# Conditions that imply closure under intersection of context-free languages

Context-free languages are not closed under intersection.

Suppose $$L_1, L_2 \in CF \setminus REG$$ (i.e., $$L_1,L_2$$ are context-free but not regular).

Are there well-known theorems (and/or whole papers/research topics) that try to shape the sufficient/necessary conditions for $$L_1,L_2$$ to make $$L_1 \cap L_2$$ context free?

• The background of this question is that it is undecidable whether the intersection of two context-free languages is context-free. – John L. May 20 '19 at 19:14
• You can probably generalize the example in this answer. We can ask more generally, which context-free language are such that their intersection with a context-free language is always context-free. – Yuval Filmus May 21 '19 at 8:41
• A related question, language whose intersection with a CFL is always a CFL. – John L. Jun 10 '19 at 9:22