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Can the uCFG probabilistic membership algorithm be improved below cubic time?

Determine whether probabilistic membership for unambiguous context-free grammars can be computed in subcubic time in the input length, for example matching O(n^ω) (Valiant’s parser) or O(n^2) (Earley’s bound for uCFGs), thereby improving the current O(n^3·|Γ|) dynamic-programming algorithm.

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Background

The authors give an O(|p|3·|Γ|) algorithm (up to arithmetic costs) for probabilistic membership on unambiguous CFGs via a CYK-style DP. They explicitly leave open whether this complexity can be lowered, pointing to classical subcubic and quadratic parsers as potential targets.

A positive result would sharpen the algorithmic frontier for probabilistic membership and align it with the best known complexities for non‑probabilistic parsing in related settings.

References

We do not claim that the cubic dependency in the probabilistic word is optimal, and leave to future work the investigation of whether the complexity can be lowered, e.g., to match the complexity of Valiant's parser, or the quadratic upper bound given by the Earley parser for unambiguous CFGs.

— On the Complexity of Language Membership for Probabilistic Words (2510.08127 - Amarilli et al., 9 Oct 2025) in Section 3, CFLs and Unambiguity (Unambiguous CFLs), immediately after Proposition prp:cyk