Goto's construction and Pascal's triangle: new insights into cellular automata synchronization
نویسنده
چکیده
Here we present a new non-recursive minimal-time solution to the Firing Squad Synchronization Problem which does not use any recursive process. In 1962, E. Goto designed an iterative algorithm which uses Minsky-McCarthy’s solutions to synchronize in minimal-time. Our solution does not use any standard recursion process, only some “fractal computation”, making it a purely iterative synchronization algorithm. Introduction The firing squad synchronization problem (FSSP for short) has been the subject of many studies since 1957 when Myhill stated it and Moore reported it (see [Mo64]). We can state the problem as follows: Does there exist a finite automaton such that a chain of n (whatever n could be) such automata would be synchronized at some time T (n) after being initiated at time t = 0? Each automaton is connected with its two neighbors and is assumed to be structurally independent of the number n. The synchronization is a configuration such that each automaton is in a socalled firing state which was never used before time T (n) and the ignition configuration is a configuration such that every automaton but the first one of the chain is in a quiescent state. Besides the fact that numerous papers were published about it and many different solutions were designed to solve the problem in various conditions, one of the very first solution made by Goto remained mythical for a long time. His courses notes are not available and Goto has not published his solution elsewhere. Many years later, Umeo (see [Um96]) was the first who tried to reconstruct it as he was able to talk to Goto himself who then gave him some old incomplete drawing. After that, Mazoyer (see [Ma98]) made a possible reconstruction of it but did not published it.
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تاریخ انتشار 2008