Kleene star

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 λ (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:$V\_0=\{lambda\},$ define recursively the set:$V\_\{i+1\}=\{wv\; :\; win\; V\_i\; mbox\{\; and\; \}\; v\; in\; V\},$ where $i\; ge\; 0,.$

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

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

Examples

Example of Kleene star applied to set of strings:: {"ab", "c"}* = {λ, "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'}* = {λ, "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", ...}

Example of Kleene star applied to the empty set::$varnothing\; ^*\; =\{lambda\}$

Example of Kleene plus applied to the empty set::$varnothing\; ^+\; =\; varnothing\; varnothing\; ^*\; =\{\}=\; varnothing$

Generalization

Strings form a monoid with concatenation as the binary operation and λ the identity element. The Kleene star is defined for any monoid, not just strings.

See also

* Kleene algebra
* Extended Backus-Naur form
* Pumping lemma
* Star height problem, generalized star height problem, star-free language
* Regular expressions