Disjunctive normal form - Definition and Overview

In Boolean logic, Disjunctive Normal Form (DNF) is a method of standardizing and normalizing logical formulas. As a normal form, it is useful in automated theorem proving. A logical formula is considered to be in DNF if and only if it is a single disjunction of one or more conjunctions of one or more literals. As in Conjunctive Normal Form, the only propositional operators in DNF are and, or, and not. The not operator can only be used as part of a literal, which means that it can only precede a propositional variable. For example, all of the following formulas are in DNF:

<math>A \or B<math>

<math>A<math>

<math>(A \and B) \or C<math>

<math>(A \and \neg B \and \neg C) \or (\neg D \and E \and F)<math>

However, the following formulas are not in DNF:

<math>\neg(A \or B)<math> — NOT is the outermost operator

<math>A \or (B \and (C \or D))<math> — an OR is nested within an AND

Converting a formula to DNF involves using logical equivalences, such as the Double Negative Law, De Morgan's laws, and the Distributive Law. Note that all logical formulas can be converted into disjunctive normal form. However, in some cases conversion to DNF can lead to an exponential explosion of the formula. For example, in DNF form, logical formulas of the following form have 2ⁿ terms:

<math>(X_1 \or Y_1) \and (X_2 \or Y_2) \and \dots \and (X_n \or Y_n)<math>

The following is a formal grammar for DNF:

1. <or> → ∨
2. <and> → ∧
3. <not> → ¬
4. <disjunct> → <conjunct>
5. <disjunct> → <disjunct> <or> <conjunct>
6. <conjunct> → <literal>
7. <conjunct> → (<conjunct> <and> <literal>)
8. <literal> → <term>
9. <literal> → <not><term>

Where <term> is any variable.

See Also:

• Conjunctive normal form
• Algebraic normal form