Question

For every symbol σ, σ 0 denotes the empty string, and for every positive integer k, σ k denotes the string of length k over the alphabet { σ }.

Using the pumping lemma for context-free languages, prove that the language {a mb mc 2m | m is an integer and m is not negative} is not context-free

Answer 1

Let us assume that L is context free.

Now, as per Pumping Lemma, let us assume that u = a^p, v = a^m-p, w = b^m, x = c^q, y = c^2m-q

Since L is CF, the string uvⁱwxⁱy should also belong to L.

For i = 2, uv²wx²y = a^m+pb^mc^4m+q

Clearly, (m + p) != m and 2*(m + p) != (4m + q)

Hence, the string uvⁱwxⁱy does not belong to L and this means that L is not a context free language.