Computing in a Simple Reversible and Conservative Cellular Automaton

Abstract

We show how to construct reversible computers in a simple reversible and conservative two-dimensional cellular space. We introduce the framework of an elementary square partitioned cellular automaton (ESPCA), and use a particular ESPCA with the hexadecimal ID number “01caef”, which is denoted by ESPCA $P_0$ for short in this paper. A cell of ESPCA $P_0$ consists of four parts each of which has two states 0 and 1. The class of ESPCAs is the simplest subclass of square partitioned cellular automata, and each of their local functions is described by only six local transition rules. ESPCA $P_0$ is not only reversible, but also conservative in the sense that the total number of state 1’s is conserved throughout its evolution process. We show that a space-moving pattern called a glider exists in $P_0$. Colliding a glider with another pattern called a blinker, several interesting phenomena appear. We give a method of constructing reversible Turing machines (RTMs) using such phenomena. We first implement a reversible logic element with memory (RLEM), rather than a reversible logic gate, using only three of these phenomena. Then, we compose reversible Turing machines out of RLEMs. In this systematic and modularized way, we can construct any RTM out of only two small patterns in a simple reversible and conservative cellular space.

Currently Professor Emeritus of Hiroshima University.