Masaya Hisaoka

A twelve-state optimum-time synchronization algorithm for two-dimensional rectangular cellular arrays

The firing squad synchronization problem has been studied extensively for more than 40 years [1-18]. The present authors are involved in research on firing squad synchronization algorithms on two-dimensional (2-D) rectangular cellular arrays. Several synchronization algorithms on 2-D arrays have been proposed, including Beyer [2], Grasselli [3], Kobayashi [4], Shinahr [10], Szwerinski [12] and Umeo et al. [13, 15]. To date, the smallest number of cell states for which an optimum-time synchronization algorithm has been developed is 14 for rectangular array, achieved by Umeo et al. [15]. In the present paper, we propose a new optimum-time synchronization algorithm that can synchronize any 2-D m × n rectangular arrays in m + n + max(m, n) –3 steps. We progressively reduce the number of internal states of each cellular automaton on rectangular arrays, achieving twelve states. This is the smallest number of states reported to date for synchronizing rectangular arrays in optimum-step.

Several new generalized linear- and optimum-time synchronization algorithms for two-dimensional rectangular arrays

We propose several new generalized synchronization algorithms for 2-D cellular arrays. Firstly, a generalized linear-time synchronization algorithm and its 14-state implementation are given. It is shown that there exists a 14-state 2-D CA that can synchronize any m × n rectangular array in m + n + max(r + s , m + n – r – s + 2) – 4 steps with the general at an arbitrary initial position (r, s),where 1 ≤ r ≤ m, 1 ≤ s ≤ n. The generalized linear-time synchronization algorithm is interesting in that it includes an optimum-step synchronization algorithm as a special case where the general is located at one corner. In addition, we propose a noveloptimum-time generalized synchronization scheme that can synchronize any m × n array in m+n+max (m, n)− min (r, m−r+1)− min (s, n−s+1)−1 optimum steps.

Some New Generalized Synchronization Algorithms and Their Implementations for Large Scale Cellular Automata

In this paper, we study a generalized synchronization problem for large scale cellular automata (CA) on one- and two- dimensional arrays. Some new generalized synchronization algorithms will be designed both on O(1)-bit and 1-bit inter-cell communication models of cellular automata. We give a 9-state and 13-state CA that can solve the generalized synchronization problem in optimum- and linear-time on O(1)-bit 1-D and 2-D CA, respectively. The number of internal states of the CA implemented is the smallest one known at present. In addition, it is shown that there exists a 1-bit inter-cell communication CA that can synchronize 1-D n cells with the general on the kth cell in n+max(k, n - k + 1) steps, which is two steps larger than the optimum time. We show that there still exist several new generalized synchronization algorithms, although more than 40 years have passed since the development of the problem.

A Comparative Study of Optimum-Time Synchronization Algorithms for One-Dimensional Cellular Automata – A Survey –

We present a survey and a comparison of the quantitative and qualitative aspects of the optimum-time synchronization algorithms developed thus far for one-dimensional cellular arrays. Several new results and viewpoints are also given.

A comparative investigation into optimum-time synchronization protocols for a large scale of one dimensional cellular automata

The firing squad synchronization problem has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed. In the present paper, we examine the state transition rule sets for the famous firing squad synchronization algorithms that give a finite-state protocol for synchronizing large-scale cellular automata. We show that the first transition rule set designed by Waksman [18] includes fundamental errors which cause unsuccessful firings and that ninety-three percent of the rules are redundant. In addition, the transition rule sets reported by Balzer [1], Gerken [2] and Mazoyer [6] are found to include several redundant rules. We also present herein a survey and a comparison of the quantitative aspects of the optimumtime synchronization algorithms developed thus far for one-dimensional cellular arrays.