# Translating between the representations of a ranked convex geometry

## TL;DR

This paper investigates the complexity of translating between implicational bases and characteristic models in acyclic convex geometries, revealing increased difficulty and providing algorithms for ranked subclasses.

## Contribution

It demonstrates the complexity of translation in acyclic convex geometries and offers quasi-polynomial algorithms for ranked subclasses, advancing understanding of this open problem.

## Key findings

- Translation problem is harder than dualization in distributive lattices.
- Provided quasi-polynomial algorithms for ranked convex geometries.
- Improved understanding of the complexity in convex geometry representations.

## Abstract

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.