Definition:Negation Normal Form

Definition
A propositional formula $P$ is in negation normal form (NNF) :
 * The only logical connectives connecting substatements of $P$ are Not, And and Or, that is, elements of the set $\left\{{\neg, \land, \lor}\right\}$;
 * The Not sign $\neg$ appears only in front of atomic statements.

That is $P$ is in negation normal form iff it consists of literals, conjunctions and disjunctions.

Examples

 * $\left({\neg p \lor q \lor r}\right) \land \left({\neg q \lor r}\right) \land \left({\neg r}\right)$

is in NNF, and also in Conjunctive Normal Form (CNF).


 * $\left({\neg p \lor q \lor r}\right) \land \left({\left({p \land \neg q}\right) \lor r}\right) \land \left({\neg r}\right)$

is in NNF, but not in CNF because there is a conjunction buried in the second disjunction.


 * $\left({\neg p \lor q \lor r}\right) \land \neg \left({\neg q \lor r}\right) \land \left({\neg r}\right)$

is not in NNF because there is a Not before the second disjunction (only atoms are allowed to be negated).


 * $\left({\neg p \land q \land r}\right) \lor \left({\neg q \land r}\right) \lor \left({\neg r}\right)$

is in NNF, and also in Disjunctive Normal Form (DNF).


 * $\left({\neg p \land q \land r}\right) \lor \left({\left({p \lor \neg q}\right) \land r}\right) \lor \left({\neg r}\right)$

is in NNF, but not in DNF because there is a disjunction buried in the second conjunction.


 * $\left({\neg p \land q \land r}\right) \lor \neg \left({\neg q \land r}\right) \lor \left({\neg r}\right)$

is not in NNF because there is a Not before the second conjunction.

Also see

 * Definition:Conjunctive Normal Form
 * Definition:Disjunctive Normal Form


 * Existence of Negation Normal Form of Statement