Multi-party Finite Computations

We consider systems consisting of a finite number of finite automata which communicate by sending messages. We consider number of messages necessary to recognize a language as a complexity measure of the language. We feel that these considerations give a new insight into computational complexity of problems computed by read-only devices in multiprocessor systems. Our considerations are related to multiparty communication complexity, but we make a realistic assumption that each party has a limited memory. We show a number of hierarchy results for this complexity measure: for each constant k there is a language, which may be recognized with k+1 messages and cannot be recognized with k-1 messages. We give an example of a language that requires Θ(log log n) messages and claim that Ω(log log(n)) messages are necessary, if a language requires more than a constant number of messages. We present a language that requires Θ(n) messages. For a large family of functions f, f(n) = ω(log log n), f(n) = o(n), we prove that there is a language which requires Θ(f(n)) messages. Finally, we present functions that require ω(n) messages.