Question

Suppose that we have a language L dened over the alphabet {a, b, c} and suppose that L is context-free. We define a new language pm(L)

to be the setof all permutations of all words in L. For example, if L = {abc, aab} then pm(L) = {abc, acb, bac, bca, cab, cba, aab, aba, baa}.

Show that pm(L) need not be context-free by giving an example of a language L that is context-free but where pm(L) is not context-free.

Answer 1

Given that,

L={a,b,c} which is context free

and

pm(L)={permutations of L}

to prove,

pm(L) is not a context free grammer

pm(L)={(L)*} =L*,where L =alphabets

L* which is not context free,because it jumps from alphabets of context free.

Hence it proved