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