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We investigate here the descriptional and the computational complexity of parallel communicating grammar systems (PCGS). A new descriptional complexity measure — the communication structure of the PCGS — is introduced and related to the communication complexity (the number of communications). Several hierarchies resulting from these complexity measures and some relations between the measures are established. The results are obtained due to the development of two lower-bound proof techniques for PCGS. The first one is a generalization of pumping lemmas from formal language theory and the second one reduces the lower bound problem for some PCGS to the proof of lower bounds on the number of reversals of certain sequential computing models.

Some hierarchies for the communication complexity measures of cooperating grammar systems