## Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication (2002)

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Citations: | 23 - 0 self |

### BibTeX

@MISC{Lee02fastcontext-free,

author = {Lillian Lee},

title = {Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication},

year = {2002}

}

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### 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.