Communication Aspects of Computation of Systems of Finite Automata

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Many computing systems can be modeled by systems of cooperating finite automata. In fact, any existing physical device is finite, even...

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Available from: Tomasz Jurdzinski, May 06, 2013
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    ABSTRACT: We consider systems consisting of a constant number of finite automata communicating via messages. We assume that the automata are asynchronous, but the answers given by the system must be always correct. We examine computational power of such systems by inspecting the number of messages exchanged. This is motivated by the fact that communication volume is one of the most important complexity measures. We show that any asynchronous system of finite automata that exchanges o(n) messages is able to recognize regular languages only. This is much different than in the case of synchronous systems considered before (where already a constant number of messages suffices to recognize some non-regular languages). We show that asynchronous and synchronous systems may differ significantly in their computational power also for tasks requiring ( n) messages. We consider a language Ltrans consisting of words of the form A#A T , where A T denotes transposition of matrix A and the matrices are written row by row. While it is easy to see thatLtrans can be recognized withO(n) messages by a synchronous system of finite automata, we show thatLtrans requires ( n 3=2 = log 2 n) messages on any asynchronous system.
    Full-text · Conference Paper · Jan 2001
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    ABSTRACT: 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.
    Full-text · Conference Paper · Jul 2001