### Abstract

In this paper, we consider functional dependencies among Boolean dependencies (BDs, for short). Armstrong relations are defined for BDs (called BD-Armstrong relations). For BDs, two necessary and sufficient conditions for the existence of BD-Armstrong relations are given. A necessary and sufficient condition for the existence of Armstrong relations for functional dependencies (FDs, for short) is given, which in some sense is more convenient than the condition given in [3]. We give an algorithm that solves the problem of deciding if two BDs imply the same set of functional dependencies. If the BDs are given in perfect disjunctive normal form, then the algorithm requires only polynomial time. Although Mannila and Räihä have shown that for some relations exponential time is needed for computing any cover of the set of FDs defined in this relation, as a consequence, we show that the problem of deciding if two relations satisfy the same set of FDs can be solved in polynomial time. Another consequence is a new correspondence of the families of functional dependencies to the families of Sperner systems. By this correspondence, the estimate of the number of databases given previously in [6] is improved. It is shown that there is a one-to-one correspondence between the closure of the FDs that hold in a BD and its so-called basic cover. As applications of basic covers, we obtain a representation of a key, the family of minimal keys and a representation of canonical covers.

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

Pages (from-to) | 83-106 |

Number of pages | 24 |

Journal | Annals of Mathematics and Artificial Intelligence |

Volume | 7 |

Issue number | 1-4 |

DOIs | |

Publication status | Published - Mar 1993 |

### Fingerprint

### Keywords

- Armstrong relations
- Boolean dependencies
- cover
- functional dependencies
- key
- Relational database
- Sperner system

### ASJC Scopus subject areas

- Applied Mathematics
- Artificial Intelligence

### Cite this

*Annals of Mathematics and Artificial Intelligence*,

*7*(1-4), 83-106. https://doi.org/10.1007/BF01556351

**Functional dependencies among Boolean dependencies.** / Demetrovics, J.; Rónyai, L.; Son, Hua Nam.

Research output: Contribution to journal › Article

*Annals of Mathematics and Artificial Intelligence*, vol. 7, no. 1-4, pp. 83-106. https://doi.org/10.1007/BF01556351

}

TY - JOUR

T1 - Functional dependencies among Boolean dependencies

AU - Demetrovics, J.

AU - Rónyai, L.

AU - Son, Hua Nam

PY - 1993/3

Y1 - 1993/3

N2 - In this paper, we consider functional dependencies among Boolean dependencies (BDs, for short). Armstrong relations are defined for BDs (called BD-Armstrong relations). For BDs, two necessary and sufficient conditions for the existence of BD-Armstrong relations are given. A necessary and sufficient condition for the existence of Armstrong relations for functional dependencies (FDs, for short) is given, which in some sense is more convenient than the condition given in [3]. We give an algorithm that solves the problem of deciding if two BDs imply the same set of functional dependencies. If the BDs are given in perfect disjunctive normal form, then the algorithm requires only polynomial time. Although Mannila and Räihä have shown that for some relations exponential time is needed for computing any cover of the set of FDs defined in this relation, as a consequence, we show that the problem of deciding if two relations satisfy the same set of FDs can be solved in polynomial time. Another consequence is a new correspondence of the families of functional dependencies to the families of Sperner systems. By this correspondence, the estimate of the number of databases given previously in [6] is improved. It is shown that there is a one-to-one correspondence between the closure of the FDs that hold in a BD and its so-called basic cover. As applications of basic covers, we obtain a representation of a key, the family of minimal keys and a representation of canonical covers.

AB - In this paper, we consider functional dependencies among Boolean dependencies (BDs, for short). Armstrong relations are defined for BDs (called BD-Armstrong relations). For BDs, two necessary and sufficient conditions for the existence of BD-Armstrong relations are given. A necessary and sufficient condition for the existence of Armstrong relations for functional dependencies (FDs, for short) is given, which in some sense is more convenient than the condition given in [3]. We give an algorithm that solves the problem of deciding if two BDs imply the same set of functional dependencies. If the BDs are given in perfect disjunctive normal form, then the algorithm requires only polynomial time. Although Mannila and Räihä have shown that for some relations exponential time is needed for computing any cover of the set of FDs defined in this relation, as a consequence, we show that the problem of deciding if two relations satisfy the same set of FDs can be solved in polynomial time. Another consequence is a new correspondence of the families of functional dependencies to the families of Sperner systems. By this correspondence, the estimate of the number of databases given previously in [6] is improved. It is shown that there is a one-to-one correspondence between the closure of the FDs that hold in a BD and its so-called basic cover. As applications of basic covers, we obtain a representation of a key, the family of minimal keys and a representation of canonical covers.

KW - Armstrong relations

KW - Boolean dependencies

KW - cover

KW - functional dependencies

KW - key

KW - Relational database

KW - Sperner system

UR - http://www.scopus.com/inward/record.url?scp=34250081015&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250081015&partnerID=8YFLogxK

U2 - 10.1007/BF01556351

DO - 10.1007/BF01556351

M3 - Article

AN - SCOPUS:34250081015

VL - 7

SP - 83

EP - 106

JO - Annals of Mathematics and Artificial Intelligence

JF - Annals of Mathematics and Artificial Intelligence

SN - 1012-2443

IS - 1-4

ER -