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In [[mathematical logic]] and [[computer science]], the '''Kleene star''' (or '''Kleene operator''' or '''Kleene closure''') is a [[unary operation]], either on [[Set (mathematics)|sets]] of [[string (computer science)|strings]] or on sets of symbols or characters. In mathematics
it is more commonly known as the [[free monoid]] construction. The application of the Kleene star to a set ''V'' is written as ''V''<sup>*</sup>. It is widely used for [[regular expression]]s, which is the context in which it was introduced by [[Stephen Kleene]] to characterise certain [[Automata theory|automata]], where it means "zero or more".


# If ''V'' is a set of strings then ''V''<sup>*</sup> is defined as the smallest [[superset]] of ''V'' that contains the empty string ε and is [[Closure (mathematics)|closed]] under the [[concatenation|string concatenation operation]].
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# If ''V'' is a set of symbols or characters then ''V''<sup>*</sup> is the set of all strings over symbols in ''V'', including the [[empty string]].
 
The set ''V''<sup>*</sup> can also be described as the set of finite-length strings that can be generated by concatenating arbitrary elements of ''V'' allowing the use of the same element multiple times. If ''V'' is a nonempty [[finite set]] then ''V''<sup>*</sup> is a [[countably infinite set]].<ref>{{cite web |url=http://nayuki.eigenstate.org/page/countable-sets-and-kleene-star |title=Countable sets and Kleene star |author=Nayuki Minase |date=10 May 2011 |work=Project Nayuki |accessdate=11 January 2012}}</ref>
 
The operators are used in [[rewrite rule]]s for [[generative grammar]]s.
 
== Definition and notation ==
Given a set ''V''
define
:''V''<sub>0</sub> = { ε } (the language consisting only of the empty string),
:''V''<sub>1</sub> = ''V''
and define recursively the set
:''V<sub>''i''+1</sub> = { ''wv'' : ''w'' ∈ ''V''<sub>''i''</sub> and ''v'' ∈ ''V'' } for each ''i''>0.
 
If ''V'' is a formal language, then ''V''<sub>''i''</sub>, the ''i''-th power of the set ''V'', is a shorthand for the [[concatenation]] of set ''V'' with itself ''i'' times. That is, ''V''<sub>''i''</sub> can be understood to be the set of all [[string (computer science)|strings]] that can be represented as the concatenation of ''i'' strings in ''V''.  
 
The definition of Kleene star on ''V'' is<ref>{{cite book |last1=Ebbinghaus |first1=H.-D. |last2=Flum |first2=J. |last3=Thomas |first3=W. |doi= |title=Mathematical Logic |url=http://www.springer.com/mathematics/book/978-0-387-94258-2 |publisher=[[Springer Science+Business Media|Springer]] |location=[[New York City|New York]] |edition=2nd |isbn=0-387-94258-0 |year=1994|page=656|quote=The '''Kleene closure''' ''L''<sup>*</sup> of ''L'' is defined to be <math>\sideset{}{_{i=0}^\infty}\bigcup L^i</math>.}}</ref>
:<math> V^*=\bigcup_{i \in \N }V_i = \{\varepsilon\} \cup V \cup V_2 \cup V_3 \cup V_4 \cup \ldots.</math>
 
== Kleene plus ==
In some [[formal language]] studies, (e.g. [[Abstract family of languages|AFL Theory]]) a variation on the Kleene star operation called the ''Kleene plus'' is used. The Kleene plus omits the ''V''<sub>0</sub> term in the above union. In other words, the Kleene plus on ''V'' is
 
:<math>V^+=\bigcup_{i \in \N \setminus \{0\}} V_i = V_1 \cup V_2 \cup V_3 \cup \ldots.</math>
 
For every set ''L'', the Kleene plus ''L''<sup>+</sup> equals the concatenation of ''L'' with ''L''<sup>*</sup>.  
Conversely, ''L''<sup>*</sup> can be written as { ε } ∪ ''L''<sup>+</sup>.  
== Examples ==
Example of Kleene star applied to set of strings:
: {"ab", "c"}<sup>*</sup> = {ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "ababc", "abcab", "abcc", "cabab", "cabc", "ccab", "ccc", ...}.
 
Example of Kleene star applied to set of characters:
: {"a", "b", "c"}<sup>*</sup> = { ε, "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}.
 
Example of Kleene star applied to the empty set:
:∅<sup>*</sup> = { ε }.
 
Example of Kleene plus applied to the empty set:
:∅<sup>+</sup> = ∅ ∅<sup>*</sup> = { }= ∅,
where concatenation is an [[associative]] and [[noncommutative]] product, sharing these properties with the [[Cartesian product]] of sets.
 
Example of Kleene plus and Kleene star applied to the singleton set containing the empty string:
:If V = {ε}, then also ''V''<sub>''i''</sub> = {ε} for each ''i'', hence V<sup>*</sup> = V<sup>+</sup> = {ε}.
 
== Generalization ==
 
Strings form a [[monoid]] with concatenation as the binary operation and ε the identity element.  The Kleene star is defined for any monoid, not just strings.
More precisely, let (''M'', ⋅) be a monoid, and ''S'' ⊆ ''M''. Then ''S''<sup>*</sup> is the smallest submonoid of ''M'' containing ''S''; that is, ''S''<sup>*</sup> contains the neutral element of ''M'', the set ''S'', and is such that if ''x'',''y'' ∈ ''S''<sup>*</sup>, then ''x''⋅''y'' ∈ ''S''<sup>*</sup>.
 
== References ==
{{Reflist}}
 
==Further reading==
*{{cite book |last1=Hopcroft |first1=John E. |authorlink1=John Hopcroft |last2=Ullman |first2=Jeffrey D. |authorlink2=Jeffrey Ullman |title=[[Introduction to Automata Theory, Languages, and Computation]] |edition=1st |publisher=[[Addison-Wesley]] |year=1979}}
 
[[Category:Formal languages]]
[[Category:Grammar]]
[[Category:Natural language processing]]

Revision as of 19:22, 28 February 2014

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