Efficient recognition algorithms for parallel multiple context-free languages and for multiple context-free languages

Parallel multiple context-free grammar (PMCFG) and multiple context-free gr ammar (MCFG) were introduced to denote the syntax of natural languages. By the known fastest algorithm, the recognition problem for multiple context-f ree language (MCFL) and parallel multiple context-free language (PMCFL) can be solved in O(n(e)) time and O(n(e+1)) time, respectively, where e is a c onstant which depends only on a given MCFG or PMCFG. In this paper. we prop ose the following two algorithms. (1) An algorithm which reduces the recogn ition problem for MCFL to the boolean matrices multiplication problem. (2) An algorithm which reduces the recognition problem for PMCFL to the recogni tion problem for MCFL. The time complexity of these algorithms is O(n(e'-3i '+1).M(n(i'))) where e' and i' are constants which depend only on a given M CFG or PMCFG, and M(k) is the time needed for multiplying two k x k boolean matrices. The proposed algorithms are faster than the known fastest algori thms unless e' = e, i' = 1 for MCFG, and e' = e, i' = 0 for PMCFG.