Closing of Kleene

See also: Closing

The closing of Kleene , sometimes called star of Kleene or iterative closing , is a unary operator used to describe the formal languages. Applied to a unit V , it has as a result the language $V^ \ star$ , defined as follows:

If V is a Alphabet, i.e. a finished whole of symbols or characters, then $V^ \ star$ is the whole of the words on V , word empties ε included.
If V is a language, then $V^ \ star$ is the smallest language which contains it, which contains {ε} and which is stable by Concaténation (the concatenation of two elements of $V^ \ star$ is also in $V^ \ star$ ).

Example of application of star of Kleene to an alphabet:

{“has”, “B”, “it} * = {ε, “has”, “B”, “C”, “aa”, “ab”, “ac”, “Ba”, “bb”, “bc”,…}

Example of application of star of Kleene to a language:

{“ab”, “C”} * = {ε, “ab”, “C”, “abab”, “ABC”, “cab”, “DC”, “ababab”, “ababc”, “abcab”, “abcc”, “cabab”, “cabc”, “ccab”, “ccc”,…}

The star of Kleene is one of the three core operators used to define a rational Expression, with the concatenation and the union ensemblist.

One often generalizes star of Kleene to all Monoïde (M.) , where $V^ \ star$ , with $V \ subset M$ , indicates the fence V by the law “. ”, to which one joint ε, the neutral element of the monoid. In other words $V^ \ star$ is the smallest unit containing $V \ cup \ {\ epsilon \}$ , and stable by “. ”. It is indeed about a generalization, because the whole of the words on an alphabet is a monoid whose Law of composition interns is the concatenation, and the neutral element is the blank word.

See too

Algebra of Kleene
wide Form of Backus-Naur

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