Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication

TL;DR

This paper establishes a theoretical equivalence between the complexity of context-free grammar parsing and Boolean matrix multiplication, explaining the difficulty of developing practical sub-cubic CFG parsers.

Contribution

It proves that any significantly faster CFG parser would imply a faster Boolean matrix multiplication algorithm, highlighting the computational barrier.

Findings

CFG parsing complexity is tightly linked to Boolean matrix multiplication speed.

Practical sub-cubic CFG parsers are unlikely without breakthroughs in matrix multiplication.

Formalization of parsing notions aids in understanding computational limits.

Abstract

In 1975, Valiant showed that Boolean matrix multiplication can be used for parsing context-free grammars (CFGs), yielding the asympotically fastest (although not practical) CFG parsing algorithm known. We prove a dual result: any CFG parser with time complexity $O(g n^{3 - \epsilson})$, where $g$ is the size of the grammar and $n$ is the length of the input string, can be efficiently converted into an algorithm to multiply $m \times m$ Boolean matrices in time $O(m^{3 - \epsilon/3})$. Given that practical, substantially sub-cubic Boolean matrix multiplication algorithms have been quite difficult to find, we thus explain why there has been little progress in developing practical, substantially sub-cubic general CFG parsers. In proving this result, we also develop a formalization of the notion of parsing.