Dictionary - Definition of Kleene star

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

$V_{i+1}=\{wv : w\in V_i \mbox{ and } v \in V\}\,$ 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 $V*=\bigcup_{i \in \N} V_i = \left \{\lambda \right\} \cup V_1 \cup V_2 \cup V_3 \cup \ldots$

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"}* = {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", ...}

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

Expressions	Domain	Definition
Kleene star	Computing	Kleene star (Or "Kleene closure", named after Stephen Kleene) The postfix "" operator used in regular expressions, Extended Backus-Naur Form, and similar formalisms to specify a match for zero or more occurrences of the preceding expression. For example, the regular expression "bet" would match the string "bt", "bet", "beet", "beeeeet", and so on. (2000-07-26). Source: The Free On-line Dictionary of Computing..
Source: compiled by the editor from various references; see credits.			Top

Specialty Expressions: Kleene star

Extended Definition: Kleene star

Kleene star

Definition and notation

Examples

Generalization

See also