# Discrete Mathematics & Theoretical Computer Science

## Volume 6 n° 2 (2004), pp. 315-338

author: | Karell Bertet and Mirabelle Nebut |
---|---|

title: | Efficient Algorithms on the Family Associated to an Implicational System |

keywords: | lattice, ordered set, Moore family, implicational system, closure operator, algorithm |

abstract: | An implication system (IS) on a finite set S is a set of rules called Σ-implications of the kind A→_{Σ}B
, with A,B ⊆ S. A subset X ⊆ S
satisfies A →_{Σ} B when ``A ⊆ X implies B
⊆ X'' holds, so ISs can be used to describe constraints
on sets of elements, such as dependency or causality. ISs are
formally closely linked to the well known notions of closure
operators and Moore families. This paper focuses on their
algorithmic aspects. A number of problems issued from an IS
Σ (e.g. is it minimal, is a given implication entailed by the
system) can be reduced to the computation of closures
φ_{Σ}(X), where φ_{Σ} is the closure
operator associated to Σ. We propose a new approach to compute
such closures, based on the characterization of the direct-optimal
IS Σ_{do} which has the following properties:
- it is equivalent to Σ
- φ
_{Σdo}(X) (thus φ_{Σ}(X)) can be computed by a single scanning of Σ_{do}-implications - it is of minimal size with respect to ISs satisfying 1. and 2.
_{do}, and from Σ_{do} closures
φ_{Σ}(X) and the Moore family associated to
φ_{Σ}.
If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files. |

reference: | Karell Bertet and Mirabelle Nebut (2004),
Efficient Algorithms on the Family Associated to an Implicational System,
Discrete Mathematics and Theoretical Computer Science 6, pp. 315-338 |

bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |

ps.gz-source: | dm060209.ps.gz (81 K) |

ps-source: | dm060209.ps (239 K) |

pdf-source: | dm060209.pdf (220 K) |

The first *source* gives you the `gzipped' PostScript, the second the plain
PostScript and the third the format for the Adobe accrobat
reader. Depending on the installation of your web browser, at least
one of these should (after some amount of time) pop up a window for
you that shows the full article. If this is not the case, you should
contact your system administrator to install your browser correctly.

Due to limitations of your local software, the two formats may show up differently on your screen. If eg you use xpdf to visualize pdf, some of the graphics in the file may not come across. On the other hand, pdf has a capacity of giving links to sections, bibliography and external references that will not appear with PostScript.

Automatically produced on Sun Jun 20 16:55:13 CEST 2004 by falk