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State-efficient One-bit-communication Solutions for Some Classical Cellular Automata Problems

Umeo H., Michisaka K., Kamikawa N., Kanazawa M.

Fundamenta Informaticae

2007

Vol. 78, nr 3

449-465

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 (CA1-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 CA1-bit that solves the early bird problem in twice real-time. Then, we give a two-dimensional CA1-bit which can synchronize any n ×n (n ^(3) 2) square and m×n (m, n ^(3) 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 the general located at an arbitrary position (r, s) in the array, where m, n ^(3) 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 CA1-bit that can generate in real-time a context-sensitive integer sequence such that {2n | n = 1, 2, 3, ... }.