Hierarchical decompositions of implicational bases for the enumeration of meet-irreducible elements

TL;DR

This paper introduces a polynomial-time hierarchical decomposition method for implicational bases to facilitate the translation to meet-irreducible elements in closure systems, especially effective in acyclic convex geometries.

Contribution

It proposes a novel recursive decomposition technique for implicational bases and provides a new characterization for meet-irreducible elements in acyclic cases.

Findings

Decomposition can be performed in polynomial time and space.

Acyclic splits enable recursive characterization of meet-irreducible elements.

New results for translation in acyclic convex geometries using hypergraph dualization.

Abstract

We are interested in the problem of translating between two representations of closure systems, namely implicational bases and meet-irreducible elements. Albeit its importance, the problem is open. Motivated by this problem, we introduce splits of an implicational base. It is a partitioning operation of the implications which we apply recursively to obtain a binary tree representing a decomposition of the implicational base. We show that this decomposition can be conducted in polynomial time and space in the size of the input implicational base. In order to use our decomposition for the translation task, we focus on the case of acyclic splits. In this case, we obtain a recursive characterization of the meet-irreducible elements of the associated closure system. We use this characterization and hypergraph dualization to derive new results for the translation problem in acyclic convex geometries.