A combinatorial min-max theorem and minimization of pure-Horn functions

Introduction A Boolean function of n variables is a mapping from {0, 1} to {0, 1}. Boolean functions naturally appear in many areas of mathematics and computer science and constitute a key concept in complexity theory. In this paper we shall study an important problem connected to Boolean functions, a so called Boolean minimization problem, which aims at finding a shortest possible representation of a given Boolean function. The formal statement of the Boolean minimization problem (BM) of course depends on how the input function is represented, and how the size of the output is measured. One of the most common representations of Boolean functions are conjunctive normal forms (CNFs). There are two usual ways how to measure the size of a CNF: the number of clauses and the total number of literals (sum of clause lengths). It is easy to see that BM is NP-hard if both input and output is a CNF (for both measures of the size of the output CNF). This is an easy consequence of the fact that BM contains the CNF satisfiability problem (SAT) as its special case (an unsatisfiable formula can be trivially recognized from its shortest CNF representation). In fact, BM was shown to be in this case probably harder than SAT: while SAT is NP-complete (i.e. Σp1-complete (Cook 1971)), BM is Σp2-complete (Umans 2001) (see also the review paper (Umans, Villa, and Sangiovanni-Vincentelli 2006) for related results). It was also shown that BM is Σp2-complete when considering Boolean functions represented by general formulas of constant depth as both the input and output for BM (Buchfuhrer and Umans 2011). Due to the above intractability result, it is reasonable to