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 <i>O</i>(<i>gn</i>^3-∈), where <i>g</i> is the size of the grammar and <i>n</i> is the length of the input string, can be efficiently converted into an algorithm to multiply <i>m</i> × <i>m</i> Boolean matrices in time <i>O</i>(<i>m</i>^3-∈/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.