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. 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 disjunction of one or more conjunctions of one or more literals. 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. For example, all of the following formulas are in DNF:

However, the following formulas are NOT in DNF:

— NOT is the outermost operator
— an OR is nested within an AND

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 disjunctive normal form. However, in some cases conversion to DNF can lead to an exponential explosion of the formula. For example, in DNF, logical formulas of the following form have 2n terms:

Any particular Boolean function can be represented by one and only one full disjunctive normal form, one of the two canonical forms.

The following is a formal grammar for DNF:

  1. disjunct → conjunct
  2. disjunct → disjunct ∨ conjunct
  3. conjunct → literal
  4. conjunct → (conjunct ∧ literal)
  5. literal → variable
  6. literal → ¬variable

Where variable is thought as any variable.

Famous quotes containing the words normal and/or form:

    “A few ideas seem to be agreed upon. Help none but those who help themselves. Educate only at schools which provide in some form for industrial education. These two points should be insisted upon. Let the normal instruction be that men must earn their own living, and that by the labor of their hands as far as may be. This is the gospel of salvation for the colored man. Let the labor not be servile, but in manly occupations like that of the carpenter, the farmer, and the blacksmith.”
    —Rutherford Birchard Hayes (1822–1893)

    “Everybody’s youth is a dream, a form of chemical madness.”
    —F. Scott Fitzgerald (1896–1940)