context “biology and water” in [16].
The corresponding concept lattice of the above formal context “biology and water” is sketched by Figure 1.
For convenience, in the representation of a concept we omit the curly braces and commas. For example, we use instead of .
Considering the stability index of concept , there are two steps.
Step 1 (find minimal generator). As is not an object concept, its minimal generator is determined by its lower neighbors and . So we must investigate the minimal generators of and , respectively. This downward recursive call continues until object concept is encountered. The integral computation process is illustrated by Figure 2, in which seven concepts are visited with concept involved.
Step 2 (compute stability index). According to Theorem 6, it follows that
Given a formal context (, , ), time complexity of concept lattice construction is (let be the number of concepts contained in concept lattice) [17]. When calculating the minimal generator of a given concept, we need to recursively traverse its subconcepts. The number of the visited concepts is apparently less than , so the time complexity of getting minimal generator of a given concept is .
Since stability index of a given concept can be derived directly by Theorems 6 and 7, the time complexity of calculating stability of a given concept is still .
Although upper estimate is the same as the already known algorithm [2], our method is more effective and there is a major reason to account for this.
Let be a concept; its integral stability index is derived by summing over stability index with level ranging from 2 to . Apparently each level must be taken into consideration which means all the subconcepts of must be visited [2]. But in our method, integral stability index is determined only by its minimal generator. When calculating the minimal generator of , instead of visiting all the subconcepts of it, the downward recursive traverse will terminate if object concept is encountered, and all the subconcepts of object concept will not be visited any more.
By using Example 1, a comparison between our method and [2] is conducted with respect to the number of concepts visited, and the result is shown in Table 2.
If we randomly select a concept from Table 2, we can see that the average number of concepts visited by our method is 50/17, whereas the number is 61/17 while using the method in [2].
Finally, our method shows its advantage when computing stability indexes of all concepts. The procedure is depicted by Algorithm 1.
| ||||||||||||||||||||||||||
Apparently, in Algorithm 1 every concept is visited only once, so time complexity of this algorithm is still . But if using the method in [2], we have to compute stability index of each concept one by one, and thus the overall time complexity is , that is, , which is greater than that of Algorithm 1.
6. Conclusions
The purpose of this paper is to find a more effective way to calculate stability of a given concept. The critical step of our method is to find minimal generator, and a recursive method is given with the irreducible elements as the ending condition. In light of minimal generator, stability can be computed by using inclusion-exclusion principle.
A valuable further work may be how to extend the results to the setting of heterogeneous information system or to the setting of fuzzy formal context.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work presented in this paper is supported by Doctorial Foundation of Henan Polytechnic University (B2011-102). The author also gratefully acknowledges the helpful comments and suggestions of the reviewers, which have improved the presentation.