Disjunctive normal form

In boolean logic, a disjunctive normal form (DNF) is a standardization (or normalization) of a logical formula which is a disjunction of conjunctive clauses; it can also be described as an OR of ANDs, a sum of products, or (in philosophical logic) a cluster concept. As a normal form, it is useful in automated theorem proving.

Definition

A logical formula is considered to be in DNF if and only if it is a disjunction of one or more conjunctions of one or more literals.^[1]^:153 A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every conjunction. As in conjunctive normal form (CNF), 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.

The following is a formal grammar for DNF:

disjunction → (conjunction ∨ disjunction)
disjunction → conjunction
conjunction → (literal ∧ conjunction)
conjunction → literal
literal → ¬variable
literal → variable

Where variable is any variable.

For example, all of the following formulas are in DNF:

$(A\land \neg B\land \neg C)\lor (\neg D\land E\land F)$
$(A\land B)\lor C$
$A\land B$
$A$

However, the following formulas are not in DNF:

$\neg (A\lor B)$ , since an OR is nested within a NOT
$A\lor (B\land (C\lor D))$ , since an OR is nested within an AND

Conversion to DNF

Karnaugh map of the disjunctive normal form (¬A∧¬B∧¬D) ∨ (¬A∧B∧C) ∨ (A∧B∧D) ∨ (A∧¬B∧¬C)

Karnaugh map of the disjunctive normal form (¬A∧C∧¬D) ∨ (B∧C∧D) ∨ (A∧¬C∧D) ∨ (¬B∧¬C∧¬D). Despite the different grouping, the same fields contain a "1" as in the previous map.

Converting a formula to DNF involves using logical equivalences, such as the double negative elimination, De Morgan's laws, and the distributive law.

All logical formulas can be converted into an equivalent disjunctive normal form.^[1]^:152-153 However, in some cases conversion to DNF can lead to an exponential explosion of the formula. For example, the DNF of a logical formula of the following form has 2ⁿ terms:

(X_{1}\lor Y_{1})\land (X_{2}\lor Y_{2})\land \dots \land (X_{n}\lor Y_{n})

Any particular Boolean function can be represented by one and only one^{[note 1]} full disjunctive normal form, one of the canonical forms. In contrast, two different plain disjunctive normal forms may denote the same Boolean function, see pictures.

Complexity issues

An important variation used in the study of computational complexity is k-DNF. A formula is in k-DNF if it is in DNF and each clause contains at most k literals.

Notes

↑ Ignoring variations based on associativity and commutativity of AND and OR.

References

1 2 B.A. Davey and H.A. Priestley (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Disjunctive normal form", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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[2] Ignoring variations based on associativity and commutativity of AND and OR.

[Davey.Priestley.1990-1] 1 2 B.A. Davey and H.A. Priestley (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.