Communication Gap for Finite Memory Devices
So far, not much is known on communication issues for computations on distributed systems, where the components are weak and simultaneously the communication bandwidth is severely limited. We consider synchronous systems consisting of finite automata which communicate by sending messages while working on a shared read-only data. We consider the number of messages necessary to recognize a language as a its complexity measure.
We present an interesting phenomenon that the systems described require either a constant number of messages or at least Ω((log log log n)c) of them (in the worst case) for input data of length n and some constant c. Thus, in the hierarchy of message complexity classes there is a gap between the languages requiring only O(1) messages and those that need a non-constant number of messages. We show a similar result for systems of one-way automata. In this case, there is no language that requires ω(1) and o(log n) messages (in the worst case). These results hold for any fixed number of automata in the system.
The lower bound arguments presented in this paper depend very much on results concerning solving systems of diophantine equations and in- equalities.
Unable to display preview. Download preview PDF.
- 5.M. Holzer, Multi-Head Finite Automata: Data-Independent Versus Data-Dependent Computations, Proc. MFCS’97, LNCS 1295, Springer Verlag, Berlin, 1997, 299–309.Google Scholar
- 6.J. Hopcroft, J.D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley, 1979.Google Scholar
- 7.T. Jurdziński, Communication Aspects of Computation of Systems of Finite Automata, Wrocław University, 2000. (http://www.ii.uni.wroc.pl/~acm/doktoraty.html)
- 11.Ju. Matijasevič, Hilbert’s tenth problem, Foundations in Computing Series, MIT Press, Cambridge, 1993.Google Scholar