Kleene star

Im Mathematische Logik und Informatik, das Kleene Star (oder Kleene operator oder Kleene -Schließung) ist ein Unary Operation, either on Sets von Saiten or on sets of symbols or characters. In mathematics it is more commonly known as the Freies Monoid Konstruktion. The application of the Kleene star to a set ${\ displayStyle v}$ ist geschrieben als ${\ displayStyle v^{*}}$. It is widely used for Reguläre Ausdrücke, which is the context in which it was introduced by Stephen Kleene to characterize certain Automaten, where it means "zero or more repetitions".

1. Wenn ${\ displayStyle v}$ is a set of strings, then ${\ displayStyle v^{*}}$ is defined as the smallest Superset von ${\ displayStyle v}$ das enthält die leerer String ${\ displayStyle \ varepsilon}$ und ist abgeschlossen unter dem string concatenation operation.
2. Wenn ${\ displayStyle v}$ is a set of symbols or characters, then ${\ displayStyle v^{*}}$ is the set of all strings over symbols in ${\ displayStyle v}$, including the empty string ${\ displayStyle \ varepsilon}$.

Der Satz ${\ displayStyle v^{*}}$ 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 ${\ displayStyle v}$, allowing the use of the same element multiple times. Wenn ${\ displayStyle v}$ ist entweder das leeres Set ∅ or the singleton set ${\displaystyle \{\varepsilon \}}$, dann ${\displaystyle V^{*}=\{\varepsilon \}}$; wenn ${\ displayStyle v}$ is any other endliche Menge oder countably infinite set, dann ${\ displayStyle v^{*}}$ is a countably infinite set.^[1] As a consequence, each formelle Sprache over a finite or countably infinite alphabet ${\ displayStyle \ sigma}$ is countable, since it is a subset of the countably infinite set ${\ displaystyle \ sigma ^{*}}$.

The operators are used in rewrite rules zum generative grammars.

Definition und Notation

Ein Satz gegeben ${\ displayStyle v}$ definieren

${\displaystyle V^{0}=\{\varepsilon \}}$ (the language consisting only of the empty string),

${\displaystyle V^{1}=V}$

and define recursively the set

${\displaystyle V^{i+1}=\{wv:w\in V^{i}{\text{ and }}v\in V\}}$ für jeden ${\displaystyle i>0}$.

Wenn ${\ displayStyle v}$ is a formal language, then ${\displaystyle V^{i}}$, das ${\ displayStyle i}$-th power of the set ${\ displayStyle v}$, is a shorthand for the Verkettung of set ${\ displayStyle v}$ mit sich selbst ${\ displayStyle i}$ mal. Das ist, ${\displaystyle V^{i}}$ can be understood to be the set of all Saiten that can be represented as the concatenation of ${\ displayStyle i}$ strings in ${\ displayStyle v}$.

The definition of Kleene star on ${\ displayStyle v}$ ist^[2]

${\displaystyle V^{*}=\bigcup _{i\geq 0}V^{i}=V^{0}\cup V^{1}\cup V^{2}\cup V^{3}\cup V^{4}\cup \cdots .}$

This means that the Kleene star operator is an idempotent unary operator: ${\displaystyle (V^{*})^{*}=V^{*}}$ für jeden Satz ${\ displayStyle v}$ of strings or characters, as ${\displaystyle (V^{*})^{i}=V^{*}}$ für jeden ${\displaystyle i\geq 1}$.

Kleene Plus

In einigen formelle Sprache studies, (e.g. AFL -Theorie) a variation on the Kleene star operation called the Kleene Plus wird genutzt. The Kleene plus omits the ${\displaystyle V^{0}}$ term in the above union. In other words, the Kleene plus on ${\ displayStyle v}$ ist

${\displaystyle V^{+}=\bigcup _{i\geq 1}V^{i}=V^{1}\cup V^{2}\cup V^{3}\cup \cdots .}$

oder

${\displaystyle V^{+}=V^{*}V}$ ^[3]

Beispiele

Example of Kleene star applied to set of strings:

{"ABC"}^* = { ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "ababc", "abcab", "abcc", "cabab", "cabc", "ccab", "ccc", ...}.

Example of Kleene plus applied to set of characters:

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

Kleene star applied to the same character set:

{"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 assoziativ und nicht kommutativ Produkt.

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

Wenn ${\displaystyle V=\{\varepsilon \}}$, then also ${\displaystyle V^{i}=\{\varepsilon \}}$ für jeden ${\ displayStyle i}$, somit ${\displaystyle V^{*}=V^{+}=\{\varepsilon \}}$.

Verallgemeinerung

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. Dann S^* is the smallest submonoid of M enthält S; das ist, S^* contains the neutral element of M, der Satz S, and is such that if x,y ∈ S^*, dann x⋅y ∈ S^*.

Furthermore, the Kleene star is generalized by including the *-operation (and the union) in the algebraische Struktur itself by the notion of complete star semiring.^[4]

Siehe auch

• Wildcard -Charakter
• Glob (Programmierung)

Verweise

Weitere Lektüre

• Hopcroft, John E.; Ullman, Jeffrey D. (1979). Einführung in die Automatentheorie, Sprachen und Berechnung (1. Aufl.). Addison-Wesley.