Kleene star

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 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, where it means "zero or more".

If V is a set of strings, then V^* is defined as the smallest superset of V that contains the empty string ε and is closed under the string concatenation operation.
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 ε.

The set V^* can also be described as the set of finite-length strings that can be generated by concatenating arbitrary elements of V, allowing the use of the same element multiple times. If V is either the empty set ∅ or the singleton set {ε}, then V^* = {ε}; if V is any other finite set, then V^* is a countably infinite set.^[1]

The operators are used in rewrite rules for generative grammars.

Definition and notation

Given a set V define

V₀ = {ε} (the language consisting only of the empty string),

V₁ = V

and define recursively the set

V_i+1 = { wv : w ∈ V_i and v ∈ V } for each i>0.

If V is a formal language, then V_i, the i-th power of the set V, is a 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 that can be represented as the concatenation of i strings in V.

The definition of Kleene star on V is^[2]

V^*=\bigcup_{i \in \N }V_i = \{\varepsilon\} \cup V \cup V_2 \cup V_3 \cup V_4 \cup \ldots.

Kleene plus

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 V₀ term in the above union. In other words, the Kleene plus on V is

V^+=\bigcup_{i \in \N \setminus \{0\}} V_i = V_1 \cup V_2 \cup V_3 \cup \ldots.

For every set L, the Kleene plus L⁺ equals the concatenation of L with L^*. Conversely, L^* can be written as {ε} ∪ L⁺.

Examples

Example of Kleene star applied to set of strings:

{"ab","c"}^* = {ε, "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"}^* = { ε, "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}.

Example of Kleene star applied to the empty set:

∅^* = {ε}.

Example of Kleene plus applied to the empty set:

∅⁺ = ∅ ∅^* = { } = ∅,

where concatenation is an associative and noncommutative product, sharing these properties with the Cartesian product of sets.

Example of Kleene plus and Kleene star applied to the singleton set containing the empty string:

If V = {ε}, then also V_i = {ε} for each i, hence V^* = V⁺ = {ε}.

Generalization

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.^[3]

References

↑ Nayuki Minase (10 May 2011). "Countable sets and Kleene star". Project Nayuki. Retrieved 11 January 2012.
↑ Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang (1994). Mathematical Logic (2nd ed.). New York: Springer. p. 656. ISBN 0-387-94258-0. The Kleene closure L^* of L is defined to be $\sideset{}{_{i=0}^\infty}\bigcup L^i$ .
↑ Droste, M.; Kuich, W. (2009). "Chapter 1: Semirings and Formal Power Series". Handbook of Weighted Automata. Monographs in Theoretical Computer Science. Springer. p. 9. ISBN 978-3-642-01491-8. doi:10.1007/978-3-642-01492-5_1.