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:

    “Cant is always rather nauseating; but before we condemn political hypocrisy, let us remember that it is the tribute paid by men of leather to men of God, and that the acting of the part of someone better than oneself may actually commit one to a course of behaviour perceptibly less evil than what would be normal and natural in an avowed cynic.”
    —Aldous Huxley (1894–1963)

    “There is a sort of veteran women of condition, who, having lived always in the grand mode, and having possibly had some gallantries, together with the experience of five and twenty or thirty years, form a young fellow better than all the rules that can be given him.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)