A parallel algorithm for counting parse trees

Abstract

A parallel algorithm for computing the number of parse trees of a given string according to a fixed context-free grammar is defined. More generally, the algorithm applies to computing the weight of a string in a weighted grammar over any semiring. The algorithm is first implemented on an arithmetic circuit of depth at most $6{({\log }_{2}n)}^2+O(\log n)$ and with $O\left(n^6\right)$ elements, where the constant factors in the big-O notation depend on the grammar. Then, the circuit is improved using fast matrix multiplication to use only $O\left(n^{5.38}\right)$ elements, while preserving depth $O\left({(\log n)}^2\right)$.