# Language whose intersection with a CFL is always a CFL (2)

This is a follow-up to this question, which asks for an example of a non-regular language $$L$$ which satisfies the following condition, intersection resilience:

If $$L'$$ is context-free then so is $$L \cap L'$$.

We know that all regular languages are intersection-resilient, and clearly every intersection-resilient language is context-free. Ginsburg and Spanier showed in 1964, in their paper Bounded ALGOL-like languages, that every context-free subset of $$x^*y^*$$ is intersection-resilient (Theorem 6.1, Corollary 2).

Which languages are intersection-resilient?

• A related question is whether it is decidable whether a given context-free language is intersection-resilient. Another related question is whether the intersection of an intersection-resilient language and an context-free language is intersection-resilient. Note that the intersection of two intersection-resilient languages is intersection-resilient. – Apass.Jack May 29 at 15:44

A necessary condition for intersection-resilience is that for all omomorphisms $$\varphi : \Sigma \to \Sigma \cup\ \{\epsilon \}$$; $$\varphi(L) \cap x^*y^*x^*$$ must be a regular language (where $$x$$ and $$y$$ are any two distinct symbols of $$\Sigma$$).