Disjunctive normal form
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In boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions; it can also be described as an OR of ANDs, a sum of products, or (in philosophical logic) a cluster concept.^{[citation needed]} As a normal form, it is useful in automated theorem proving.
Definition[edit]
A logical formula is considered to be in DNF 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 contextfree grammar for DNF:
 DNF → (Conjunction) DNF
 DNF → (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:
However, the following formulas are not in DNF:
 , since an OR is nested within a NOT
 , since an AND is nested within a NOT
 , since an OR is nested within an AND
The formula is in DNF, but not in full DNF; an equivalent fullDNF version is .
Conversion to DNF[edit]
Converting a formula to DNF involves using logical equivalences, such as double negation 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, converting the formula to DNF yields a formula with 2^{n} terms.
Every 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.
Computational complexity[edit]
The Boolean satisfiability problem on conjunctive normal form formulas is NPhard; by the duality principle, so is the falsifiability problem on DNF formulas. Therefore, it is coNPhard to decide if a DNF formula is a tautology.
Variants[edit]
An important variation used in the study of computational complexity is kDNF. A formula is in kDNF if it is in DNF and each conjunction contains at most k literals.
See also[edit]
 Algebraic normal form — other normal forms for logical formulas
 Blake canonical form — a special case of DNF
 Propositional logic
 Quine–McCluskey algorithm — obtains a minimal DNF for a given Boolean function
 Truth table
Notes[edit]
 ^ Ignoring variations based on associativity and commutativity of AND and OR.
References[edit]
 ^ ^{a} ^{b} B.A. Davey and H.A. Priestley (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.
 David Hilbert; Wilhelm Ackermann (1999). Principles of Mathematical Logic. American Mathematical Soc. ISBN 9780821820247.
 J. Eldon Whitesitt (24 May 2012). Boolean Algebra and Its Applications. Courier Corporation. ISBN 9780486158167.
 Colin Howson (11 October 2005). Logic with Trees: An Introduction to Symbolic Logic. Routledge. ISBN 9781134785506.
 David Gries; Fred B. Schneider (22 October 1993). A Logical Approach to Discrete Math. Springer Science & Business Media. pp. 67–. ISBN 9780387941158.
External links[edit]
 "Disjunctive normal form", Encyclopedia of Mathematics, EMS Press, 2001 [1994]