## Karell Bertet ; Mirabelle Nebut - Efficient Algorithms on the Family Associated to an Implicational System

dmtcs:330 - Discrete Mathematics & Theoretical Computer Science, January 1, 2004, Vol. 6 no. 2 - https://doi.org/10.46298/dmtcs.330
Efficient Algorithms on the Family Associated to an Implicational System

Authors: Karell Bertet ; Mirabelle Nebut

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: \beginenumerate ıtemit is equivalent to Σ ıtemφ _Σ _do(X) (thus φ _Σ (X)) can be computed by a single scanning of Σ _do-implications ıtemit is of minimal size with respect to ISs satisfying 1. and 2. \endenumerate We give algorithms that compute Σ _do, and from Σ _do closures φ _Σ (X) and the Moore family associated to φ _Σ .

Volume: Vol. 6 no. 2
Published on: January 1, 2004
Imported on: March 26, 2015
Keywords: lattice,ordered set,Moore family,implicational system,closure operator,algorithm,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]