Course Title: Theory of Automata (CS402)

Lecture No. 2: Kleene Star Closure

Introduction

Recursive definition of languages

Summary

     Introduction

Given Σ, then the Kleene Star Closure of the alphabet Σ, denoted by Σ^*, is the collection of all strings defined over Σ, including Λ.

It is to be noted that Kleene Star Closure can be defined over any set of strings.

Examples

If Σ = {x}

Then Σ^* = {Λ, x, xx, xxx, xxxx, ….}

If Σ = {0,1}

Then Σ^* = {Λ, 0, 1, 00, 01, 10, 11, ….}

If Σ = {aaB, c}

Then Σ^* = {Λ, aaB, c, aaBaaB, aaBc, caaB, cc, ….}

Note

Languages generated by Kleene Star Closure of set of strings, are infinite languages. (By infinite language, it is supposed that the language contains infinite many words, each of finite length).

PLUS Operation (⁺)

Plus Operation is same as Kleene Star Closure except that it does not generate  Λ (null string), automatically.

Example

If Σ = {0,1}

Then Σ⁺ = {0, 1, 00, 01, 10, 11, ….}

If Σ = {aab, c}

Then Σ⁺ = {aab, c, aabaab, aabc, caab, cc, ….}

Remark

It is to be noted that Kleene Star can also be operated on any string i.e. a^* can be considered to be all possible strings defined over {a}, which shows that a^* generates Λ, a, aa, aaa, …

It may also be noted that a⁺ can be considered to be all possible non empty strings defined over {a}, which shows that a⁺ generates a, aa, aaa, aaaa, …

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