What is difference between closure and kleene closure

In 1951, S Kleene was studying a mathematical model of neurons. These are like Finite State machines in that they do not have scratch memory. He noted patterns to the languages that are recognized by such devices. He realized that in the Theory of computation,  languages are not only set having standard set operations such as union, intersection, and complementation which are applied to the languages.  Apart from those sets of operations, he proposed two more important operations that were closed under such languages as Concatenation and closure. Concatenation is termed as Kleene Plus denoted by Σ+. And closure which is termed as Kleene star or Kleene closure denoted by Σ∗.

Kleene Star Closure

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).

Kleen PLUS Operation ( + )

Kleen 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, ...

The following is the Difference between Closure and Kleene’s Closure

·    The closure is nothing but the iteration of 0 to ∞ times, but Kleene’s closure is the set including Λ.

·     The closure is applied on RE, but Kleene’s closure is applied on Σ.

·   If R = 01, then the closure on R denoted by R* is Λ, 01, 0101, 010101…etc., i.e., the iteration of the same string 01.

·     If Σ = {0, 1}, then Kleene’s closure is denoted by Σ* = {Λ, 0, 1, 01, 00, 11, 010, 011, 100…}, i.e., the set of any combinations of 0 and 1 including Λ.

 Also, Study about: 

• Mealy Machine
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• Rice Theorem
• Brief about Turing Machine
• Multi Dimension Turing Machine
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• Parikh Theorem in TOC
• Parse Tree or Derivation Tree
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• Know Where we use Automata
• What do you understand by Derivation of Grammar
• What is Context-free Grammar
• Example based on CFG
• Regular Expression in TOC
• What is Two-way infinite Turing machine?
• Example to generate Parse tree from the given grammar

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