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 is written as . 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".
If is a set of strings, then is defined as the smallest superset of that contains the empty string and is closed under the string concatenation operation.
If is a set of symbols or characters, then is the set of all strings over symbols in , including the empty string .
The set 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. If is either the empty set ∅ or the singleton set , then ; if 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 is countable, since it is a subset of the countably infinite set .
The operators are used in rewrite rules for generative grammars.
^Nayuki Minase (10 May 2011). "Countable sets and Kleene star". Project Nayuki. Retrieved 11 January 2012.
In mathematical logic and computer science, the Kleenestar (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or...
Kleene hierarchy, Kleene algebra, the Kleenestar (Kleene closure), Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular...
language {a } is a regular language. If A is a regular language, A* (Kleenestar) is a regular language. Due to this, the empty string language {ε} is...
symbols is called a "word over A", and the free monoid A∗ is called the "Kleenestar of A". Thus, the abstract study of formal languages can be thought of...
In mathematics, a Kleene algebra (/ˈkleɪni/ KLAY-nee; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed...
State complexity is an area of theoretical computer science dealing with the size of abstract automata, such as different kinds of finite automata. The...
expressions began in the 1950s, when the American mathematician Stephen Cole Kleene formalized the concept of a regular language. They came into common use...
of all words over an alphabet Σ is usually denoted by Σ* (using the Kleenestar). The length of a word is the number of letters it is composed of. For...
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and regular expressions. In a complete star semiring, the star operator behaves more like the usual Kleenestar: for a complete semiring we use the infinitary...
expressed using regular expressions of limited star height, i.e. with a limited nesting depth of Kleene stars. Specifically, is a nesting depth of one...
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N)^{*}\rightarrow (\Sigma \cup N)^{*}} where ∗ {\displaystyle {}^{*}} is the Kleenestar operator and ∪ denotes set union, so ( Σ ∪ N ) ∗ {\displaystyle (\Sigma...
class PSPACE is closed under operations union, complementation, and Kleenestar. An alternative characterization of PSPACE is the set of problems decidable...
then the following languages are recursively enumerable as well: the Kleenestar L ∗ {\displaystyle L^{*}} of L the concatenation L ∘ P {\displaystyle...
alphabet A using only the standard operators set union, concatenation, and Kleenestar. Generalized regular expressions are defined just as regular expressions...
{\mathcal {P}}(Q\times \Gamma ^{*})} where ∗ {\displaystyle *} is the Kleenestar, meaning that Γ ∗ {\displaystyle \Gamma ^{*}} is "the set of all finite...
given by the regular expression 1*( 0 (1*) 0 (1*) )*, where "*" is the Kleenestar, e.g., 1* denotes any non-negative number (possibly zero) of symbols...
Formal grammar Formal language Formal system Generalized star height problem Kleene algebra Kleenestar L-attributed grammar LR-attributed grammar Myhill-Nerode...
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