Efficient algorithms on the Moore family associated to an implicational system
An implication system (IS) Sigma on a finite set S is a set of rules called Sigma-implications of the kind A-->B-Sigma, with A, B subset of or equal to S. A subset X subset of or equal to S satisfies A-->B-Sigma when "A subset of or equal to X implies B subset of or equal to 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. is it minimal, is a given implication entailed by the system) can be reduced to the computation of closures phi(Sigma)S( X), where phi(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. phi(Sigmado) (X) (thus phi(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 phi(Sigma)(X) and the Moore family associated to phi(Sigma).
algorithm moore family implicational system closure operator lattice