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.

Definition and notation

Given

V_0={epsilon},

define recursively the set

V_{i+1}={wv : win V_i mbox{ and }  v in V},

where

i > 0,.

If $V$ is a formal language, then the $i$ -th power of the set $V$ is shorthand for the concatenation of set $V$ with itself $i$ times. That is, $V_i$ can be understood to be the set of all strings of length $i$ , formed from the symbols in $V$ .

The definition of Kleene star on $V$ is $V^*=bigcup_{i in N} V_i = left {epsilon right} cup V_1 cup V_2 cup V_3 cup ldots$

That is, it is the collection of all possible finite-length strings generated from the symbols in $V$ .

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 $V^+=bigcup_{i in N^{star}} V_i = V_1 cup V_2 cup V_3 cup ldots$

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", ...}

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) $forall a,b,c in M:~ (a circ b) circ c = a circ (b circ c)$

Kleene star

Definition and notation

Examples

Generalization

See also