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Concept
Kleene star
Summary
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. 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.
Given a set ,
define
(the language consisting only of the empty string),
and define recursively the set
for each .
If is a formal language, then , the -th power of the set , is a shorthand for the concatenation of set with itself times. That is, can be understood to be the set of all strings that can be represented as the concatenation of strings in .
The definition of Kleene star on is
This means that the Kleene star operator is an idempotent unary operator: for any set of strings or characters, as for every .
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 term in the above union.
Official source
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