Translating between Horn Representations and their Characteristic Models

TL;DR

This paper investigates the computational complexity of translating between Horn expressions and their characteristic models, revealing that these translation problems are as hard as the hypergraph transversal problem, with implications for AI and database theory.

Contribution

It establishes the polynomial equivalence of translation and decision problems between Horn representations and characteristic models, linking them to the hypergraph transversal problem.

Findings

Translation problems are polynomially equivalent to decision problems.

These problems are at least as hard as the hypergraph transversal problem.

In some cases, they are equivalent to the hypergraph transversal problem.

Abstract

Characteristic models are an alternative, model based, representation for Horn expressions. It has been shown that these two representations are incomparable and each has its advantages over the other. It is therefore natural to ask what is the cost of translating, back and forth, between these representations. Interestingly, the same translation questions arise in database theory, where it has applications to the design of relational databases. This paper studies the computational complexity of these problems. Our main result is that the two translation problems are equivalent under polynomial reductions, and that they are equivalent to the corresponding decision problem. Namely, translating is equivalent to deciding whether a given set of models is the set of characteristic models for a given Horn expression. We also relate these problems to the hypergraph transversal problem, a well known problem which is related to other applications in AI and for which no polynomial time algorithm is known. It is shown that in general our translation problems are at least as hard as the hypergraph transversal problem, and in a special case they are equivalent to it.