The Cook Book

Recipe for the week of January 1 - 7

Stalking Class IV: Langton's Lambda Applet

Happy New Year to all! This week's cooking uses a slick new Web-based experimentation platform for one-dimensional cellular automata that has been developed by Chris Langton, Howard Gutowitz, and the folks at the Santa Fe Institute. Their simulator can be accessed from the Artificial Life link in the Kitchen Sink, or directly from Exploring the Space of Cellular Automata. One can choose among 4 forms (elementary, lambda, stochastic lambda, and complex) that generate space-time images based on CA rules of increasing generality. Our soup was concocted using the lambda form, with 8 colors, 640 cells, 480 time steps, a random initial seed, and a range-two CA rule with a lambda value of approximately 0.3. One can also obtain a text file with the exact rule specifications, although they are not terribly illuminating in this case.

In a nutshell, the lambda parameter provides a rule of thumb for quantifying the complexity of CA rules on the integers. A celebrated empirical scheme of Stephen Wolfram partitions such systems into four Classes: I, II, III, and IV, according to their limiting (ergodic) behavior from disordered initial states. Class IV rules are the rarest -- roughly, those quasi-periodic dynamics that are delicately perched between order and disorder. As Langton et al. explain, the lambda parameter can be tweaked to discover ostensible Class IV rules with increased efficiency.

From a mathematical point of view, I think it's fair to say that the Wofram classification has never been precisely delineated, so there is really no way of deciding definitively whether a given rule is Class IV. Moreover, the classification was proposed largely on the basis of one-dimensional experimentation, and the extent to which it extends to higher dimensions is not at all clear. Indeed, some of the exotic self-organizational phenomena documented here in the Kitchen seem to defy the four categories. That said, the Langton-Gutowitz Suggested Reading link is well worth pursuing. And our soup just might be Class IV.

I also like the Exploring page as a little preview of the much-touted paradigm shift coming round the bend. Applets like this will be commonplace real soon now, when Java reigns supreme, our PCs and workstations have all been scrapped in favor of $50 Web boxes, we all enjoy infinite bandwidth, and the Microsoft juggernaut finally crashes and burns...

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