#CFG and #DNNF admit FPRAS

TL;DR

This paper introduces the first fully polynomial-time randomized approximation schemes for counting words generated by context-free grammars and satisfying assignments of DNNF circuits, solving longstanding open problems.

Contribution

It provides the first efficient randomized approximation algorithms for these two fundamental counting problems, surpassing previous quasi-polynomial or restricted fragment solutions.

Findings

First FPRAS for counting words of a given length in CFGs.

First FPRAS for counting satisfying assignments in DNNF circuits.

Overcomes longstanding open problems in counting complexity.

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 $\Sigma$, 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).