Notes

Chapter 11: The Notion of Computation

Section 8: The Rule 110 Cellular Automaton


History [of universality in 1D cellular automata]

The fact that 1D cellular automata can be universal was discussed by Alvy Ray Smith in 1970—who set up an 18-color nearest-neighbor cellular automaton rule capable of emulating Marvin Minsky's 7-state 4-color universal Turing machine (see page 706). (Roger Banks also mentioned in 1970 a 17-color cellular automaton that he believed was universal.) But without any particular reason to think it would be interesting, almost nothing was done on finding simpler universal 1D cellular automata. In 1984 I suggested that cellular automata showing what I called class 4 behavior should be universal—and I identified some simple rules (such as k=2, r=2 totalistic code 20) as explicit candidates. A piece published in Scientific American in 1985 describing my interest in finding simple 1D universal cellular automata led me to receive a large number of proofs of the fact (already well known to me) that 1D cellular automata can in principle emulate Turing machines. In 1989 Kristian Lindgren and Mats Nordahl constructed a 7-color nearest-neighbor cellular automaton that could emulate Minsky's 7,4 universal Turing machine, and showed that in general a rule with s + k + 2 colors could emulate an s-state k-color Turing machine (compare page 658). Following my ideas about class 4 cellular automata I had come by 1985 to suspect that rule 110 must be universal. And when I started working on the writing of this book in 1991, I decided to try to establish this for certain. The general outline of what had to be done was fairly clear—but there were an immense number of details to be handled, and I asked a young assistant of mine named Matthew Cook to investigate them. His initial results were encouraging, but after a few months he became increasingly convinced that rule 110 would never in fact be proved universal. I insisted, however, that he keep on trying, and over the next several years he developed a systematic computer-aided design system for working with structures in rule 110. Using this he was then in 1994 successfully able to find the main elements of the proof. Many details were filled in over the next year, some mistakes were corrected in 1998, and the specific version in the note below was constructed in 2001. Like most proofs of universality, the final proof he found is conceptually quite straightforward, but is filled with many excruciatingly elaborate details. And among these details it is certainly possible that a few errors still remain. But if so, I believe that they can be overcome by the same general methods that have been used in the proof so far. Quite probably a somewhat simpler proof can be given, but as discussed on page 722 it is essentially inevitable that proofs of universality must be at least somewhat complicated. In the future it should be possible to give a proof in a form that can be checked completely by computer. (The initial conditions in the note below quite soon become too large to run explicitly on any existing computer.) And in addition, with sufficient effort, I believe one should be able to construct an automated system that will allow many universality proofs of this general kind to be found almost entirely by computer (compare page 810).

From Stephen Wolfram: A New Kind of Science [citation]