Disjunctive normal form#Disjunctive Normal Form Theorem

A logical formula is considered to be in DNF if it is a disjunction of one or more conjunctions of one or more literals.{{sfn|Davey|Priestley|1990|page=153}}{{sfn|Gries|Schneider|1993|page=67}}{{sfn|Whitesitt|2012|pages=33-37}} 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 ($\backslash wedge$), or ($\backslash vee$), and not ($\backslash neg$). 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 $\backslash ,\; \backslash to\; \backslash ,$ Conjunct $\backslash ,\; \backslash mid\; \backslash ,$ Conjunct $\backslash ,\; \backslash lor\; \backslash ,$ DNF

: Conjunct $\backslash ,\; \backslash to\; \backslash ,$ Literal $\backslash ,\; \backslash mid\backslash ,$ Literal $\backslash ,\; \backslash land\; \backslash ,$ Conjunct

: Literal $\backslash ,\; \backslash to\; \backslash ,$ Variable $\backslash ,\; \backslash mid\; \backslash ,$ $\backslash ,\; \backslash neg\; \backslash ,$ Variable

Where Variable is any variable.

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

• $(A\; \backslash land\; \backslash neg\; B\; \backslash land\; \backslash neg\; C)\; \backslash lor\; (\backslash neg\; D\; \backslash land\; E\; \backslash land\; F\; \backslash land\; D\; \backslash land\; F)$
• $(A\; \backslash land\; B)\; \backslash lor\; (C)$
• $(A\; \backslash land\; B)$
• $(A)$

The formula $A\; \backslash lor\; B$ is in DNF, but not in full DNF; an equivalent full-DNF version is $(A\; \backslash land\; B)\; \backslash lor\; (A\; \backslash land\; \backslash lnot\; B)\; \backslash lor\; (\backslash lnot\; A\; \backslash land\; B)$.

The following formulas are not in DNF:

• $\backslash neg(A\; \backslash lor\; B)$, since an OR is nested within a NOT
• $\backslash neg(A\; \backslash land\; B)\; \backslash lor\; C$, since an AND is nested within a NOT
• $A\; \backslash lor\; (B\; \backslash land\; (C\; \backslash lor\; D))$, since an OR is nested within an ANDHowever, this one is in negation normal form.

In classical logic each propositional formula can be converted to DNF{{sfn|Davey|Priestley|1990|page=152-153}} ...

File:Karnaugh map KV 4mal4 18.svg of the disjunctive normal form {{color|#800000|(¬A∧¬B∧¬D)}} ∨ {{color|#000080|(¬A∧B∧C)}} ∨ {{color|#008000|(A∧B∧D)}} ∨ {{color|#800080|(A∧¬B∧¬C)}}]]

File:Karnaugh map KV 4mal4 19.svg

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 $\backslash \{\backslash land,\backslash lor,\backslash lnot\backslash \}$Formulas with other connectives can be brought into negation normal form first. can be converted to DNF by the following canonical term rewriting system:{{sfn|Dershowitz|Jouannaud|1990|page=270|loc=Sect.5.1}}

:$\backslash begin\{array\}\{rcl\}$

(\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}

The full DNF of a formula can be read off its truth table.{{harvnb|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."{{sfn|Sobolev|2020}} For example, consider the formula

:$\backslash phi\; =\; ((\backslash lnot\; (p\; \backslash land\; q))\; \backslash leftrightarrow\; (\backslash lnot\; r\; \backslash uparrow\; (p\; \backslash oplus\; q)))$.$\backslash phi$ = ((NOT (p AND q)) IFF ((NOT r) NAND (p XOR q)))

The corresponding truth table is

:

• The full DNF equivalent of $\backslash phi$ is

:$$

( 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 $\backslash lnot\; \backslash phi$ is

:$$

( 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)

A propositional formula can be represented by one and only one full DNF.{{refn|name="noreps"|It is assumed that repetitions and variationslike $(a\; \backslash land\; b)\; \backslash lor\; (b\; \backslash land\; a)\; \backslash lor\; (a\; \backslash land\; b\; \backslash land\; b)$ based on the commutativity and associativity of $\backslash lor$ and $\backslash land$ do not occur.}} In contrast, several plain DNFs may be possible. For example, by applying the rule $((a\; \backslash land\; b)\; \backslash lor\; (\backslash lnot\; a\; \backslash land\; b))\; \backslash rightsquigarrow\; b$ three times, the full DNF of the above $\backslash phi$ can be simplified to $(\backslash lnot\; p\; \backslash land\; \backslash lnot\; q)\; \backslash lor\; (\backslash lnot\; p\; \backslash land\; r)\; \backslash lor\; (\backslash lnot\; q\; \backslash land\; r)$. However, there are also equivalent DNF formulas that cannot be transformed one into another by this rule, see the pictures for an example.

It is a theorem that all consistent formulas in propositional logic can be converted to disjunctive normal form.{{Cite book |last=Halbeisen |first=Lorenz |title=Gödel´s theorems and zermelo´s axioms: a firm foundation of mathematics |last2=Kraph |first2=Regula |date=2020 |publisher=Birkhäuser |isbn=978-3-030-52279-7 |location=Cham |pages=27}}{{Cite book |last=Cenzer |first=Douglas |title=Set theory and foundations of mathematics: an introduction to mathematical logic |last2=Larson |first2=Jean |last3=Porter |first3=Christopher |last4=Zapletal |first4=Jindřich |date=2020 |publisher=World Scientific |isbn=978-981-12-0192-9 |location=New Jersey |pages=19–21}}{{Cite book |last=Halvorson |first=Hans |title=How logic works: a user's guide |date=2020 |publisher=Princeton University Press |isbn=978-0-691-18222-3 |location=Princeton Oxford |pages=195}} This is called the Disjunctive Normal Form Theorem. The formal statement is as follows:

Disjunctive Normal Form Theorem: Suppose $X$ is a sentence in a propositional language $\backslash mathcal\{L\}$ with $n$ sentence letters, which we shall denote by $A\_1,...,A\_n$. If $X$ is not a contradiction, then it is truth-functionally equivalent to a disjunction of conjunctions of the form $\backslash pm\; A\_1\; \backslash land\; ...\; \backslash land\; \backslash pm\; A\_n$, where $+A\_i=A\_i$, and $-A\_i=\; \backslash neg\; A\_i$.{{Cite book |last=Howson |first=Colin |title=Logic with trees: an introduction to symbolic logic |date=1997 |publisher=Routledge |isbn=978-0-415-13342-5 |location=London; New York |pages=41}}

Suppose $X$ is a sentence in a propositional language whose sentence letters are $A,\; B,\; C,\; \backslash ldots$. For each row of $X$'s truth table, write out a corresponding conjunction $\backslash pm\; A\; \backslash land\; \backslash pm\; B\; \backslash land\; \backslash pm\; C\; \backslash land\; \backslash ldots$, where $\backslash pm\; A$ is defined to be $A$ if $A$ takes the value $T$ at that row, and is $\backslash neg\; A$ if $A$ takes the value $F$ at that row; similarly for $\backslash pm\; B$, $\backslash pm\; C$, etc. (the alphabetical ordering of $A,\; B,\; C,\; \backslash ldots$ in the conjunctions is quite arbitrary; any other could be chosen instead). Now form the disjunction of all these conjunctions which correspond to $T$ rows of $X$'s truth table. This disjunction is a sentence in $\backslash mathcal\{L\}[A,\; B,\; C,\; \backslash ldots;\; \backslash land,\; \backslash lor,\; \backslash neg]$,That is, the language with the propositional variables $A,\; B,\; C,\; \backslash ldots$ and the connectives $\backslash \{\backslash land,\; \backslash lor,\; \backslash neg\backslash \}$. which by the reasoning above is truth-functionally equivalent to $X$. This construction obviously presupposes that $X$ takes the value $T$ on at least one row of its truth table; if $X$ doesn’t, i.e., if $X$ is a contradiction, then $X$ is equivalent to $A\; \backslash land\; \backslash neg\; A$, which is, of course, also a sentence in $\backslash mathcal\{L\}[A,\; B,\; C,\; \backslash ldots;\; \backslash land,\; \backslash lor,\; \backslash neg]$.

Any propositional formula is built from $n$ variables, where $n\; \backslash ge\; 1$.

There are $2n$ possible literals: $L\; =\; \backslash \{\; p\_1,\; \backslash lnot\; p\_1,\; p\_2,\; \backslash lnot\; p\_2,\; \backslash ldots,\; p\_n,\; \backslash lnot\; p\_n\backslash \}$.

$L$ has $(2^\{2n\}\; -1)$ non-empty subsets.$\backslash left|\backslash mathcal\{P\}(L)\backslash right|\; =\; 2^\{2n\}$

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

A full DNF can have up to $2^\{n\}$ conjunctions, one for each row of the truth table.

Example 1

Consider a formula with two variables $p$ and $q$.

The longest possible DNF has $2^\{(2\; \backslash times\; 2)\}\; -1\; =\; 15$ conjunctions:

:$$

\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 $(\backslash neg\; p\; \backslash lor\; p)$ or to $(\backslash neg\; q\; \backslash lor\; q)$, which are also tautologies, as well as valid DNFs.

Example 2

Each DNF of the e.g. formula $(X\_1\; \backslash lor\; Y\_1)\; \backslash land\; (X\_2\; \backslash lor\; Y\_2)\; \backslash land\; \backslash dots\; \backslash land\; (X\_n\; \backslash lor\; Y\_n)$ has $2^n$ conjunctions.

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.

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.{{sfn|Arora|Barak|2009}}

• Algebraic normal form – an XOR of AND clauses
• Blake canonical form – DNF including all prime implicants
• Quine–McCluskey algorithm – algorithm for calculating prime implicants
• Conjunction/disjunction duality
• Propositional logic
• Truth table

{{reflist}}

{{sfn whitelist |CITEREFSobolev2020}}

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• {{cite book | isbn=0-444-88074-7 | editor-first=Jan |editor-last= Van Leeuwen | editor-link=Jan van Leeuwen | title=Formal Models and Semantics | publisher=Elsevier | series=Handbook of Theoretical Computer Science | volume=B | year=1990 | first1=Nachum |last1=Dershowitz |author1-link=Nachum Dershowitz |first2= Jean-Pierre |last2=Jouannaud |author2-link= Jean-Pierre Jouannaud | contribution=Rewrite Systems | pages=243–320 }}
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• {{cite book|last1= Hilbert|first1=David|author1-link=David Hilbert|last2= Ackermann|first2=Wilhelm |author2-link=Wilhelm Ackermann|title=Principles of Mathematical Logic|url=https://books.google.com/books?id=45ZGMjV9vfcC&q=%22disjunctive+normal+form%22|year=1999|publisher=American Mathematical Soc.|isbn=978-0-8218-2024-7}}
• {{cite book|last=Howson|first=Colin|author-link=Colin Howson|title=Logic with trees: an introduction to symbolic logic|url=https://books.google.com/books?id=Y4WGAgAAQBAJ&q=%22disjunctive+normal+form%22|date=11 October 2005|orig-date=1997|publisher=Routledge|isbn=978-1-134-78550-6}}
• {{Cite journal|last=Post|first=Emil|author-link=Emil Post|title=Introduction to a General Theory of Elementary Propositions|date=July 1921|volume=43|issue=3|pages= 163–185| journal=American Journal of Mathematics|doi=10.2307/2370324|jstor=2370324}}
• {{Cite book|last=Smullyan|first=Raymond M.|author-link=Raymond Smullyan |title=First-Order Logic|year=1968|edition=1st edition, Second Printing 1971 |publisher=Springer-Verlag|pages=160|location=New York Heidelberg Berlin|doi=10.1007/978-3-642-86718-7|isbn= 978-3-642-86718-7|series= Ergebnisse der Mathematik und ihrer Grenzgebiete|volume=43}}
• {{SpringerEOM|title=Disjunctive normal form|author-last1=Sobolev|author-first1=S.K.|oldid=54535|date=2020}}
• {{cite book|first=J. Eldon |last=Whitesitt|title=Boolean Algebra and Its Applications|url=https://books.google.com/books?id=20Un1T78GlMC&q=%22disjunctive+normal+form%22|date=24 May 2012|orig-year=1961|publisher=Courier Corporation|isbn=978-0-486-15816-7}}

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