A Trade-off in Firing Squad Synchronization Protocols for Communication-restricted Cellular Automata

Cellular automata (CA) are considered to be a good model of complex systems in which an infinite one-dimensional array of finite state machines (cells) updates itself in a synchronous manner according to a uniform local rule. In the long history of the study of CA, generally speaking, the number of internal states of each cell is finite and the local state transition rule is defined in a such way that the state of each cell depends on the previous states of itself and its neighboring cells. Thus, in the finite state description of the CA, the number of communication bits exchanged in one step between neighboring cells is assumed to be O(1) bits. However, such inter-cell bitinformation is hidden under the definition of the conventional automata-theoretic finite state description. In the present paper, we study a firing squad synchronization problem on a very restricted model of cellular automata, CAk−bit, for which inter-cell communication at one step is restricted to k-bit, where k is any positive integer such that k ≥ 1. We hereinafter refer to the model as k-bit CA. The number of internal states of CAk−bit is assumed to be finite in the usual sense. The next state of each cell is determined by the present state of the cell and two binary k-bit inputs from its leftand rightneighbor cells. Thus, the 1-bit CA, where k = 1, can be thought of as being one of the simplest CAs to have a low computational complexity. On the k-bit CA we consider the firing squad synchronization problem that has been studied extensively on the conventional CA model and propose several optimum-time firing squad synchronization protocols together with its implementation on a computer. Although many researchers have examined various aspects of the conventional cellular automata, studies focusing on the amount of bit-information exchanged in inter-cell communications are few. Mazoyer [4] first studied this model under the name of CAs with channels and proposed a time-optimum firing squad synchronization algorithm in which only one-bit information is exchanged. Umeo and Kamikawa [9, 11] showed that infinite non-regular sequences such as {2n|n = 1, 2, 3, ..}, {n2|n = 1, 2, 3, ..}, Fibonacci and the prime sequences can be generated in real-time by CA1−bit. Worsch [13] established a computational hierarchy between one-way 1-bit CAs. In this paper, we study a trade-off between internal states and communication bits in firing squad synchronization problems on communication-restricted cellular automata and propose several time-optimum bit-transfer-based synchronization protocols. Several state-efficient implementations are also given. For example, we establish the following theorems. [Theorem] There exists a 6-state CA2−bit that can synchronize any n cells in 2n−2 optimum-step. [Theorem] There exists a 4-state CA3−bit that can synchronize any n cells in 2n−2 optimum-step.