Horn Extensions of a Partially Defined Boolean Function

Given a partially defined Boolean function (pdBf) (T,F), we investigate in this paper how to find a Horn extension $f: \{0,1\}^n \mapsto \{0,1\}$, which is consistent with (T,F), where $T \subseteq \{0,1\}^n$ denotes a set of true Boolean vectors (or positive examples) and $F \subseteq \{0,1\}^n$ denotes a set of false Boolean vectors (or negative examples). Given a pdBf (T,F), it is known that the existence of a Horn extension can be checked in polynomial time. As there are many Horn extensions, however, we consider those extensions f which have maximal and minimal sets T(f) of the true vectors of f, respectively. For a pdBf (T,F), there always exists the unique maximal (i.e., maximum) Horn extension, but there are in general many minimal Horn extensions. We first show that a polynomial time membership oracle can be constructed for the maximum extension, even if its disjunctive normal form (DNF) can be very long. Our main contribution is to show that checking if a given Horn DNF represents a minimal extension and generating a Horn DNF of a minimal Horn extension can both be done in polynomial time. We also can check in polynomial time if a pdBf (T,F) has the unique minimal Horn extension. However, the problems of finding a Horn extension f with the smallest |T(f)| and of obtaining a Horn DNF, whose number of literals is smallest, are both NP-hard.