Disjunctive normal form

In boolean logic, a disjunctive normal form (DNF) is a 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] The disjunctive normal form and its counterpart, the conjunctive normal form, are the most common standardized ways of representing boolean expressions. They are widely used in various applications such as circuit design or automated theorem proving.

Definition

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 ( $\wedge$ ), or (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vee} ), and not (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg} ). 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:

DNF Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \to \,} (Disjunct) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \mid \, } (Disjunct) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \lor \,} DNF

Disjunct Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \to \,} Literal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \mid\, } Literal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \land \,} Disjunct

Literal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \to \,} Variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \mid \,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \, \neg \,} Variable

Where Variable is any variable.

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \land \neg B \land \neg C) \lor (\neg D \land E \land F \land D \land F)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \land B) \lor (C)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \land B)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A)}

The formula Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \lor B} is in DNF, but not in full DNF; an equivalent full-DNF version is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \land B) \lor (A \land \lnot B) \lor (\lnot A \land B)} .

The following formulas are not in DNF:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg(A \lor B)} , since an OR is nested within a NOT
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg(A \land B) \lor C} , since an AND is nested within a NOT
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \lor (B \land (C \lor D))} , since an OR is nested within an AND^[5]

Conversion to DNF

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

File:Karnaugh map KV 4mal4 18.svg

Karnaugh map of the disjunctive normal form (¬A∧¬B∧¬D) ∨ (¬A∧B∧C) ∨ (A∧B∧D) ∨ (A∧¬B∧¬C)

File:Karnaugh map KV 4mal4 19.svg

Karnaugh map of the disjunctive normal form (¬A∧C∧¬D) ∨ (B∧C∧D) ∨ (A∧¬C∧D) ∨ (¬B∧¬C∧¬D). Despite the different grouping, the same fields contain a "1" as in the previous map.

... by syntactic means

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\land,\lor,\lnot\}} ^[7] can be converted to DNF by the following canonical term rewriting system:^[8]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} (\lnot \lnot x) & \rightsquigarrow & x \\ (\lnot (x \lor y)) & \rightsquigarrow & ((\lnot x) \land (\lnot y)) \\ (\lnot (x \land y)) & \rightsquigarrow & ((\lnot x) \lor (\lnot y)) \\ (x \land (y \lor z)) & \rightsquigarrow & ((x \land y) \lor (x \land z)) \\ ((x \lor y) \land z) & \rightsquigarrow & ((x \land z) \lor (y \land z)) \\ \end{array}}

... by semantic means

The full DNF of a formula can be read off its truth table.^[9]^[10] For example, consider the formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi = ((\lnot (p \land q)) \leftrightarrow (\lnot r \uparrow (p \oplus q)))} .^[11]

The corresponding truth table is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lnot}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p \land q)}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leftrightarrow}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lnot r}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \uparrow}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p \oplus q)}
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( p \land \lnot q \land r) \lor (\lnot p \land q \land r) \lor (\lnot p \land \lnot q \land r) \lor (\lnot p \land \lnot q \land \lnot r) }

The full DNF equivalent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lnot \phi} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( p \land q \land r) \lor ( p \land q \land \lnot r) \lor ( p \land \lnot q \land \lnot r) \lor (\lnot p \land q \land \lnot r) }

Remark

A propositional formula can be represented by one and only one full DNF.^[13] In contrast, several plain DNFs may be possible. For example, by applying the rule Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ((a \land b) \lor (\lnot a \land b)) \rightsquigarrow b} three times, the full DNF of the above Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} can be simplified to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\lnot p \land \lnot q) \lor (\lnot p \land r) \lor (\lnot q \land r)} . 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.^[14]^[15]^[16]^[17] This is called the Disjunctive Normal Form Theorem.^[14]^[15]^[16]^[17] The formal statement is as follows:

Disjunctive Normal Form Theorem: Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a sentence in a propositional language Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} sentence letters, which we shall denote by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1,...,A_n} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is not a contradiction, then it is truth-functionally equivalent to a disjunction of conjunctions of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm A_1 \land ... \land \pm A_n} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +A_i=A_i} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -A_i= \neg A_i} .^[15]

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

Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a sentence in a propositional language whose sentence letters are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B, C, \ldots} . For each row of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} 's truth table, write out a corresponding conjunction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm A \land \pm B \land \pm C \land \ldots} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm A} is defined to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} takes the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} at that row, and is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg A} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} takes the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} at that row; similarly for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm B} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm C} , etc. (the alphabetical ordering of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B, C, \ldots} in the conjunctions is quite arbitrary; any other could be chosen instead). Now form the disjunction of all these conjunctions which correspond to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} rows of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} 's truth table. This disjunction is a sentence in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}[A, B, C, \ldots; \land, \lor, \neg]} ,^[18] which by the reasoning above is truth-functionally equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . This construction obviously presupposes that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} takes the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} on at least one row of its truth table; if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} doesn’t, i.e., if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a contradiction, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \land \neg A} , which is, of course, also a sentence in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}[A, B, C, \ldots; \land, \lor, \neg]} .^[15]

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\land, \lor, \neg\}} is functionally complete.^[15]

Maximum number of conjunctions

Any propositional formula is built from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} variables, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \ge 1} .

There are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} possible literals: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \{ p_1, \lnot p_1, p_2, \lnot p_2, \ldots, p_n, \lnot p_n\}} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2^{2n} -1)} non-empty subsets.^[19]

This is the maximum number of conjunctions a DNF can have.^[13]

A full DNF can have up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{n}} conjunctions, one for each row of the truth table.

Example 1

Consider a formula with two variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} .

The longest possible DNF has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{(2 \times 2)} -1 = 15} conjunctions:^[13]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcl} (\lnot p) \lor (p) \lor (\lnot q) \lor (q) \lor \\ (\lnot p \land p) \lor \underline{(\lnot p \land \lnot q)} \lor \underline{(\lnot p \land q)} \lor \underline{( p \land \lnot q)} \lor \underline{( p \land q)} \lor (\lnot q \land q) \lor \\ (\lnot p \land p \land \lnot q) \lor (\lnot p \land p \land q) \lor (\lnot p \land \lnot q \land q) \lor ( p \land \lnot q \land q) \lor \\ (\lnot p \land p \land \lnot q \land q) \end{array}}

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

This formula is a tautology. It can be simplified to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\neg p \lor p)} or to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\neg q \lor q)} , which are also tautologies, as well as valid DNFs.

Example 2

Each DNF of the e.g. formula Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_1 \lor Y_1) \land (X_2 \lor Y_2) \land \dots \land (X_n \lor Y_n)} has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^n} conjunctions.

Computational complexity

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

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.^[20]

Notes

↑ Post 1921.
↑ Davey & Priestley 1990, p. 153.
↑ Gries & Schneider 1993, p. 67.
↑ Whitesitt 2012, pp. 33–37.
↑ However, this one is in negation normal form.
↑ Davey & Priestley 1990, p. 152-153.
↑ Formulas with other connectives can be brought into negation normal form first.
↑ Dershowitz & Jouannaud 1990, p. 270, Sect.5.1.
↑ Smullyan 1968, p. 14: "Make a truth-table for the formula. Each line of the table which comes out "T" will yield one of the basic conjunctions of the disjunctive normal form."
↑ Sobolev 2020.
↑ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} = ((NOT (p AND q)) IFF ((NOT r) NAND (p XOR q)))
↑ like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \land b) \lor (b \land a) \lor (a \land b \land b)}
↑ ^13.0 ^13.1 ^13.2 It is assumed that repetitions and variations^[12] based on the commutativity and associativity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lor} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \land} do not occur.
↑ ^14.0 ^14.1 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.
↑ ^15.0 ^15.1 ^15.2 ^15.3 ^15.4 Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. p. 41. ISBN 978-0-415-13342-5.
↑ ^16.0 ^16.1 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.
↑ ^17.0 ^17.1 Halvorson, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B, C, \ldots} and the connectives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\land, \lor, \neg\}} .
↑ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\mathcal{P}(L)\right| = 2^{2n}}
↑ Arora & Barak 2009.

References

Template:Sfn whitelist

Arora, Sanjeev; Barak, Boaz (20 April 2009). Computational Complexity: A Modern Approach. Cambridge University Press. p. 579. doi:10.1017/CBO9780511804090. ISBN 9780521424264.
Davey, B.A.; Priestley, H.A. (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.
Dershowitz, Nachum; Jouannaud, Jean-Pierre (1990). "Rewrite Systems". In Van Leeuwen, Jan (ed.). Formal Models and Semantics. Handbook of Theoretical Computer Science. B. Elsevier. pp. 243–320. ISBN 0-444-88074-7.
Gries, David; Schneider, Fred B. (22 October 1993). A Logical Approach to Discrete Math. Springer Science & Business Media. ISBN 978-0-387-94115-8.
Hilbert, David; Ackermann, Wilhelm (1999). Principles of Mathematical Logic. American Mathematical Soc. ISBN 978-0-8218-2024-7.
Howson, Colin (11 October 2005) [1997]. Logic with trees: an introduction to symbolic logic. Routledge. ISBN 978-1-134-78550-6.
Post, Emil (July 1921). "Introduction to a General Theory of Elementary Propositions". American Journal of Mathematics. 43 (3): 163–185. doi:10.2307/2370324. JSTOR 2370324.
Smullyan, Raymond M. (1968). First-Order Logic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 43 (1st edition, Second Printing 1971 ed.). New York Heidelberg Berlin: Springer-Verlag. p. 160. doi:10.1007/978-3-642-86718-7. ISBN 978-3-642-86718-7.
Template:SpringerEOM
Whitesitt, J. Eldon (24 May 2012) [1961]. Boolean Algebra and Its Applications. Courier Corporation. ISBN 978-0-486-15816-7.

Template:Normal forms in logic

[FOOTNOTEPost1921-1] Post 1921.

[FOOTNOTEDaveyPriestley1990153-2] Davey & Priestley 1990, p. 153.

[FOOTNOTEGriesSchneider199367-3] Gries & Schneider 1993, p. 67.

[FOOTNOTEWhitesitt201233–37-4] Whitesitt 2012, pp. 33–37.

[5] However, this one is in negation normal form.

[FOOTNOTEDaveyPriestley1990152-153-6] Davey & Priestley 1990, p. 152-153.

[7] Formulas with other connectives can be brought into negation normal form first.

[FOOTNOTEDershowitzJouannaud1990270Sect.5.1-8] Dershowitz & Jouannaud 1990, p. 270, Sect.5.1.

[9] Smullyan 1968, p. 14: "Make a truth-table for the formula. Each line of the table which comes out "T" will yield one of the basic conjunctions of the disjunctive normal form."

[FOOTNOTESobolev2020-10] Sobolev 2020.

[11] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} = ((NOT (p AND q)) IFF ((NOT r) NAND (p XOR q)))

[12] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \land b) \lor (b \land a) \lor (a \land b \land b)}

[noreps-13] 13.0 ^13.1 ^13.2 It is assumed that repetitions and variations^[12] based on the commutativity and associativity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lor} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \land} do not occur.

[:0-14] 14.0 ^14.1 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.

[:13-15] 15.0 ^15.1 ^15.2 ^15.3 ^15.4 Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. p. 41. ISBN 978-0-415-13342-5.

[:1-16] 16.0 ^16.1 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.

[:2-17] 17.0 ^17.1 Halvorson, Hans (2020). How logic works: a user's guide. Princeton Oxford: Princeton University Press. p. 195. ISBN 978-0-691-18222-3.

[18] That is, the language with the propositional variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B, C, \ldots} and the connectives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\land, \lor, \neg\}} .

[19] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\mathcal{P}(L)\right| = 2^{2n}}

[FOOTNOTEAroraBarak2009-20] Arora & Barak 2009.

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Disjunctive normal form

Contents

Definition

Conversion to DNF

... by syntactic means

... by semantic means

Remark

Disjunctive Normal Form Theorem

Maximum number of conjunctions

Computational complexity

Variants

See also

Notes

References

Navigation menu

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lnot}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p \land q)}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leftrightarrow}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lnot r}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \uparrow}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p \oplus q)}
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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lnot}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p \land q)}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leftrightarrow}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lnot r}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \uparrow}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p \oplus q)}
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

Disjunctive normal form

Definition

Conversion to DNF

... by syntactic means

... by semantic means

Remark

Disjunctive Normal Form Theorem

Maximum number of conjunctions

Computational complexity

Variants

See also

Notes

References

Navigation menu

Search

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lnot}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p \land q)}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leftrightarrow}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lnot r}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \uparrow}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p \oplus q)}
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