### Abstract

Closure system is a fundamental concept appearing in several areas such as databases, formal concept analysis, artificial intelligence, etc. It is well-known that there exists a connection between a closure operator on a set and the lattice of its closed sets. Furthermore, the closure system can be replaced by a set of implications but this set has usually a lot of redundancy inducing non desired properties. In the literature, there is a common interest in the search of the minimality of a set of implications because of the importance of bases. The well-known Duquenne-Guigues basis satisfies this minimality condition. However, several authors emphasize the relevance of the optimality in order to reduce the size of implications in the basis. In addition to this, some bases have been defined to improve the computation of closures relying on the directness property. The efficiency of computation with the direct basis is achieved due to the fact that the closure is computed in one traversal. In this work, we focus on the D-basis, which is ordered-direct. An open problem is to obtain it from an arbitrary implicational system, so it is our aim in this paper. We introduce a method to compute the D-basis by means of minimal generators calculated using the Simplification Logic for implications.

Original language | English |
---|---|

Title of host publication | CEUR Workshop Proceedings |

Publisher | CEUR-WS |

Pages | 217-228 |

Number of pages | 12 |

Volume | 1466 |

ISBN (Print) | 9782954494807 |

Publication status | Published - 2015 |

Event | 12th International Conference on Concept Lattices and Their Applications, CLA 2015 - Clermont-Ferrand, France Duration: Oct 13 2015 → Oct 16 2015 |

### Other

Other | 12th International Conference on Concept Lattices and Their Applications, CLA 2015 |
---|---|

Country | France |

City | Clermont-Ferrand |

Period | 10/13/15 → 10/16/15 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*CEUR Workshop Proceedings*(Vol. 1466, pp. 217-228). CEUR-WS.

**From an implicational system to its corresponding D-basis.** / Rodríguez-Lorenzo, Estrella; Adaricheva, Kira; Cordero, Pablo; Enciso, Manuel; Mora, Angel.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*CEUR Workshop Proceedings.*vol. 1466, CEUR-WS, pp. 217-228, 12th International Conference on Concept Lattices and Their Applications, CLA 2015, Clermont-Ferrand, France, 10/13/15.

}

TY - GEN

T1 - From an implicational system to its corresponding D-basis

AU - Rodríguez-Lorenzo, Estrella

AU - Adaricheva, Kira

AU - Cordero, Pablo

AU - Enciso, Manuel

AU - Mora, Angel

PY - 2015

Y1 - 2015

N2 - Closure system is a fundamental concept appearing in several areas such as databases, formal concept analysis, artificial intelligence, etc. It is well-known that there exists a connection between a closure operator on a set and the lattice of its closed sets. Furthermore, the closure system can be replaced by a set of implications but this set has usually a lot of redundancy inducing non desired properties. In the literature, there is a common interest in the search of the minimality of a set of implications because of the importance of bases. The well-known Duquenne-Guigues basis satisfies this minimality condition. However, several authors emphasize the relevance of the optimality in order to reduce the size of implications in the basis. In addition to this, some bases have been defined to improve the computation of closures relying on the directness property. The efficiency of computation with the direct basis is achieved due to the fact that the closure is computed in one traversal. In this work, we focus on the D-basis, which is ordered-direct. An open problem is to obtain it from an arbitrary implicational system, so it is our aim in this paper. We introduce a method to compute the D-basis by means of minimal generators calculated using the Simplification Logic for implications.

AB - Closure system is a fundamental concept appearing in several areas such as databases, formal concept analysis, artificial intelligence, etc. It is well-known that there exists a connection between a closure operator on a set and the lattice of its closed sets. Furthermore, the closure system can be replaced by a set of implications but this set has usually a lot of redundancy inducing non desired properties. In the literature, there is a common interest in the search of the minimality of a set of implications because of the importance of bases. The well-known Duquenne-Guigues basis satisfies this minimality condition. However, several authors emphasize the relevance of the optimality in order to reduce the size of implications in the basis. In addition to this, some bases have been defined to improve the computation of closures relying on the directness property. The efficiency of computation with the direct basis is achieved due to the fact that the closure is computed in one traversal. In this work, we focus on the D-basis, which is ordered-direct. An open problem is to obtain it from an arbitrary implicational system, so it is our aim in this paper. We introduce a method to compute the D-basis by means of minimal generators calculated using the Simplification Logic for implications.

UR - http://www.scopus.com/inward/record.url?scp=84950149462&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84950149462&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84950149462

SN - 9782954494807

VL - 1466

SP - 217

EP - 228

BT - CEUR Workshop Proceedings

PB - CEUR-WS

ER -