Fast context-free parsing requires fast Boolean matrix multiplication

Authors:
Lillian Lee
Affiliations:
Harvard University, Cambridge, MA
Venue:
ACL '98 Proceedings of the 35th Annual Meeting of the Association for Computational Linguistics and Eighth Conference of the European Chapter of the Association for Computational Linguistics
Year:
1997

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Abstract

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 tmie 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.