#CFG and #DNNF admit FPRAS
<|reference_start|>#CFG and #DNNF admit FPRAS: We provide the first fully polynomial-time randomized approximation scheme for the following two counting problems: 1. Given a Context Free Grammar $G$ over alphabet $\Sigma$, count the number of words of length exactly $n$ generated by $G$. 2. Given a circuit $\varphi$ in Decomposable Negation Normal Form (DNNF) over the set of Boolean variables $X$, compute the number of assignments to $X$ such that $\varphi$ evaluates to 1.   Finding polynomial time algorithms for the aforementioned problems has been a longstanding open problem. Prior work could either only obtain a quasi-polynomial runtime (SODA 1995) or a polynomial-time randomized approximation scheme for restricted fragments, such as non-deterministic finite automata (JACM 2021) or non-deterministic tree automata (STOC 2021).<|reference_end|>
@article{meel2024#cfg,
    title={#CFG and #DNNF admit FPRAS},
    author={Kuldeep S. Meel and Alexis de Colnet},
    journal={arXiv preprint arXiv:2406.18224},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.18224},
    primaryClass={cs.DS}
}