Efficient Algorithms on the Moore Family Associated to an Implicational System

An implication system (IS) Σ on a finite setS is a set of rules called Σ-implications of the kindA→Σ B, with A,B⊆ S. A subsetX⊆SsatisfiesA→Σ B when “A⊆X impliesB⊆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 φΣ(X), whereφΣ 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 Σdo which has the following properties: 1. it is equivalent to Σ 2. φΣdo (X) (thusφΣ(X)) can be computed by a single scanning of Σdo-implications 3. it is of minimal size with respect to ISs satisfying 1. and 2. We give algorithms that compute Σdo , and fromΣdo closuresφΣ(X) and the Moore family associated toφΣ.