I have to prove that $\left \{ a, b \right \}^{\ast} - \left \{ a^ib^i | i\geq 0 \right \}$ is a context-free language and it's not regular.
So far I've got that this language is not regular because regular languages are closed under complementation and $\left \{ a^ib^i | i\geq 0 \right \}$ is a well know context free language (by the way do you have a link with a formal proof that $\left \{ a^ib^i | i\geq 0 \right \}$ is context free?)
For the context-free I'm a little confused I've thought about building a PDA that recognize $\left \{ a^ib^i | i\geq 0 \right \}$ (pushing a's on the stack and popping a's when a b is read works?) I think is deterministic CF so i can use the same reasoning (complement of a DCF is a DCF language?)