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单词 KleeneStar
释义

Kleene star


If Σ is an alphabet (a set of symbols), then the Kleene star of Σ, denoted Σ*, is the set of all strings of finite length consisting of symbols in Σ, including the empty string λ. * is also called the asterate.

If S is a set of strings, then the Kleene star of S,denoted S*, is the smallest supersetMathworldPlanetmath of S that contains λ andis closed underPlanetmathPlanetmath the string concatenation operation. That is, S* is theset of all strings that can be generated by concatenating zero or more strings in S.

The definition of Kleene star can be generalized so that it operates on anymonoid (M,++), where ++ is a binary operationMathworldPlanetmath on the set M.If e is the identity elementMathworldPlanetmath of (M,++)and S is a subset of M, then S* is the smallest superset of S thatcontains e and is closed under ++.

Examples

  • *={λ}, since there are no strings of finite length consisting of symbols in , so λ is the only element in *.

  • If E={λ}, then E*=E, since λa=aλ=a by definition, so λλ=λ.

  • If A={a}, then A*={λ,a,aa,aaa,}.

  • If Σ={a,b}, then Σ*={λ,a,b,aa,ab,ba,bb,aaa,}

  • If S={ab,cd}, then S*={λ,ab,cd,abab,abcd,cdab,cdcd,ababab,}

For any set S, S* is the free monoid generated by S.

Remark. There is an associated operation, called the Kleene plus, is defined for any set S, such that S+ is the smallest set containing S such that S+ is closed under the concatenationMathworldPlanetmath. In other words, S+=S*-{λ}.

TitleKleene star
Canonical nameKleeneStar
Date of creation2013-03-22 12:26:58
Last modified on2013-03-22 12:26:58
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id9
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 20M35
Classificationmsc 68Q70
Synonymasterate
Related topicAlphabet
Related topicString
Related topicRegularExpression
Related topicKleeneAlgebra
Related topicLanguagePlanetmathPlanetmath
Related topicConvolution2
Related topicWeightStrings
Related topicWeightEnumerator
DefinesKleene plus
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更新时间:2025/5/4 5:12:30