Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication
<|reference_start|>Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication: 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.<|reference_end|>
@article{lee2001fast,
    title={Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix
  Multiplication},
    author={Lillian Lee},
    journal={Journal of the ACM 49(1), pp. 1--15, January 2002},
    year={2001},
    archivePrefix={arXiv},
    eprint={cs/0112018},
    primaryClass={cs.CL cs.DS}
}