My work on cellular automata

I began serious work on cellular automata in the middle of 1981. I had been thinking for some time about how complicated patterns could arise in natural systems—in apparent violation of the Second Law of Thermodynamics. I had been particularly interested in self-gravitating gases where the basic physics seemed clear, but where complex phenomena like galaxy formation seemed to occur. I had also been interested in neural networks, where there had been fairly simple models developed by Warren McCulloch and Walter Pitts in the 1940s. I came up with cellular automata as an attempt to capture the essential features of a range of systems, from self-gravitating gases to neural networks. I wanted to find models that had a simple structure like the Ising model in statistical mechanics (studied since the 1920s), but which had definite rules for time evolution and could easily be simulated on a computer. Ironically enough, while cellular automata are good for many things, they turn out to be rather unsuitable for modelling either self-gravitating gases or neural networks. (See page 1021). But by the time I realized this, it was clear that cellular automata were of great interest for many other purposes.

I did my first major computer experiments on cellular automata late in 1981 (see page 19). Two features initially struck me most. First, that starting from random initial conditions, cellular automata could organize themselves to produce complex patterns. And second, that in cases like rule 90 simple initial conditions led to nested or fractal patterns. During the first half of 1982, I worked hard to analyze the behavior of cellular automata using ideas from statistical mechanics, dynamical systems theory and discrete mathematics. And in June 1982, I finished my first paper on cellular automata, entitled "Statistical Mechanics of Cellular Automata". Published in the journal *Reviews of Modern Physics* in July 1983, this paper already presents in raw form many of the key ideas that led to the development of the science described in this book. It discusses the fact that by not using traditional mathematical equations, simple models can potentially be made to reproduce complex phenomena, and it mentions some of the consequences of viewing models like cellular automata as computational systems. The paper also contained a small picture of rule 30 started from a single black cell. But at the time, I did not study this picture in detail, and I tacitly assumed that whenever I saw randomness it must come from the random initial conditions that I used. (See page 112.)

It was some time in the fall of 1981 that I first found out (at a dinner with some then-young MIT computer scientists) that a version of the systems I had invented had been studied before under the name of "cellular automata". (I had been aware of the Game of Life, but its recreational emphasis had put me off studying it.) Knowing the name cellular automata, I was able to track down quite a number of relevant papers from the 1950s and 1960s. But I found that active research on what had been called cellular automata had more or less petered out (with the slight exception of a group at MIT at that time mainly concerned with building special-purpose hardware for 2D cellular automata). By late 1982 preprints of my paper on cellular automata had created quite a stir, and I got involved in organizing a conference held in March 1983 at Los Alamos to bring together many people newly interested in cellular automata with earlier workers in the field.

As part of preparing for that conference, I decided to use the graphics capabilities of the new workstation computer I had just obtained (a very early unit from Sun Microsystems) to investigate in a systematic way the behavior of a large collection of different cellular automata. And after spending several weeks looking at screen after screen of patterns—and trying to analyze their properties—I came to the conclusion that one could identify in the behavior of cellular automata with random initial conditions just four basic classes, each with its own characteristic features (see page 231).

In 1982 and early 1983, my efforts to analyze cellular automata were mainly based on ideas from discrete mathematics and dynamical systems theory. In the course of 1983, I also began to make serious use of formal language theory and the theory of computation. But for the most part I concentrated on characterizing behavior obtained from all possible initial conditions. And in fact I still vaguely assumed that if simple initial conditions were used, only fairly simple behavior would be obtained. Several of my papers had actually shown quite detailed pictures where this was not the case. I had noticed them, but they had never been among the examples I had studied in depth, partly for the superficial reason that the rules they involved were not symmetrical, or inevitably led to patterns that were otherwise not convenient for display. I do not know exactly what made me start looking more carefully at simple initial conditions, though I believe that I first systematically generated high-resolution pictures of all the k = 2, r = 1 cellular automata as an exercise for an early laserprinter—probably at the beginning of 1984. And I do know that for example on June 1, 1984 I printed out pictures of rule 30, rule 110 and k = 2, r = 2 totalistic code 10 (see note below), took them with me on a flight from New York to London, and a few days later was in Sweden talking about randomness in rule 30 and its potential significance.

A month or so later, writing an article for *Scientific American*—nominally on the subject of software in science and mathematics—led me to think more carefully about basic issues of computation and modelling, and to describe for the first time the idea of computational irreducibility (see page 737). In the fall of 1984 I began to investigate some of the implications of what I had discovered about cellular automata for foundational questions in science. And by early 1985 I had written what I consider to be my two most fundamental (if excessively short) papers from the period: one on undecidability and intractability in theoretical physics, and the other on intrinsic randomness generation and the origins of randomness in physical systems.

In the early summer of 1985 I was doing consulting at a startup company called Thinking Machines Corporation, which had developed a massively parallel computer called the Connection Machine that was fairly well suited to cellular automaton simulation. Partly as an application for this computer I then ended up making a detailed study of rule 30 and its randomness—among other things proposing it as a practical random sequence generator and cryptosystem.

I had always thought that cellular automata could be a way to get at foundational questions in thermodynamics and hydrodynamics. And in mid-1985, partly in an attempt to find uses for the Connection Machine, I devised a practical scheme for doing fluid mechanics with cellular automata (see page 378). Then over the course of that winter and the following spring I analyzed the scheme and worked out its correspondence to the traditional continuum approach.

By 1986, however, I felt that I had answered at least the first round of obvious questions about cellular automata, and it increasingly seemed that it would not be easier to go further with the computational tools available. In June 1986 I organized one last conference on cellular automata—then in August 1986 essentially left the field to begin the development of Mathematica.

Over the years, I have come back to look at cellular automata again and again, and every time I have been amazed and delighted by the richness of the phenomena they exhibit. As I argue in this book, a vast range of systems must in the end show the same basic phenomena. But cellular automata—and especially 1D ones—make the phenomena particularly clear, which is why even after investigating all sorts of other systems 1D cellular automata are still the most common examples that I use in this book.