Fast ContextFree Grammar Parsing Requires Fast Boolean Matrix Multiplication
Abstract
In 1975, Valiant showed that Boolean matrix multiplication can be used for parsing contextfree 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 subcubic Boolean matrix multiplication algorithms have been quite difficult to find, we thus explain why there has been little progress in developing practical, substantially subcubic general CFG parsers. In proving this result, we also develop a formalization of the notion of parsing.
 Publication:

arXiv eprints
 Pub Date:
 December 2001
 arXiv:
 arXiv:cs/0112018
 Bibcode:
 2001cs.......12018L
 Keywords:

 Computer Science  Computation and Language;
 Computer Science  Data Structures and Algorithms;
 I.2.7;
 F.2.2
 EPrint:
 To appear in Journal of the ACM