×

Efficient algorithms on the Moore family associated to an implicational system. (English) Zbl 1062.06004

Summary: An implication system (IS) \(\Sigma\) on a finite set {\({S}\)} is a set of rules called \(\Sigma\)-implications of the kind \(A\rightarrow_\Sigma B\), with \(A,B\subseteq S\). A subset \(X\subseteq S\) satisfies \(A \rightarrow_\Sigma B\) when “\(A\subseteq X\) implies \(B\subseteq X\)” 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 \(\Sigma\) (e.g. whether it is minimal, or whether a given implication is entailed by the system) can be reduced to the computation of closures \(\varphi _\Sigma(X)\), where \(\varphi _\Sigma\) is the closure operator associated to \(\Sigma \). We propose a new approach to compute such closures, based on the characterization of the direct-optimal IS \(\Sigma _{do}\) which has the following properties: 1. it is equivalent to \(\Sigma\); 2. \(\varphi _{\Sigma _{do}}(X)\) (thus \(\varphi _\Sigma(X)\)) can be computed by a single scanning of \(\Sigma _{do}\)-implications; 3. it is of minimal size with respect to ISs satisfying 1. and 2. We give algorithms that compute \(\Sigma _{do}\), and from \(\Sigma _{do}\) closures \(\varphi_\Sigma(X)\) and the Moore family associated to \(\varphi_\Sigma\).

MSC:

06A15 Galois correspondences, closure operators (in relation to ordered sets)
68P15 Database theory
68W05 Nonnumerical algorithms
PDFBibTeX XMLCite
Full Text: EuDML EMIS