Turing Universality of the Game of Life

Abstract

This chapter describes a Turing machine built from patterns in the Conway’s Game of Life cellular automaton. It outlines the architecture of the construction, the structure of its parts and explains how the machine works. It also illustrates the principle choices made during the design0 [17]. Background information about Turing machines, minimal universal Turing machines (those that simulate any other Turing machine) and non-erasing Turing machines can be found in [20,7,15,14,18]. The importance of Turing machines is the existence of universal Turing machines. Thus a machine that can simulate any Turing machine can simulate a universal Turing machine. It has been proved that Turing machines can be simulated by many types of machine: cellular automata (as one can see in this and other chapters of this book), random access machines [4], register machines [1] and others. In particular Minsky [16] describes a register machine which can simulate a Turing machine. The registers have the unusual property of being able to store positive numbers of any size. Remarkably, a long time ago Conway described [1] a method of constructing a register of this form in the Game of Life. This is discussed later in this chapter.