Kleene star

 ( Formal Languages ) 

In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a set to generate a set of all finite-length strings that are composed of zero or more repetitions of members from . It was named after American mathematician Stephen Cole Kleene, who first introduced and widely used it to characterize Automata theory for regular expressions. In mathematics, it is more commonly known as the free monoid construction.

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Definition Given a set $V$, define

$V^\{0\}=\backslash \{\backslash varepsilon\backslash \}$ (the set consists only of the empty string),

$V^\{1\}=V,$

and define recursively the set

$V^\{i+1\}=\backslash \{wv:\; w\backslash in\; V^\{i\}\; \backslash text\{\; and\; \}\; v\backslash in\; V\; \backslash \}$ for each $i>0.$

$V^i$ is called the $i$-th power of $V$, it is a shorthand for the concatenation of $V$ by itself $i$ times. That is, $V^i$ can be understood to be the set of all strings that can be represented as the concatenation of $i$ members from $V$.

The definition of Kleene star on $V$ is

$V^*=\backslash bigcup\_\{i\; \backslash ge\; 0\; \}V^i\; =\; V^0\; \backslash cup\; V^1\; \backslash cup\; V^2\; \backslash cup\; V^3\; \backslash cup\; V^4\; \backslash cup\; \backslash cdots.$

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Kleene plus In some formal language studies, (e.g. AFL theory) a variation on the Kleene star operation called the Kleene plus is used. The Kleene plus omits the $V^\{0\}$ term in the above union. In other words, the Kleene plus on $V$ is

$V^+=\backslash bigcup\_\{i\; \backslash geq\; 1\}\; V^i\; =\; V^1\; \backslash cup\; V^2\; \backslash cup\; V^3\; \backslash cup\; \backslash cdots,$

or

$V^+\; =\; V^*V.$

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Examples Example of Kleene star applied to a set of strings:

{"ab","c"}^* = { ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "ababc", "abcab", "abcc", "cabab", "cabc", "ccab", "ccc", ...}.

Example of Kleene star applied to a set of strings without the prefix property:

{"a","ab","b"}^* = { ε, "a", "ab", "b", "aa", "aab", "aba", "abab", "abb", "ba", "bab", "bb", ...};
e.g. the string "aab" can be obtained in several different ways. The Sardinas-Patterson algorithm can be used to check for a given V whether any member of V^* can be obtained in more than one way.

Example of Kleene and Kleene plus applied to a set of characters:

{"a", "b", "c"}^* = { ε, "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}.

{"a", "b", "c"}⁺ = { "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}.

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Properties

• If $V$ is any finite set or countably infinite set, then $V^*$ is a countably infinite set. As a result, each formal language over a finite or countably infinite alphabet $\backslash Sigma$ is countable, since it is a subset of the countably infinite set $\backslash Sigma^\{*\}$.
• $(V^\{*\})^\{*\}=V^\{*\}$, which means that the Kleene star operator is an idempotent unary operator, as $(V^\{*\})^\{i\}=V^\{*\}$ for every $i\backslash geq\; 1$.
• $V^\{*\}=\backslash \{\backslash varepsilon\backslash \}$, if $V$ is either the empty set ∅ or the singleton set $\backslash \{\backslash varepsilon\backslash \}$.

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Generalization Strings form a monoid with concatenation as the binary operation and ε the identity element. In addition to strings, the Kleene star is defined for any monoid. More precisely, let ( M, ⋅) be a monoid, and S ⊆ M. Then S^* is the smallest submonoid of M containing S; that is, S^* contains the neutral element of M, the set S, and is such that if x, y ∈ S^*, then x⋅ y ∈ S^*.

Furthermore, the Kleene star is generalized by including the *-operation (and the union) in the algebraic structure itself by the notion of complete star semiring.

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See also

• Wildcard character
• Glob (programming)

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Notes

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Further reading

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