Kleene star

The clover ash shell (also called finite degree, Kleene * 's degree, chaining shell or star shell ) of an alphabet or a formal language is the set of words that by any concatenation (linking ) of symbols of the alphabet and words of the language are formed may, with the empty word is included. It is named after the American mathematician and logician Stephen Cole Kleene. In contrast, the positive envelope (also called Kleene degree ) of an alphabet or a formal language, the set of all words that can be formed from the symbols of, or consisting of words and of which only contains the empty word if the positive envelope is applied to a language that itself contains the empty word as an element.

The operator of clover eschen shell is the Kleene star " ". Thus, the representation of the clover eschen shell of an alphabet and a language equal is equal. In contrast, the operator of the positive envelope is the plus sign " " so that the positive envelope of an alphabet and with a language is shown with.

Following the Kleene * operator on the languages ​​of the * operator in regular expressions is also called Kleene * operator. The number of nested Kleene * - operators determines the star height of a regular expression.

• 2.1 alphabets
• 2.2 languages

Definition

Closure operator for alphabets

The clover ash shell of an alphabet is a language that contains all words over the alphabet. They can be defined by means of structural induction. In the induction base is defined, first, that the empty word is contained in the clover eschen shell, and the induction step is defined, that is, for each word, the element of clover eschen shell, the concatenations for all symbols are elements of clover ash shell:

• Induction base:
• Induction step:

The positive hull of an alphabet is defined as the clover ash shell of this alphabet without the empty word:

Starting from the clover eschen shell itself can be defined subsets of words of fixed length.

Alternatively, be defined as the fold Cartesian product of the alphabet, so

Then:

Closure operator for Languages

The clover ash shell of a language is the union of all its potency languages ​​(repeated concatenation of the languages):

Where and.

The positive hull of a language is defined similarly, it is the union of all powers of greater than or equal to 1:

Examples

Alphabets

The clover ash shell of the alphabet contains the empty word, the word, and therefore the word and so on. This is.

For the alphabet, and so on. This is.

Languages

The clover ash shell of the language is the set of all words that are composed of and, and the empty word:

The positive envelope is accordingly:

The clover ash shell of the empty language and the language of the empty word only contains the empty word:

The positive hull of the empty language is empty, the language of the empty word only contains the empty word:

Features

• The clover ash shell and the positive cover (if the latter contains the empty word ), the amount of support the monoid with concatenation of words as an operator and the empty word as a neutral element, respectively. The clover ash shell and the positive envelope are completed against the concatenation.

• The clover ash and the positive envelope are infinite for all languages ​​that contain at least one non - empty word, is countable:

• If a language contains the empty word, the clover ash and the positive envelope of the same; the converse also holds: