A New Approach of Knowledge Reduction in Knowledge Context Based on Boolean Matrix
Abstract
:1. Introduction
2. Preliminaries
2.1. Formal Concept Analysis Theory
- There exist with and ;
- There are satisfying and .
- ;
- .
2.2. Knowledge Space Theory
3. Molecular Lattice
- 1.
- .
- 2.
- .
- 1.
- , i.e., ,
- 2.
- and for with such that ,
- 1.
- , i.e., ,
- 2.
- and for with such that ,
- 1.
- For any , is an upper set, that is , there is with .
- 2.
- is an intersection–union mapping, that is suppose , then .
- 1.
- For any , is a lower set, that is, , there is with .
- 2.
- is a union-preserved mapping, that is, suppose , then .
- 1.
- and , .
- 2.
- , .
- 1.
- and , such that and ;
- 2.
- , that is, , such that .
- 1.
- ;
- 2.
- For any , there exists a minimal element in such that .
- 1.
- is called a complement-prime element if implies or .
- 2.
- is defined a join-irreducible element if implies or .
- 3.
- is seen as a prime element if implies or .
- 4.
- is called a meet-irreducible element if implies or .
- 1.
- is a complement-prime element if and only if is a join-irreducible element;
- 2.
- is a prime element if and only if is a meet-irreducible element.
4. Knowledge Reduction of Knowledge Context Based on FCA
4.1. Classification of Concepts
- ,
- ,
- ,
- ,
- ,
- ,
- .
4.2. Representing a Concept Lattice Based on a Boolean Matrix
- 1.
- if and only if for , ;
- 2.
- , ;
- 3.
- , where ;
- 4.
- , .
- 1.
- There does not exist such that , and , i.e., ;
- 2.
- and .
- 1.
- There does not exist such that , , that is, ;
- 2.
- and .
- 1.
- There exists such that and ;
- 2.
- There exist and with such that , i.e., can be represented by the union of its two sub-concepts.
Algorithm 1: Concept reduction in formal contexts |
Input: A knowledge context , where and Output://the collection of intents of all concepts 1: Generate the relation matrix 2: Complement of object relation matrix ← 3: for ton 4: if and 5: ←, , ← // delete and in M 6: end if 7: end for 8: for ton 9: if and 10: find corresponding to 11: ←, , ← 12: end if 13: end for 14: while 15: if // empty matrix 16: 17: make uniquely so that each pair of rows in is different 18: ← 19: for tox 20: find the th, ⋯, th rows in M corresponding to 21: L ← 22: end for 23: ←, N← 24: ← 25: else 26: end if 27 end while 28: ← |
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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p | q | r | s | |
---|---|---|---|---|
0 | 0 | 1 | 1 | |
1 | 0 | 1 | 1 | |
0 | 1 | 0 | 0 | |
1 | 1 | 1 | 1 | |
1 | 1 | 1 | 0 |
I | a | b | c | d | e | f |
---|---|---|---|---|---|---|
0 | 1 | 1 | 1 | 1 | 0 | |
1 | 1 | 1 | 0 | 0 | 0 | |
1 | 1 | 0 | 0 | 0 | 1 | |
0 | 0 | 0 | 1 | 1 | 0 | |
0 | 0 | 1 | 1 | 0 | 0 | |
1 | 0 | 0 | 0 | 0 | 0 |
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Yang, L.; Li, J.; Zhang, C.; Lin, Y. A New Approach of Knowledge Reduction in Knowledge Context Based on Boolean Matrix. Symmetry 2022, 14, 850. https://doi.org/10.3390/sym14050850
Yang L, Li J, Zhang C, Lin Y. A New Approach of Knowledge Reduction in Knowledge Context Based on Boolean Matrix. Symmetry. 2022; 14(5):850. https://doi.org/10.3390/sym14050850
Chicago/Turabian StyleYang, Liying, Jinjin Li, Chengling Zhang, and Yidong Lin. 2022. "A New Approach of Knowledge Reduction in Knowledge Context Based on Boolean Matrix" Symmetry 14, no. 5: 850. https://doi.org/10.3390/sym14050850
APA StyleYang, L., Li, J., Zhang, C., & Lin, Y. (2022). A New Approach of Knowledge Reduction in Knowledge Context Based on Boolean Matrix. Symmetry, 14(5), 850. https://doi.org/10.3390/sym14050850