Efficient 1-Bit-Communication Cellular Algorithms

Summary

We propose several efficient algorithms for a large scale of cellular automata having 1-bit inter-cell communications (CA_1-bit). A 1-bit inter-cell communication model studied in this paper is a new class of cellular automata (CA) whose inter-cell communication is restricted to 1-bit. We call the model 1-bit CA in short. The number of internal states of the 1-bit CA is assumed to be finite in a usual way. The next state of each cell is determined by the present state of itself and two binary 1-bit inputs from its left and right neighbor cells. Thus the 1-bit CA can be thought to be one of the most powerless and simplest models in a variety of CAs.

We study a sequence generation problem, a firing squad synchronization problem and an early bird problem, all of which are known as the classical and fundamental problems in cellular automata.

First we consider the sequence generation problem. It is shown that there exists a 1-state CA_1-bit that can generate in real-time a context-sensitive sequence such that {2ⁿ|n = 1, 2, 3, …}. Prime sequence can also be generated in real-time by CA_1-bit with 34 states. Secondary, we study the firing squad synchronization problem on two-dimensional CA_1-bit. We give a two-dimensional CA_1-bit which can synchronize any n × n square and m × n rectangular arrays in 2n − 1 and m + n + max(m, n) steps, respectively. In addition, we propose a generalized synchronization algorithm that operates in linear steps on two-dimensional rectangular arrays with the general located at an arbitrary position of the array. The time complexities for the first two algorithms developed are one to two steps larger than optimum ones proposed for O(1)-bit communication model. In the last, we give a 1-bit implementation for an early bird problem. It is shown that there exists a 12-state CA_1-bit that solves the early bird problem in linear time.