Firing squad synchronization problem in number-conserving cellular automata

• Authors: Katsunobu Imai;Kenichi Morita;Kenji Sako
• Venue: Fundamenta Informaticae - Special issue on cellular automata
• Year: 2002

We study the Firing Squad Synchronization Problem (FSSP) on a cellular automaton (CA) having number-conservation property. In a number-conserving CA, all states of cells are represented by (tuples of) non-negative integers and the total number of its configuration is conserved throughout its computing processes. But, if we use a usual framework of CA in which each state of a cell is represented by a single integer, it is not possible to make every cell to be in the same firing state, which should be different from the soldier state, under the usual FSSP condition without violating the number-conservativeness. So, we employ the framework of a partitioned cellular automaton, and define a number-conserving partitioned cellular automaton (NC-PCA). Its cell is divided into three parts, and hence each cell is represented by a triple of non-negative integers. In NC-PCA, only the constraint that the local transition function should satisfy a number-conserving condition is supposed. Thus, it makes relatively easy to construct an NC-PCA. Because each cell can hold three non-negative integers, it is possible to represent different states even if the sum of three numbers are equal. Using this technique, we show that Minsky's 3n time solution can be embedded into an NC-PCA, having an integer at most 9 in each part of a cell.