Disjunctive Normal Form Logic Formula

A Disjunctive Normal Form Logic Formula is a Logic Formula described in terms of a Disjunction of Conjunctive Clauses.

• AKA: DNF, Disjunctive Normal Form, Disjunctive Normal Form Formula, Disjunctive Normal Form Propositional Formula, Disjunctive Normal Form Logic Sentence, DNF Formula.
• Context:
  • 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.
  • The only propositional operators in DNF are and, or, and not.
  • The not operator can only be used as part of a literal
  • The Logic Grammar of a Disjunctive Normal Form Logic Formula is:
    • Disjunct → Conjunct
    • Disjunct → Disjunct Or Conjunct
    • Conjunct → Logic Literal
    • Conjunct → (Conjunct And Logic Literal)
    • Logic Literal → Logic Term
    • Logic Literal → Not Logic Term.
• Example(s):
  • A
  • A or B
  • (A and B) or C
  • (A and not B) or (not C and D).
  • …
• Counter-Example(s):
  • Not (A or B)
  • A or (B and (C or D))
• See: Conjunctive Normal Form Logic Formula; De Morgan's Theorem.

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References

2011

• (Pfahringer, 2011b) ⇒ Bernhard Pfahringer. (2011). “Disjunctive Normal Form.” In: (Sammut & Webb, 2011) p.289

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/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. The following is a formal grammar for DNF: (where <term> is any variable)

  1. <or> → ∨
  2. <and> → ∧
  3. <not> → ¬
  4. <disjunct> → <conjunct>
  5. <disjunct> → <disjunct> <or> <conjunct>
  6. <conjunct> → <literal>
  7. <conjunct> → (<conjunct> <and> <literal>)
  8. <literal> → <term>
  9. <literal> → <not><term>

• http://dms.irb.hr/tutorial/tut_glosary.php
  • is a disjunction of clauses, where clauses are conjunctions of attribute-value conditions. For example, (color=red and shape=rectangular) or (color=green and shape=rectangular) is a formula in Disjunctive Normal Form (DNF).

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