Cutting the Firing Squad Synchronization

The firing squad synchronization problem on Cellular Automata (CA) has been studied extensively for many years, and a rich variety of synchronization algorithms have been proposed. From Mazoyer’s paper it is known that a minimal-time solution with 6 states exists. The firing squad synchronization problem has also been studied for defective CA where a defective cell can still transmit information without processing it. In the present paper, we consider defective CA where the dynamic defects are such that a defective cell totally fails. The failures are permanent and may occur at any time in the computation. In this way the array is cut into two parts. The question addressed is how many cells in each part can still be synchronized and at which time steps. It is analyzed how many cells are synchronized, where and when this happens and how these three characteristics are connected with the position of the defective cell and the time at which the cell fails. Based on Mazoyer’s 6-state algorithm, a solution for one-dimensional CA is proposed that synchronizes the maximal possible number of cells in each part.