Give an example of a language that is not context free but that acts like a CFL in the pumping lemma. Prove that your example works. See the analogous example for regular languages in Problem 1.54.

Problem 1.54.

Consider the language F = {aⁱb^jc^k| i, j, k ≥ 0 and if i = 1 then j = k}.

a. Show that F is not regular.

b. Show that F acts like a regular language in the pumping lemma. In other words, give a pumping length p and demonstrate that F satisfies the three conditions of the pumping lemma for this value of p.

c. Explain why parts (a) and (b) do not contradict the pumping lemma.