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Chapter 0: 15E (page 1)

Give a counter example to show that the following construction fails to prove that the class of context-free languages is closed under star. Let A be a CFL G=(V,∑,R,S)that is generated by the CFG . Add the new rule S→SSand call the resulting grammar. This grammar is supposed to generate A* .

Short Answer

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The construction fails for L=a

Step by step solution

01

Define Context Free Language

The context free language is generated by context free grammar. These languages are accepted by Pushdown Automata. These are the superset of regular languages.

Consider context-free languages L1described as G1=(V1'S,R1'S1)

Consider context-free language L2described as G2=(V2'S,R2'S2)

02

Determine the counter example

Consider the language L=a

The given grammar and the language is in context free grammar in the form of .G=(V,∑,R,S)

Note that L*={t,a,aa,aaa,…}

A grammar of LisG=({S},{a},{(S→a)},S)

The construction gives: G'=({S},{a},{(S→a),(S→SS)},S)

However, {L(G)does not contain ε so the construction doesn’t produce a grammar for L*.

Hence, the construction fails for L = {a}

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