How can one prove that the language below is not context-free using the pumping lemma?
$$\{ a^i b^m a^j b^m a^k b^m \mid i,j,k,m \geq 0 \}$$
How can one prove that the language below is not context-free using the pumping lemma?
$$\{ a^i b^m a^j b^m a^k b^m \mid i,j,k,m \geq 0 \}$$
Suppose that $a^* b^m a^* b^m a^* b^m$ is context-free. Applying the inverse of the homomorphism defined by $a\mapsto a$ and $b,c,d \mapsto b$, we see that $a^* (b+c+d)^m a^* (b+c+d)^m a^* (b+c+d)^m$ is also context-free. Intersecting with the regular language $b^* a c^* a d^*$, we see that $b^m a c^m a d^m$ is also context-free. Applying the homomorphism defined by $a \mapsto \epsilon$ and $\sigma \mapsto \sigma$ for $\sigma=b,c,d$, we see that $b^mc^md^m$ is also context-free. But we know that it's not context-free.