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arXiv · 2510.08127

On the Complexity of Language Membership for Probabilistic Words

Antoine Amarilli·Mikaël Monet·Paul Raphaël·Sylvain Salvati
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Abstract

We study the membership problem to context-free languages (CFLs) on probabilistic words, that specify for each position a probability distribution on the letters. Our task is to compute, given a probabilistic word, what is the probability that a word drawn according to the distribution belongs to the language $L$. This problem generalizes the problem of counting how many words of length $n$ belong to $L$, or of counting how many completions of a partial word belong to $L$. We show that this problem is in polynomial time for unambiguous context-free languages (uCFLs), but can be #P-hard already for unions of two linear uCFLs. More generally, we show that the problem is in polynomial time for so-called poly-slicewise-unambiguous languages, where given a length $n$ we can tractably compute an uCFL for the words of length $n$ in the language. This class includes some inherently ambiguous languages, and implies the tractability of bounded CFLs and of languages recognized by unambiguous polynomial-time counter automata. We then introduce classes of circuits from knowledge compilation which we use for tractable counting, and show that this covers the tractability of poly-slicewise-unambiguous languages and of some CFLs that are not poly-slicewise-unambiguous. Extending these circuits with negation further allows us to show tractability for the language of primitive words, and for the language of concatenations of two palindromes. We also show that, when the target language is given as input, our problem is intractable already when the language asks whether there is a factor that matches one partial word; however, it becomes tractable when the language is given as a $k$-ambiguous automaton for any fixed $k>0$. We finally show the conditional undecidability of the meta-problem that asks, given a CFG, whether the probabilistic membership problem for that CFG is tractable or #P-hard.

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Antoine Amarilli, Mikaël Monet, Paul Raphaël, Sylvain Salvati. 2025-10-09. On the Complexity of Language Membership for Probabilistic Words. https://arxiv.org/abs/2510.08127

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