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 $Σ$, 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).

翻译：我们为以下两个计数问题提供了首个完全多项式时间随机近似方案：1. 给定字母表 $\Sigma$ 上的上下文无关文法 $G$，计算由 $G$ 生成的恰好长度为 $n$ 的单词数量。2. 给定布尔变量集合 $X$ 上的可分解否定范式电路 $\varphi$，计算使得 $\varphi$ 求值为 1 的赋值数量。为上述问题寻找多项式时间算法一直是一个长期存在的开放问题。先前的工作要么只能获得拟多项式运行时间（SODA 1995），要么只能针对受限片段（例如非确定性有限自动机（JACM 2021）或非确定性树自动机（STOC 2021））获得多项式时间随机近似方案。