Disjunctive normal form

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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 The conjunctions of a DNF formula are sometimes[citation needed] called its clauses.[note 1] A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every clause. 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:

  1. disjunction → (conjunctiondisjunction)
  2. disjunctionconjunction
  3. conjunction → (literalconjunction)
  4. conjunctionliteral
  5. literal → ¬variable
  6. literalvariable

Where variable is any variable.

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

However, the following formulae are not in DNF:

  • , since an OR is nested within a NOT
  • , since an OR is nested within an AND

Conversion to DNF

Karnaugh map of the disjunctive normal form A∧¬B∧¬D)ABC)(ABD)(A∧¬B∧¬C)
Karnaugh map of the disjunctive normal form AC∧¬D)(BCD)(A∧¬CD)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:

Any particular Boolean function can be represented by one and only one[note 2] 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. Dually to CNFs, the problem of deciding whether a given DNF is true for every variable assignment is NP-complete, the same holds if only k-DNFs are considered.

See also

Notes

  1. ^ This notion shouldn't be confused with the (more common) notion of the clauses of a conjunctive normal form, which means its disjunctions.
  2. ^ Ignoring variations based on associativity and commutativity of AND and OR.

References

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

External links

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