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.
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
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
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.
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
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]
^ a b cIt is assumed that repetitions and variations[11] based on the commutativity and associativity of and do not occur.
^ a bHalbeisen, 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.
^ a b c d eHowson, Colin (1997). Logic with trees: an introduction to symbolic logic. London ; New York: Routledge. p. 41. ISBN 978-0-415-13342-5.
^ a bCenzer, 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.
^ a bHalvorson, Hans (2020). How logic works: a user's guide. Princeton Oxford: Princeton University Press. p. 195. ISBN 978-0-691-18222-3.
^That is, the language with the propositional variables and the connectives .
^
^Arora & Barak 2009.
References
Arora, Sanjeev; Barak, Boaz (20 April 2009). Computational Complexity: A Modern Approach. Cambridge University Press. p. 579. doi:10.1017/CBO9780511804090. ISBN 9780521424264.