  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.
