Two-face horn extensions

Abstract

A partially defined Boolean function (pdBf) (T, F) generalizes a Boolean function, by allowing that the function values on some inputs are unknown, where T, F ⊆ {0,1}ⁿ are disjoint sets of true and false vectors, respectively. A pdBf (T, F) often arises in conjunction with data analysis, and in such a case, it is natural and important to decide if (T, F) has an extension f : {0,1}ⁿ → {0,1} such that f(v) = 1 (resp., 0) for all χ ε T (resp., χ ε F). It may also be required that such an extension f is Horn, because the false set of a Horn function can be described by Horn rules (whose satisfiability problem can be solved in polynomial time). In this paper, we introduce two interesting restrictions of Horn functions, namely double and bidual Horn functions, because both true and false sets of such functions can be described by Horn-like rules, and furthermore, abduction, one of the basic operation of expert systems, for such functions can be done in polynomial time, whereas abduction for general Horn functions is NP-hard. Our main results show that deciding if a pdBf has extensions of such functions can be done in polynomial time.