Parsing Boolean grammars over a one-letter alphabet using online convolution

: Okhotin A, Reitwiessner C

Publisher: ELSEVIER SCIENCE BV

: 2012

: Theoretical Computer Science

: THEORETICAL COMPUTER SCIENCE

: THEOR COMPUT SCI

: 457

: 149

: 157

: 9

: 0304-3975

DOI: https://doi.org/10.1016/j.tcs.2012.06.032

Consider context-free grammars generating strings over a one-letter alphabet. For the membership problem for such grammars, stated as "Given a grammar G and a string a(n), determine whether a(n) is generated by G", only a naive O(vertical bar G vertical bar . n(2))-time algorithm is known. This paper develops a new algorithm solving this problem, which is based upon fast multiplication of integers, works in time vertical bar G vertical bar . n log(3) n . 2(O(log)* (n)) and is applicable to context-free grammars augmented with Boolean operations, known as Boolean grammars. For unambiguous grammars, the running time of the algorithm is reduced to vertical bar G vertical bar . n log(2) n . 2(O(log)* (n)). The algorithm is based upon (a simplification of) the online integer multiplication algorithm by Fischer and Stockmeyer [M.J. Fischer, L.J. Stockmeyer, Fast on-line integer multiplication, Journal of Computer and System Sciences 9 (3) (1974) 317-331]. (C) 2012 Elsevier B.V. All rights reserved.