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In mathematical logic, a formula is in negation normal form if the negation operator ( $\neg$ , 28 year-old Aircraft Maintenance Manufacture (Avionics) Cameron Lester from Port Coquitlam, usually spends time with hobbies which include mountain biking, property developers in singapore and train collecting. Loves to discover unknown towns and spots like Athens.

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Here is my web page; test.wbxonline.com) and disjunction ( $\lor$ , 28 year-old Aircraft Maintenance Manufacture (Avionics) Cameron Lester from Port Coquitlam, usually spends time with hobbies which include mountain biking, property developers in singapore and train collecting. Loves to discover unknown towns and spots like Athens.

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Negation normal form is not a canonical form: for example, $a \land (b \lor \neg c)$ and $(a \land b) \lor (a \land \neg c)$ are equivalent, and are both in negation normal form.

In classical logic and many modal logics, every formula can be brought into this form by replacing implications and equivalences by their definitions, using De Morgan's laws to push negation inwards, and eliminating double negations. This process can be represented using the following rewrite rules:

\neg (\forall x . G) \to \exists x . \neg G

\neg (\exists x . G) \to \forall x . \neg G

\neg \neg G \to G

\neg (G_{1} \land G_{2}) \to (\neg G_{1}) \lor (\neg G_{2})

\neg (G_{1} \lor G_{2}) \to (\neg G_{1}) \land (\neg G_{2})

A formula in negation normal form can be put into the stronger conjunctive normal form or disjunctive normal form by applying distributivity.

Examples and counterexamples

The following formulae are all in negation normal form:

(A \lor B) \land C

(A \land (\neg B \lor C) \land \neg C) \lor D

A \lor \neg B

A \land \neg B

The first example is also in conjunctive normal form and the last two are in both conjunctive normal form and disjunctive normal form, but the second example is in neither.

The following formulae are not in negation normal form:

A \Rightarrow B

\neg (A \lor B)

\neg (A \land B)

\neg (A \lor \neg C)

They are however respectively equivalent to the following formulae in negation normal form:

\neg A \lor B

\neg A \land \neg B

\neg A \lor \neg B

\neg A \land C

References

Alan J.A. Robinson and Andrei Voronkov, Handbook of Automated Reasoning 1:203ff (2001) ISBN 0444829490.

External links

Java applet for converting logical formula to Negation Normal Form, showing laws used

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+In [[mathematical logic]], a formula is in '''negation normal form''' if the [[negation]] operator (<math>\lnot</math>, {{smallcaps|not}}) is only applied to variables and the only other allowed [[Boolean algebra|Boolean operators]] are [[logical conjunction|conjunction]] (<math>\land</math>, {{smallcaps|and}}) and [[logical disjunction|disjunction]] (<math>\lor</math>, {{smallcaps|or}}).
+Negation normal form is not a canonical form: for example, <math>a \land (b\lor \lnot c)</math> and <math>(a \land b) \lor (a \land \lnot c)</math> are equivalent, and are both in negation normal form.
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+In classical logic and many [[modal logic]]s, every formula can be brought into this form by replacing implications and equivalences by their definitions, using [[De Morgan's laws]] to push negation inwards, and eliminating double negations.  This process can be represented using the following [[rewrite rule]]s:
+:<math>\lnot (\forall x. G) \to \exists x. \lnot G</math>
+:<math>\lnot (\exists x. G) \to \forall x. \lnot G</math>
+:<math>\lnot \lnot G \to G</math>
+:<math>\lnot (G_1 \land G_2) \to (\lnot G_1) \lor (\lnot G_2)</math>
+:<math>\lnot (G_1 \lor G_2) \to (\lnot G_1) \land (\lnot G_2)</math>
+A formula in negation normal form can be put into the stronger [[conjunctive normal form]] or [[disjunctive normal form]] by applying [[Distributive property|distributivity]].
+==Examples and counterexamples==
+The following formulae are all in negation normal form:
+:<math>(A \vee B) \wedge C</math>
+:<math>(A \wedge (\lnot B \vee C) \wedge \lnot C) \vee D</math>
+:<math>A \vee \lnot B</math>
+:<math>A \wedge \lnot B</math>
+The first example is also in [[conjunctive normal form]] and the last two are in both [[conjunctive normal form]] and [[disjunctive normal form]], but the second example is in neither.
+The following formulae are not in negation normal form:
+:<math>A \Rightarrow B</math>
+:<math>\lnot (A \vee B)</math>
+:<math>\lnot (A \wedge B)</math>
+:<math>\lnot (A \vee \lnot C)</math>
+They are however respectively equivalent to the following formulae in negation normal form:
+:<math>\lnot A \vee B</math>
+:<math>\lnot A \wedge \lnot B</math>
+:<math>\lnot A \vee \lnot B</math>
+:<math>\lnot A \wedge C</math>
+==References==
+* Alan J.A. Robinson and Andrei Voronkov, ''Handbook of Automated Reasoning'' '''1''':203''ff''  (2001) ISBN 0444829490.
+==External links==
+* [http://www.izyt.com/BooleanLogic/applet.php Java applet for converting logical formula to Negation Normal Form, showing laws used]
+[[Category:Propositional calculus]]
+[[Category:Normal forms (logic)]]

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