Disjunctive normal form

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.[1] As a normal form, it is useful in automated theorem proving.

Definition

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A logical formula is considered to be in DNF if it is a disjunction of one or more conjunctions of one or more literals.[2][3][4] A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every conjunction and each conjunction appears at most once (up to the order of variables). 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 context-free grammar for DNF:

  1. DNF → (Conjunction)   DNF
  2. DNF → (Conjunction)
  3. ConjunctionLiteral   Conjunction
  4. ConjunctionLiteral
  5. Literal Variable
  6. LiteralVariable

Where Variable is any variable.

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

  •  
  •  
  •  
  •  

The formula   is in DNF, but not in full DNF; an equivalent full-DNF version is  .

The following formulas are not in DNF:[5]

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

Conversion to DNF

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In classical logic each propositional formula can be converted to DNF[6] ...

 
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.

... by syntactic means

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The conversion involves using logical equivalences, such as double negation elimination, De Morgan's laws, and the distributive law. Formulas built from the primitive connectives  [7] can be converted to DNF by the following canonical term rewriting system:[8]

 

... by semantic means

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The full DNF of a formula can be read off its truth table.[9] For example, consider the formula

 .[10]

The corresponding truth table is

                         
T T T F T F F T F
T T F F T F T T F
T F T T F T F T T
T F F T F F T F T
F T T T F T F T T
F T F T F F T F T
F F T T F T F T F
F F F T F T T T F
  • The full DNF equivalent of   is
 
  • The full DNF equivalent of   is
 

Remark

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A propositional formula can be represented by one and only one full DNF.[12] In contrast, several plain DNFs may be possible. For example, by applying the rule   three times, the full DNF of the above   can be simplified to  . However, there are also equivalent DNF formulas that cannot be transformed one into another by this rule, see the pictures for an example.

Disjunctive Normal Form Theorem

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It is a theorem that all consistent formulas in propositional logic can be converted to disjunctive normal form.[13][14][15][16] This is called the Disjunctive Normal Form Theorem.[13][14][15][16] The formal statement is as follows:

Disjunctive Normal Form Theorem: Suppose   is a sentence in a propositional language   with   sentence letters, which we shall denote by  . If   is not a contradiction, then it is truth-functionally equivalent to a disjunction of conjunctions of the form  , where  , and  .[14]

The proof follows from the procedure given above for generating DNFs from truth tables. Formally, the proof is as follows:

Suppose   is a sentence in a propositional language whose sentence letters are  . For each row of  's truth table, write out a corresponding conjunction  , where   is defined to be   if   takes the value   at that row, and is   if   takes the value   at that row; similarly for  ,  , etc. (the alphabetical ordering of   in the conjunctions is quite arbitrary; any other could be chosen instead). Now form the disjunction of all these conjunctions which correspond to   rows of  's truth table. This disjunction is a sentence in  ,[17] which by the reasoning above is truth-functionally equivalent to  . This construction obviously presupposes that   takes the value   on at least one row of its truth table; if   doesn’t, i.e., if   is a contradiction, then   is equivalent to  , which is, of course, also a sentence in  .[14]

This theorem is a convenient way to derive many useful metalogical results in propositional logic, such as, trivially, the result that the set of connectives   is functionally complete.[14]

Maximum number of conjunctions

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Any propositional formula is built from   variables, where  .

There are   possible literals:  .

  has   non-empty subsets.[18]

This is the maximum number of conjunctions a DNF can have.[12]

A full DNF can have up to   conjunctions, one for each row of the truth table.

Example 1

Consider a formula with two variables   and  .

The longest possible DNF has   conjunctions:[12]

 

The longest possible full DNF has 4 conjunctions: they are underlined.

This formula is a tautology.

Example 2

Each DNF of the e.g. formula   has   conjunctions.

Computational complexity

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The Boolean satisfiability problem on conjunctive normal form formulas is NP-complete. By the duality principle, so is the falsifiability problem on DNF formulas. Therefore, it is co-NP-hard to decide if a DNF formula is a tautology.

Conversely, a DNF formula is satisfiable if, and only if, one of its conjunctions is satisfiable. This can be decided in polynomial time simply by checking that at least one conjunction does not contain conflicting literals.

Variants

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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 conjunction contains at most k literals.[19]

See also

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Notes

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  1. ^ Post 1921.
  2. ^ Davey & Priestley 1990, p. 153.
  3. ^ Gries & Schneider 1993, p. 67.
  4. ^ Whitesitt 2012, pp. 33–37.
  5. ^ However, they are in negation normal form.
  6. ^ Davey & Priestley 1990, p. 152-153.
  7. ^ Formulas with other connectives can be brought into negation normal form first.
  8. ^ Dershowitz & Jouannaud 1990, p. 270, Sect.5.1.
  9. ^ Sobolev 2020.
  10. ^   = ((NOT (p AND q)) IFF ((NOT r) NAND (p XOR q)))
  11. ^ like  
  12. ^ a b c It is assumed that repetitions and variations[11] based on the commutativity and associativity of   and   do not occur.
  13. ^ a b Halbeisen, Lorenz; Kraph, Regula (2020). Gödel´s theorems and zermelo´s axioms: a firm foundation of mathematics. Cham: Birkhäuser. p. 27. ISBN 978-3-030-52279-7.
  14. ^ a b c d e Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. p. 41. ISBN 978-0-415-13342-5.
  15. ^ a b Cenzer, Douglas; Larson, Jean; Porter, Christopher; Zapletal, Jindřich (2020). Set theory and foundations of mathematics: an introduction to mathematical logic. New Jersey: World Scientific. pp. 19–21. ISBN 978-981-12-0192-9.
  16. ^ a b Halvorson, Hans (2020). How logic works: a user's guide. Princeton Oxford: Princeton University Press. p. 195. ISBN 978-0-691-18222-3.
  17. ^ That is, the language with the propositional variables   and the connectives  .
  18. ^  
  19. ^ Arora & Barak 2009.

References

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