  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 the 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.
