In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters.
In mathematics, it is more commonly known as the free monoid construction.
The application of the Kleene star to a set
It is widely used for regular expressions, which is the context in which it was introduced by Stephen Kleene to characterize certain automata, where it means "zero or more repetitions".
can also be described as the set containing the empty string and all finite-length strings that can be generated by concatenating arbitrary elements of
, allowing the use of the same element multiple times.
is any other finite set or countably infinite set, then
is a countably infinite set.
[1] As a consequence, each formal language over a finite or countably infinite alphabet
The operators are used in rewrite rules for generative grammars.
, is a shorthand for the concatenation of set
can be understood to be the set of all strings that can be represented as the concatenation of
The definition of Kleene star on
is[2] This means that the Kleene star operator is an idempotent unary operator:
In some formal language studies, (e.g. AFL theory) a variation on the Kleene star operation called the Kleene plus is used.
is or Example of Kleene star applied to set of strings: Example of Kleene plus applied to set of characters: Kleene star applied to the same character set: Example of Kleene star applied to the empty set: Example of Kleene plus applied to the empty set: where concatenation is an associative and noncommutative product.
Example of Kleene plus and Kleene star applied to the singleton set containing the empty string: Strings form a monoid with concatenation as the binary operation and ε the identity element.
The Kleene star is defined for any monoid, not just strings.
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.