The " n "-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number of 2 " n " & minus; 1 monomials of length " n " and all conjunctive normal forms have the maximal number of 2 " n " & minus; 1 clauses of length " n ".
32.
For example, he showed the problem 3SAT ( the Boolean satisfiability problem for expressions in conjunctive normal form with exactly three variables or negations of variables per clause ) to be NP-complete by showing how to reduce ( in polynomial time ) any instance of SAT to an equivalent instance of 3SAT . ( First you modify the proof of the Cook Levin theorem, so that the resulting formula is in conjunctive normal form, then you introduce new variables to split clauses with more than 3 atoms.
33.
For example, he showed the problem 3SAT ( the Boolean satisfiability problem for expressions in conjunctive normal form with exactly three variables or negations of variables per clause ) to be NP-complete by showing how to reduce ( in polynomial time ) any instance of SAT to an equivalent instance of 3SAT . ( First you modify the proof of the Cook Levin theorem, so that the resulting formula is in conjunctive normal form, then you introduce new variables to split clauses with more than 3 atoms.
34.
Papadimitriou and Yannakakis go on to complete this class by defining "'MaxSNP "', the class of all problems with an L-reduction ( " linear reduction ", not " log-space reduction " ) to problems in "'MaxSNP 0 "', and show that it has a natural complete problem : given an instance of 3CNFSAT ( the boolean satisfiability problem with the formula in conjunctive normal form and at most 3 literals per clause ), find an assignment satisfying as many clauses as possible.
