Question

4. (Closure) Show that the class of context-free languages is closed under the star operation.

Answer 1

Solution: Star operation basically involves a unary operator * that operates on a set of symbols or strings, ∑, and produces an infinite set of all possible strings of all possible lengths as output. As far as Context-Free Languages are concerned, they are generated by the Context-Free Grammars of form C -> ρ (where C ∈ N and ρ ∈ (T ∪ N)* and N is a non-terminal and T is a terminal).

Proof: Let us say there is a language L1 that is context-free, therefore its star closure L1* would always be context-free as shown below.

L1 = { aⁿbⁿ | n >= 0 }
L1* = { aⁿbⁿ | n >= 0 }* is also context free.

L1* is context-free because it is nothing but a superset of all the strings that can be created out of the elements present in context-free language L1 and it is already known that L1 is a collection of the strings which follow the context-free grammar. Therefore L1* is a context-free language as well.

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