# Definition:Negation Normal Form

## Definition

A propositional formula $P$ is in negation normal form (NNF) if and only if:

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.