A 4-states algebraic solution to linear cellular automata synchronization
Information Processing Letters
About 4-States Solutions to the Firing Squad Synchronization Problem
ACRI '08 Proceedings of the 8th international conference on Cellular Automata for Reseach and Industry
| Hi-index | 0.00 |
The firing synchronization problem concerns a one-dimensional array of $n$ finite automata. All automata are identical, except the ones on either end of the array, one of which is the initiator for the synchronization. The goal is to define the set of states and transition rules for the automata so that all machines enter a special fire state for the first time and simultaneously at some time $t(n)$. The generalized firing synchronization problem differs from the original problem in that the initiator of the synchronization can be located anywhere in the array. In this paper we present two non-minimal time solutions to the firing synchronization problem. The first is a 6-state solution to the original problem which allows the initiator to be located at either the right or left endpoint. This automaton has two fewer states than R.~Balzer''s 8-state minimal-time automaton \cite{b}, the minimal-time automaton with the smallest number of states that allows the initiator to be located at either endpoint. We also give a 7-state non-minimal time solution to the generalized problem. No previous non-minimal time solutions to the generalized firing synchronization problem have been published, and the smallest minimal-time automaton for the generalized problem requires 9 states \cite{s9}. Both automata are based on the 6-state minimal-time solution to a restricted version of the original problem given by J.~Mazoyer \cite{m6}. Finally, we provide a proof of correctness for each of the automata.