Parsing Unary Boolean Grammars Using Online Convolution

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 nave O(jGj 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 jGj n log 3 n 2 O(log n) , and is applicable to the more general conjunctive and Boolean grammars. The algorithm is based upon (a simplication of) the online integer multiplication algorithm by Fischer and Stockmeyer (\Fast on-line integer multiplication", Journal of Computer and System Sciences, 1974).

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