Kleene star

In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a set V to generate a set V* of all finite-length strings^{[note 1]} that are composed of zero or more repetitions of members from V. It was named after American mathematician Stephen Cole Kleene, who first introduced and widely used it to characterize automata for regular expressions. In mathematics, it is more commonly known as the free monoid construction.

Definition

Given a set ${\displaystyle V}$, define

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

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

and define recursively the set

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

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

The definition of Kleene star on ${\displaystyle V}$ is^[1]

${\displaystyle V^{*}=\bigcup _{i\geq 0}V^{i}=V^{0}\cup V^{1}\cup V^{2}\cup V^{3}\cup V^{4}\cup \cdots .}$

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 ${\displaystyle V^{0}}$ term in the above union. In other words, the Kleene plus on ${\displaystyle V}$ is

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

or

${\displaystyle V^{+}=V^{*}V.}$^{[note 2]}

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", ...}.

Properties

• If ${\displaystyle V}$ is any finite or countably infinite set, then ${\displaystyle V^{*}}$ is a countably infinite set.^[2] As a result, each formal language over a finite or countably infinite alphabet ${\displaystyle \Sigma }$ is countable, since it is a subset of the countably infinite set ${\displaystyle \Sigma ^{*}}$.
• ${\displaystyle (V^{*})^{*}=V^{*}}$, which means that the Kleene star operator is an idempotent unary operator, as ${\displaystyle (V^{*})^{i}=V^{*}}$ for every ${\displaystyle i\geq 1}$.
• ${\displaystyle V^{*}=\{\varepsilon \}}$, if ${\displaystyle V}$ is either the empty set ∅ or the singleton set ${\displaystyle \{\varepsilon \}}$.

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.^[3]

See also

• Wildcard character
• Glob (programming)

Notes

1. It is called "strings" for historical reasons, since Kleene invented it in the context of automata theory, but the idea has been generalized such that each symbol in a string is not necessarily a single character (see § Generalization).
2. This equation holds because every member of V⁺ can be generated by first picking a member from V*, and then picking a member from V for appending. This two-step process does not generate ε since the second step never pick an ε.