The calculation of formal concepts is a very important part in the theory of formal concept analysis (FCA); however, within the framework of FCA, computing all formal concepts is the main challenge because of its exponential complexity and difficulty in visualizing the calculating process. With the basic idea of Depth First Search, this paper presents a visualization algorithm by the attribute topology of formal context. Limited by the constraints and calculation rules, all concepts are achieved by the visualization global formal concepts searching, based on the topology degenerated with the fixed start and end points, without repetition and omission. This method makes the calculation of formal concepts precise and easy to operate and reflects the integrity of the algorithm, which enables it to be suitable for visualization analysis.
FCA, a branch of lattice theory, presented by Ganter and Wille [
Computing all formal concepts and the concept lattice is the most basic issue that has been investigated by many domestic and foreign researchers from different angles. Various forms of algorithms in concepts computing and concept lattice generation can be generally divided into three categories: batch processing algorithm [
However, the structure of concept lattice is relatively complex and the relationship among attributes of formal context cannot be visually represented. Attribute topology [
Based on the new representation, Zhang et al. present a calculation algorithm of formal concepts by attribute topology [
Global formal concepts searching of attribute topology is proposed in this paper with the basic idea of Depth First Search. This method, firstly, adds two vertices, global start and end points, and edges to the original attribute topology, degenerating it into the topology with the fixed start and end points and then, limited by the constraints and calculation rules, explores and backtracks the vertices, sorted formerly, repetitively, completing the traversal of paths. All formal concepts are achieved by the path traversing between the global start and end points. This method constructs the topology into a complete whole, avoiding the decomposition process of the whole topology, and reflects the integrity of the algorithm. Calculation process is demonstrated intuitively in the process of traversing paths, reflecting the good visibility.
Formal context, which acts as the research object and date representation, is an important basic aspect in FCA. Here are a few notions about formal context [
A formal context can be expressed as
Let
If
The concept, with the whole attributes as the intension and the corresponding object set as the extension (called global concept of whole attribute) or the whole objects as the extension and the corresponding attribute set as the intension (called global concept of whole object), is called global concept. There exist only two global concepts in
Concept lattice, the core date structure of FCA, is a complete lattice denoted by all concepts and the generalization-specialization relationships between them. Hasse diagram, equipped with the partial order of concept lattice simply and effectively, is the best way and common method to represent the concept lattice, which can express the relationships between all concepts intuitively and integrally.
Every vertex in Hasse diagram is a concept and all vertices respect the whole concepts in formal context. The entire Hasse diagram is constructed by all the concepts together with the order inclusion relation between two formal concepts, and each layer is arranged in descending order of the extension (ascending order of the intension).
Formal contexts mentioned in this paper are all simplified contexts [
From the perspective of graph theory, the attribute topological representation is the weighted graph which concerns the relationship among attributes. So it can use the storage method of graph, that is, adjacency matrix sequence. In [
Suppose
In this paper, adjacency matrix, induced in [
Here are a few notions about attribute topology [
Global attribute is the attribute that possess all objects. In the formal context
Empty attribute is the attribute that does not possess any object in formal context. In
According to the lattice theory, the global objects and global attributes only emerge on the top and at the bottom of the concept lattice, and they will not influence the structure of the concept lattice, so the global objects and global attributes can be reduced in concept lattice [
In attribute topology, if any
Edges connected to the top-attributes in attribute topology are bidirectional or unidirectional pointing to the outside.
In
In
A top-attribute is definitely not the child-attribute.
Global formal concepts searching of attribute topology are induced in the paper in order to construct the relationship between concepts while calculating all the concepts. Limited by the constraints and calculation rules, a series of paths are formed through exploring and backtracking the vertices of attributes sorted formerly while computing all formal concepts, in which removes the relevant edges of topology simultaneously, based on the topology degenerated with the fixed start and end points.
According to [
This paper, firstly, establishes the global attribute
Topological degeneration model mentioned above is described as follows: vertices
The establishment of the topological degeneration model can be divided into the following two cases.
The topological degeneration model is established by the following example: Table
A formal context that contains child-attributes.
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0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
2 | 1 | 1 | 1 | 0 | 1 | 1 | 0 |
3 | 1 | 1 | 1 | 0 | 0 | 1 | 1 |
4 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
5 | 1 | 1 | 1 | 1 | 0 | 0 | 1 |
6 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |
7 | 1 | 0 | 0 | 0 | 0 | 1 | 1 |
8 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
The attribute topology of the formal context from Table
In the formal context (see Table
The topological degeneration model from the topology shown in Figure
The degeneration of the topology (see Figure
The degeneration of the topology (see Figure
Above analysis shows that the added two vertices and edges have no influence on the original association and correlation intensity between attributes. That is, the structure of the original topology has no changes, still containing the association and correlation intensity between all attributes and objects. The association and correlation intensity required in calculating the concepts are not damaged, and subsequent calculation of concepts based on the degenerated topology has not been affected.
From the perspective of graph theory, a complete diagram with fixed start and end points, suitable for the following path searching, is achieved by the degeneration.
The traversal on path, the case of exploring and backtracking vertices, is fundamentally carried from the global start point
For
Let
According to Definition
Let
According to the above analysis,
In the formal context shown in Table
where
for for if
If
For
If
According to Definition for similarly, for then for that is,
similarly, for that is,
For
Suppose since so
according to Definition obviously, Formulae ( So
In the formal context presented in Table
As Definition
where
As shown in Definition
According to the above analysis, the path can be represented by
Suppose adding a new vertex
Consider the following:
Consider the following:
Consider the following:
and then
since so for
According to Formulae (
So
This algorithm begins with the first element, that is, start point
Suppose
Flowchart of the process of exploring attribute
As seen in Figure
for
Figure
It is needed to traverse the attribute next to attribute
Then we analyze the exploring process through an example. For the formal context shown in Table
This algorithm begins with the start point, that is,
The process above can be demonstrated by Figure
The updated path from the path
Figure
Additionally, suppose
That is,
Suppose the current path is
Suppose the current attribute explored
Simultaneously, the set
If
If
If
On the contrary, updates on the two aspects described above will not be induced if the current attribute explored
Then we analyze the date updated through an example.
For the formal context shown in Table
The path updated from the path listed in Figure
The updated data is shown as follows.
The update on set
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Additionally, Suppose
The current path
Suppose the current attribute explored
The path updated from the path listed in Figure
The updated data is shown as the following.
Table
The update on set
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({0, 2, 3, 5, 7, 8 | |
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Suppose the set of all attributes of formal context is
Backtracking the vertex occurs if the current attribute explored
Suppose the current path is
Combined with the description above, the process of backtracking vertices is listed in Figure
Flowchart of the process of backtracking vertices.
The process of backtracking vertices is finished when it meets the condition
Then, the process of backtracking vertices is illustrated by the example.
Suppose the current path is shown in Figure
Suppose the current attribute explored
Then a judge on
The path updated from the path listed in Figure
Repeating the judge on
Flowchart of the complete algorithm is listed in Figure
Flowchart of the complete algorithm.
As seen in Figure
Figure
The update on set updated topology is shown in Figure updated set
The updated attribute topology after the end of the algorithm.
Each two-tuples
The traversal process of all paths between the start and end points is illustrated by the tree structure shown in Figure
The path traversal tree of the attribute topology presented in Figure
Figure
Any node
As seen in Figure
The concept tree of the attribute topology presented in Figure
Figure
Hasse diagram of the formal context presented in Table
The concepts successively represented by c(0)–c(47) are (012345678,ø), (01345678,
Under the constrains and calculation rules, the process of computing all formal concepts is achieved by traversing vertices successively. The path is formed by the traversed vertices and all concepts are induced in the process of traversing all paths. As shown in Figures
As seen in Figure
Global formal concepts searching of attribute topology is proposed in this paper based on the concepts of calculating with subtopologies. With the basic idea of Depth First Search, this algorithm, beginning with the global start point, employs the constraints and calculation rules to explore and backtrack the attributes of the degenerated topology repeatedly until traversal of all paths is achieved. The set of concepts is updated throughout the process and all concepts are obtained ultimately. This method avoids the decomposition process of the whole topology, which reflects the integrity of the algorithm. Visualization features of the calculation process are enhanced by the concept tree. This method makes the whole process more logical and feasible, easy to implement, and suitable for large-scale data sets. Attribute topology provides a new approach of representation of the formal context. The further step is to refine and optimize the method and put it into application.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is partially supported by the National Natural Science Foundation of China (nos. 61273019 and 81373767) and National Social Science Foundation of China (no. 12BYY121). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.