Learning Definite Horn Formulas from Closure Queries

TL;DR

This paper introduces a new query type called closure queries for learning definite Horn formulas, presenting a polynomial-time algorithm that leverages these queries and relates to existing models and bases.

Contribution

It proposes closure queries as a novel extension for learning definite Horn formulas and provides a polynomial-time algorithm utilizing these queries.

Findings

The algorithm learns conjunctions of definite Horn clauses in polynomial time.

Closure queries extend membership queries and relate to correction queries.

Connections between different query models are established through reductions and algorithm analysis.

Abstract

A definite Horn theory is a set of n-dimensional Boolean vectors whose characteristic function is expressible as a definite Horn formula, that is, as conjunction of definite Horn clauses. The class of definite Horn theories is known to be learnable under different query learning settings, such as learning from membership and equivalence queries or learning from entailment. We propose yet a different type of query: the closure query. Closure queries are a natural extension of membership queries and also a variant, appropriate in the context of definite Horn formulas, of the so-called correction queries. We present an algorithm that learns conjunctions of definite Horn clauses in polynomial time, using closure and equivalence queries, and show how it relates to the canonical Guigues-Duquenne basis for implicational systems. We also show how the different query models mentioned relate to each other by either showing full-fledged reductions by means of query simulation (where possible), or by showing their connections in the context of particular algorithms that use them for learning definite Horn formulas.