State-Efficient One-Bit Communication Solutions for Some Classical Cellular Automata Problems

• Authors: Hiroshi Umeo;Koshi Michisaka;Naoki Kamikawa;Masaru Kanazawa
• Venue: Fundamenta Informaticae - Special issue on DLT'04
• Year: 2007

We present several state-efficient implementations on 1-bit inter-cell communication cellular automata for some classical cellular automata problems. The 1-bit inter-cell communication cellular automaton model (CA$_{1-bit}$) studied in this paper is a subclass of cellular automata (CA) whose inter-cell communication at one step is restricted to 1-bit. We study an early bird problem, a firing squad synchronization problem and an integer sequence generation problem, all of which are known as the classical, fundamental problems in cellular automata. Firstly, it is shown that there exists a 37-state CA$_{1-bit}$ that solves the early bird problem in twice real-time. Then, we give a two-dimensional CA$_{1-bit}$ which can synchronize any n × n (n⩾ 2) square and m × n (m, n ⩾ 2) rectangular arrays in 2n − 1 and m + n + max(m, n) steps, respectively. In addition, we propose a generalized linear-time synchronization algorithm that operates in m+n+max(r+s,m+n−r−s+2)+O(1) steps on two-dimensional rectangular arrays of size m×n with a general located at an arbitrary position (r, s) in the array, where m, n ⩾ 2, 1⩽ r ⩽ m and 1 ⩽ s⩽ n. The time complexities for the first two algorithms developed are one to two steps larger than optimum ones proposed for O(1)-bit conventional communication model. In the last, it is shown that there exists a 1-state CA$_{1-bit}$ that can generate in real-time a context-sensitive integer sequence such that {2$^n$∣ n = 1, 2, 3, ...}.