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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}}).
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| 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:
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| :<math>\lnot (\forall x. G) \to \exists x. \lnot G</math>
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| :<math>\lnot (\exists x. G) \to \forall x. \lnot G</math>
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| :<math>\lnot \lnot G \to G</math> | |
| :<math>\lnot (G_1 \land G_2) \to (\lnot G_1) \lor (\lnot G_2)</math>
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| :<math>\lnot (G_1 \lor G_2) \to (\lnot G_1) \land (\lnot G_2)</math>
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| A formula in negation normal form can be put into the stronger [[conjunctive normal form]] or [[disjunctive normal form]] by applying [[Distributive property|distributivity]].
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| ==Examples and counterexamples==
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| The following formulae are all in negation normal form:
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| :<math>(A \vee B) \wedge C</math>
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| :<math>(A \wedge (\lnot B \vee C) \wedge \lnot C) \vee D</math>
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| :<math>A \vee \lnot B</math> | |
| :<math>A \wedge \lnot B</math>
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| 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.
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| The following formulae are not in negation normal form:
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| :<math>A \Rightarrow B</math>
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| :<math>\lnot (A \vee B)</math>
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| :<math>\lnot (A \wedge B)</math>
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| :<math>\lnot (A \vee \lnot C)</math>
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| They are however respectively equivalent to the following formulae in negation normal form:
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| :<math>\lnot A \vee B</math>
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| :<math>\lnot A \wedge \lnot B</math>
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| :<math>\lnot A \vee \lnot B</math>
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| :<math>\lnot A \wedge C</math>
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| ==References==
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| * Alan J.A. Robinson and Andrei Voronkov, ''Handbook of Automated Reasoning'' '''1''':203''ff'' (2001) ISBN 0444829490.
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| ==External links==
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| * [http://www.izyt.com/BooleanLogic/applet.php Java applet for converting logical formula to Negation Normal Form, showing laws used]
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| [[Category:Propositional calculus]]
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| [[Category:Normal forms (logic)]]
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Anybody who wrote the blog is called Eusebio. South Carolina is michael's birth place. The most beloved hobby for him and as well his kids is up to fish and he's been really doing it for a while. Filing has been his profession although. Go to his website to identify a out more: http://prometeu.net
My web-site; clash of clans unlimited gems