Myhill-Nerode style characterization of CFL?
David S. Wise provides in his paper A strong pumping lemma for context-free languages a strong pumping lemma which is equivalent to being context-free. He also provides an additional equivalent condition (property 3 on page 362) which he claims could be viewed as an analog of the Myhill-Nerode theorem. As an application of the latter, he shows that $\{ a^nba^{mn} : m,n &gt; 0 \}$ cannot be expressed as a finite intersection of context-free languages.

More information on the strong pumping lemma appears in one of my answers.