Hilbert's tenth problem
Communication-Space Tradeoffs for UnrestrictedProtocols
SIAM Journal on Computing
An efficient incremental algorithm for solving systems of linear Diophantine equations
Information and Computation
On the degree of communication in parallel communicating finite automata systems
Journal of Automata, Languages and Combinatorics
Introduction To Automata Theory, Languages, And Computation
Introduction To Automata Theory, Languages, And Computation
Multi-Head Finite Automata: Data-Independent Versus Data-Dependent Computations
MFCS '97 Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science
Multi-party finite computations
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
Two-party Watson-Crick computations
CIAA'10 Proceedings of the 15th international conference on Implementation and application of automata
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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 inequalities.