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In mathematical logic, a formula is in negation normal form if the negation operator ($\mathrm{\neg }$, 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.

Here is my web page; test.wbxonline.com) is only applied to variables and the only other allowed Boolean operators are conjunction ($\wedge$, 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.

Here is my web page; test.wbxonline.com) and disjunction ($\vee$, 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.

Here is my web page; test.wbxonline.com).

Negation normal form is not a canonical form: for example, $a\wedge (b\vee \mathrm{\neg }c)$ and $(a\wedge b)\vee (a\wedge \mathrm{\neg }c)$ 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:

$\mathrm{\neg }(\mathrm{\forall }x.G)\to \mathrm{\exists }x.\mathrm{\neg }G$

$\mathrm{\neg }(\mathrm{\exists }x.G)\to \mathrm{\forall }x.\mathrm{\neg }G$

$\mathrm{\neg }\mathrm{\neg }G\to G$

$\mathrm{\neg }(G_1\wedge G_2)\to (\mathrm{\neg }{G}_1)\vee (\mathrm{\neg }{G}_2)$

$\mathrm{\neg }(G_1\vee G_2)\to (\mathrm{\neg }{G}_1)\wedge (\mathrm{\neg }{G}_2)$

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:

$(A\vee B)\wedge C$

$(A\wedge (\mathrm{\neg }B\vee C)\wedge \mathrm{\neg }C)\vee D$

$A\vee \mathrm{\neg }B$

$A\wedge \mathrm{\neg }B$

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:

$A\Rightarrow B$

$\mathrm{\neg }(A\vee B)$

$\mathrm{\neg }(A\wedge B)$

$\mathrm{\neg }(A\vee \mathrm{\neg }C)$

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

$\mathrm{\neg }A\vee B$

$\mathrm{\neg }A\wedge \mathrm{\neg }B$

$\mathrm{\neg }A\vee \mathrm{\neg }B$

$\mathrm{\neg }A\wedge C$

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