An extremely interesting paper has now made its public appearance:
The "cellular automata that compute" used as a baseline here are the ones grown as part of the Evolving Cellular Automata project. I happen to have worked on that project for a few years (making only the smallest of contributions), so I'm now going to talk your ear off about how cool my collaborators' work was, and how cool this new work is. I've explained cellular automata elsewhere, so I won't repeat myself here.
The motivation for the EvCA work was that there are some computations which are very easy if you have a centralized processor or memory, but very hard for decentralized systems. Suppose, for instance, I give you a string of random bits, and ask whether most of the bits are 0 or most of them are 1 ("density classification"). This is easy, if you can count and store your results in a central place; it's very hard if you can only do local operations, looking at, say, a bit and its immediate neighbors in the string. EvCA used genetic algorithms to evolve local rules --- cellular automata --- for these problems, measuring their fitness by (in essence) how likely they were to solve random instances of the task. The tasks themselves were completely trivial and uninteresting (a fact which eluded some people); what was interesting was seeing the tricks which evolution found to solve the problems in a decentralized fashion. (I won't draw any B.S. morals about politics or society from all this, but I will say I've long been tempted to send a pile of reprints to the Discovery Institute and the Templeton Foundation. Unfortunately that'd just make the honest ones' heads explode, and leave us to deal with the conscious charlatans --- a clear case of adverse selection.)
One of the more robust positive results was that, for many, many different tasks, the cellular automata evolve a system of computation based on "domains" and "particles". Domains are extended regions in space and time where the CA is repeating the same pattern over and over, possibly with a little unimportant noise incorporated into the pattern. (More exactly, domains are patterns which are left unchanged by the CA rule. For a yet more exact definition, see here or here.) Domains contain very little information, and do no computation. Particles are propagating boundaries between domains, which repeat a given cycle of states over and over and move through space at some constant velocity. They're like solitons (if you're a physicist) or gliders (if you're an afficinado of Conway's Game of Life). When particles meet, they interact, producing new particles or annihilating (looking rather like a Feynman diagram). Particles have a very high density of information, and particle interactions are what end up doing the computation. This is emergent computation: you can describe what's going on in terms of the domains and particles, and doing so is vastly simpler and nearly as accurate as evaluating the low-level CA dynamics, but the rules of the CA don't explicitly encode domains or particles at all; they're generated entirely by local interactions.
The idea of using particles to do computation goes back at least to the Game of Life in the 1970s, and it's a pretty standard trick in studying CAs. (Here is a non-EvCA paper which does it, for instance.) Probably the most mathematically elaborate use of particle computation is Matthew Cook's proof that one of the most basic cellular automata rules (called "rule 110") is as powerful as a universal Turing machine. Universal Turing machines are equivalent to a kind of string-manipulation system called a Post tag system (named after the logician Emil Post, who invented them in the 1920s). Cook devised a variation on Post tag systems he called cyclic tag systems, and showed that they, too, are computationally universal. Finally, he showed how, if you engineer the proper arrangement of particles and domains just so in rule 110, the particles implement a cyclic tag system; and there you are. (You can read a sketch of Cook's proof in Stephen Wolfram's book A New Kind of Science, and doing so is the only good reason to read that monstrosity.) What was new in EvCA, and in my humble opinion neat, was (a) showing that particle computation is something which reliably gets evolved to solve this kind of problem, and (b) showing that you could define particles endogenously, as it were, in terms of domains. It's worth noting, too, that the particles in the EvCA rules don't require this careful engineering; they form spontaneously, from completely random initial conditions. You'd have to engineer things to keep them from forming.
(An earlier negative finding of the evolving CA project was that the idea systems must "evolve to the edge of chaos" to perform powerful, adaptive computations is bollocks --- see here or here. But once a notion so conformable to the zeitgeist becomes endemic among the lumpenprofessoriat, it's hard for any amount of actual science to uproot it.)
Now what does this have to with plants? Well, as the abstract of Peak et al. explains, plants face an essentially computational problem as well, in deciding how to balance carbon dioxide uptake against water loss. This is really something they need to do globally, but the sensors and the decision-making are all local; so it's a distributed problem. What Peak et al. have shown is that the statistical properties of the plant stomata look very like those of CA evolved to solve global coordination problems, and that there seem to be persistent, localized, moving patches of activity --- particles --- involved in establishing the coordination. (In a two-dimensional system, like a leaf, one might also get extended lines or boundaries as emergent structures, as well as point-like particles, but they don't report such things.) This is very suggestive of some kind of distributed computation. One would like very much to be able to identify the computation more directly; I think I know a way to do that...
For another take, see this news piece in Nature (may require subscription), in which the usually-reliable Philip Ball (or his copy editor?) manages (1) to confuse the emergent particles with the basic cells of a CA, and (2) to say that Wolfram was the first to show cellular automata can "mimic computers", something established by John von Neumann and Stanislaw Ulam before Wolfram was born. (Via Melanie Mitchell, in e-mail.)
Update while waiting for code to run, 3 February 2004. I think this is my most popular post ever. These are the links I know of which haven't been picked up by trackback (below): Brad DeLong; Metafilter; GR3M; The Abstract Factory; Tobi Schwarz; Language Log. --- Ray Girvan writes to point out that Philip Ball seems to have an unaccountable fondness for Wolfram's tome, here attributing to it ideas originating long before with Ed Fredkin and collaborators. I'm surprised, because most of Ball's pieces for Nature are actually well-informed, and his book on pattern formation, The Self-Made Tapestry, is truly excellent. So I'm puzzled about what's gone wrong here.
Posted at January 21, 2004 22:10 | permanent link