Definition:Negation Normal Form

Definition
A logical formula $$P$$ is in negation normal form (NNF) iff:
 * 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.