#CFG and #DNNF admit FPRAS

Abstract

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 $\hat{I}\pounds$, count the number of words of length exactly $n$ generated by $G$. 2. Given a circuit $\ddot{I}†$ in Decomposable Negation Normal Form (DNNF) over the set of Boolean variables $X$, compute the number of assignments to $X$ such that $\ddot{I}†$ 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).

Full version of the paper published at SODA 2026