On minimum representations of matched formulas

Abstract

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. We present here two results for matched CNFs: The first result is a shorter and simpler proof of the fact that Boolean minimization remains complete for the second level of polynomial hierarchy even if the input is restricted to matched CNFs. The second result is structural - we show that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.