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Complexity of PTL separability for context-free languages

Ascertain the computational complexity of deciding, given two context-free languages K and L, whether there exists a piecewise testable language R such that K ⊆ R and L ∩ R = ∅ (separability by piecewise testable languages).

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Background

Separability by piecewise testable languages (PTL) is tightly connected to unboundedness properties and downward closures, and is known to be decidable for several language classes. Despite substantial progress, the exact complexity for context-free languages has resisted classification.

The paper notes broad equivalences and inter-reductions among various verification problems but emphasizes that, even when emptiness complexity is known (P-complete for CFL), the complexity of PTL separability for CFL remains an open problem.

References

For some classes of infinite-state systems, the complexity of some of the problems (\labelcref{sup-decidable})--(\labelcref{pdc-computable}) even remains open, whereas the complexity of emptiness is known. For example, the complexity of separability by piecewise testable languages is not known for context-free languages, whereas emptiness is well-known to be $P$-complete.

— Verifying Unboundedness via Amalgamation (2405.10296 - Anand et al., 16 May 2024) in Section 6, Conclusion (Complexity)