Constrained generation aims to sample from language models (LMs) conditioned on hard constraints. Existing constrained-generation techniques for nondeterministic finite automaton (NFA) constraints either distort the distribution or sacrifice efficiency. Theoretically, this task reduces to counting the length-$n$ sequences accepted by an NFA (#NFA), and the exact #NFA problem is #P-complete. Recent work has shown that #NFA admits a fully polynomial randomized approximation scheme (FPRAS). Inspired by this result, we propose NFA-LM, a polynomial-time engine for NFA-constrained generation with theoretical guarantees under mild assumptions. Experiments show that NFA-LM efficiently generates high-quality outputs with theoretically bounded approximation error.

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