Fast Context-Free Parsing Requires Fast Boolean Matrix Multiplication 


Valiant showed that Boolean matrix multiplication (BMM) can be used for CFG parsing. We prove a dual result: CFG parsers running in time O(|G||w|3-e) on a grammar G and a string w can be used to multiply m x m Boolean matrices in time O(m3-e/3). In the process we also provide a formal definition of parsing motivated by an informal notion due to Lang. Our result establishes one of the first limitations on general CFG parsing: a fast, practical CFG parser would yield a fast, practical BMM algorithm, which is not believed to exist. 