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.
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 (
The not operator can only be used as part of a literal, which means that it can only precede a propositional variable.
The following formulas are not in DNF: In classical logic each propositional formula can be converted to DNF[6] ...
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] The full DNF of a formula can be read off its truth table.
[9] For example, consider the formula The corresponding truth table is A propositional formula can be represented by one and only one full DNF.
three times, the full DNF of the above
However, there are also equivalent DNF formulas that cannot be transformed one into another by this rule, see the pictures for an example.
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 not a contradiction, then it is truth-functionally equivalent to a disjunction of conjunctions of the form
[14]The proof follows from the procedure given above for generating DNFs from truth tables.
's truth table, write out a corresponding conjunction
in the conjunctions is quite arbitrary; any other could be chosen instead).
Now form the disjunction of all these conjunctions which correspond to
[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
[18] This is the maximum number of conjunctions a DNF can have.
conjunctions, one for each row of the truth table.
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.
This can be decided in polynomial time simply by checking that at least one conjunction does not contain conflicting literals.
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.
