Efficient parallel and incremental parsing of practical context-free languages

We present a divide-and-conquer algorithm for parsing context-free languages efficiently. Our algorithm is an instance of Valiant’s (1975), who reduced the problem of parsing to matrix multiplications. We show that, while the conquer step of Valiant’s is O(n), it improves to O(log n) under certain conditions satisfied by many useful inputs, and if one uses a sparse representation of matrices. The required conditions occur for example in program texts written by humans. The improvement happens because the multiplications involve an overwhelming majority of empty matrices. This result is relevant to modern computing: divide-and-conquer algorithms with a polylogarithmic conquer step can be parallelised relatively easily.