Kleene star

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In mathematical logic and computer science, the Kleene star (or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. The application of the Kleene star to a set V is written as V*. It is widely used for regular expressions, which is the context in which it was introduced by Stephen Kleene to characterise certain automata.

If V is a set of strings then V* is defined as the smallest superset of V that contains ε (the empty string) and is closed under the string concatenation operation. This set can also be described as the set of strings that can be made by concatenating zero or more strings from V.
If V is a set of symbols or characters then V* is the set of all strings over symbols in V, including the empty string.

1 Definition and notation
2 Examples
3 Generalization
4 See also

[edit] Definition and notation

Given

$V_0=\{\epsilon\}\,$
$V_{i+1}=\{wv : w\in V_i \wedge v \in V\}\,$ where $i > 0\,$

If $V$ is a formal language, the $i$ -th power of set $V$ is shorthand for the concatenation of set $V$ with itself i times. More information about set concatenation for languages is here.

Therefore the definition of Kleene star on $V$ is

$V^*=\bigcup_{i=0}^{\infty} V_i = \left \{\varepsilon \right\} \cup V \cup V^2 \cup V^3 \cup \ldots$

[edit] Examples

Example of Kleene star applied to 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 set of characters:

{'a', 'b', 'c'}* = {ε, "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", ...}

[edit] Generalization

The Kleene star is often generalized for any monoid (M, $\circ$ ), that is, a set M and binary operation $\circ$ on M such that

(closure) $\forall a,b \in M:~ a \circ b \in M$
(associativity) $\forall a,b,c \in M:~ (a \circ b) \circ c = a \circ (b \circ c)$
(identity) $\exists \epsilon \in M:~ \forall a \in M:~ a \circ \epsilon = a = \epsilon \circ a$

If V is a subset of M, then V* is defined as the smallest superset of V that contains ε (the empty string) and is closed under the operation. V* is then itself a monoid, and is called the monoid generated by V. This is a generalization of the Kleene star discussed above since the set of all strings over some set of symbols forms a monoid (with string concatenation as binary operation).