Translating between the representations of a ranked convex geometry [article]

Oscar Defrain and Lhouari Nourine and Simon Vilmin
2021 arXiv   pre-print
It is well known that every closure system can be represented by an implicational base, or by the set of its meet-irreducible elements. In Horn logic, these are respectively known as the Horn expressions and the characteristic models. In this paper, we consider the problem of translating between the two representations in acyclic convex geometries. Quite surprisingly, we show that the problem in this context is already harder than the dualization in distributive lattices, a generalization of
more » ... well-known hypergraph dualization problem for which the existence of an output quasi-polynomial time algorithm is open. In light of this result, we consider a proper subclass of acyclic convex geometries, namely ranked convex geometries, as those that admit a ranked implicational base analogous to that of ranked posets. For this class, we provide output quasi-polynomial time algorithms based on hypergraph dualization for translating between the two representations. This improves the understanding of a long-standing open problem.
arXiv:1907.09433v2 fatcat:zshldvesr5d6hjoyvdf4dpxb7m