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Are all 2‑ambiguous CFLs unions of two unambiguous CFLs?

Determine whether every 2-ambiguous context-free language can be written as the union of two unambiguous context-free languages.

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Background

The paper proves #P-hardness of probabilistic membership already for the union of two linear unambiguous CFLs, and remarks on the relationship to 2-ambiguous CFLs. Whether every 2-ambiguous CFL decomposes as a union of two unambiguous CFLs is a longstanding ambiguity-structure question.

References

We do not know whether this is a generalization, because it is not known whether all 2-ambiguous CFLs can be written as the union of two uCFLs, cf Section 9.3, point (iv).

— On the Complexity of Language Membership for Probabilistic Words (2510.08127 - Amarilli et al., 9 Oct 2025) in Section 3, Intractability for 2-ambiguous linear CFLs, footnote following Proposition prp:hard2