### Efficient Algorithms on the Family Associated to an Implicational System

*Karell Bertet, Mirabelle Nebut*

#### 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 φΣ(X), where φΣ 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:

- it is equivalent to Σ
- φΣdo(X) (thus φΣ(X)) can be computed by a single scanning of Σdo-implications
- it is of minimal size with respect to ISs satisfying 1. and 2.

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