S CIENCE’ S C OMPASS BOOKS ET AL. 65 nomena as well as fundamental philosophi- 64 BOOKS: DOING SCIENCE cal questions. Each of these ideas warrants 63 careful consideration. 62 61 Is the Universe a Universal Computer? Complexity from Simple Programs 60 Melanie Mitchell Throughout the book, Wolfram presents 59 many beautiful, striking pictures of the be- 58 istory has seen the development of troller and limited communication among havior of various cellular automata. These 57 many new sciences but very few components. Originally defined and stud- demonstrate that even elementary cellular 56 Hnew kinds of science. New kinds of ied by the mathematicians Stanislaw Ulam automata can produce patterns ranging 55 science involve radical changes in think- and John von Neumann, cellular automata from simple to quite complicated, highly 54 ing, such as the shift from Aristotelian tra- have been used as models for such natural ordered to seemingly random. The great 53 ditions to experi- phenomena as earthquakes, turbulent flow, diversity of these patterns and the fact that mental methods biological pigmentation, and tumor growth. such simple rules can produce such appar- A New Kind and the description They have also been applied in computer ently complex behavior is viewed by Wol- of Science of natural phenom- science as idealizations of massively paral- fram as deeply significant. by Stephen Wolfram ena in mathemati- lel, non-centralized computation. That simple rules can produce complex Wolfram Media, Cham- cal terms—revolu- Wolfram grounds his approach on six behavior is a very important idea, and it un- paign, IL, 2002. 1280 pp. tions associated principal claims: Simple programs (i.e., ele- derlies the science of complex systems. $44.95, £40, C$69.95. with names like mentary cellular automata) can produce However, Wolfram implies that the notion ISBN 1-57955-008-8. Galileo and New- highly complex and random-looking behav- was his discovery and the field of complex ton. Thus it is with systems his invention—an 52 no small risk of hubris that Stephen Wol- absurd claim. The idea of 51 fram titles his account of his approach to simple rules leading to 50 explaining the natural world A New Kind complex behavior under- 49 of Science. At over 1200 pages, his book lies much of dynamical 48 rivals the combined lengths of Galileo’s systems theory and par- 47 Dialogues and Newton’s Principia. But ticularly the subset often 46 does it fulfill the promise of its title and known as “chaos theory.” 45 heft? Not very well. I don’t know when the 44 The book’s central idea is that the sim- idea was first articulated, 43 ple computer programs, namely cellular Image not but by the early 1970s 42 automata, can explain natural phenomena available for Nicholas Metropolis, Paul 41 that have so far eluded “traditional” mathe- online use. Stein, and Myron Stein 40 matical approaches such as differential (1) had provided a de- 39 equations. A cellular automaton, in its sim- tailed explanation of the 38 plest incarnation, is a one-dimensional line complex behavior of sim- 37 of sites or cells, each of which is either ple iterated maps (such as 36 black or white. The color (state) of the cell the famous “logistic 35 can change over time. At each discrete time map”). In the late 1960s, 34 step, every cell updates its state—either re- John Conway developed 33 taining or flipping its color from the previ- his “Game of Life,” a 32 ous step—as a function of its previous state simple two-dimensional 31 and those of its two nearest neighbors. The cellular automaton capa- 30 cellular automaton rule comprises the list ble of highly complex be- 29 of functions that map each three-cell Width doublers. These examples of three-color cellular automata havior (2). Around the 28 neighborhood to the update state for the that achieve the purpose of doubling the width of their initial con- same time, Aristid Lin- 27 center cell. In his theoretical work, Wol- dition were taken from the 4277 cases found in an exhaustive denmayer invented what 26 fram typically considers the lines of cells search of all of the more than 7.625 x 1012 possible rules. are now called L-systems, 25 limitless to ensure there is no ambiguity at simple rules that give rise 24 the boundaries. Such one-dimensional, ior. Such programs, implemented in natural to extremely lifelike pictures of plants and 23 two-state, two-neighbor cellular automata systems, give rise to most of the complexity other natural forms (3). One of Wolfram’s 22 are called elementary; more complicated and randomness that we observe in nature. own early contributions was observing and 21 versions can have additional states per cell, These programs lead to better models of classifying such complex behavior in ele- 20 larger neighborhoods to determine update complex systems than do traditional mathe- mentary cellular automata. 19 states, and additional dimensions. matical approaches. Computation in cellu- 18 Cellular automata are perhaps the most lar automata and similar simple programs Origins of Complexity in Nature 17 idealized models of complex systems: they provide a new framework for understanding In Wolfram’s view, cellular automata or simi- 16 consist of large numbers of simple compo- complex systems. Elementary cellular au- lar simple rules are responsible for most of 15 nents (here, cells) with no central con- tomata can exhibit the ability to perform the randomness and complexity seen in na- 14 any computable procedure. This computa- ture. To support this claim, he notes that ran- 13 tional universality gives rise to the principle domness and complexity are very common 12 The author is at the Department of Computer Sci- ence and Engineering, OGI School of Science and En- of “computational equivalence,” which in the behavior of simple rules and that some 11 gineering, Oregon Health and Sciences University, Wolfram claims is a new law of nature that complex and random-looking phenomena in 10 CREDIT: STEPHEN WOLFRAM Beaverton, OR 97006, USA. E-mail: [email protected] illuminates many aspects of natural phe- nature have visual features similar to those www.sciencemag.org SCIENCE VOL 298 4 OCTOBER 2002 65 S CIENCE’ S C OMPASS 65 produced by simple cellular automata. He therefore, natural selection is not needed to to capture the essence of what is going on— 64 offers an important and useful discussion of create complexity in biology. In his view, even though traditional efforts have been 63 the terms “random” and “complex.” The biological systems are complex merely be- quite unsuccessful.” But as far as I can tell, 62 meaning of both terms depends on the fea- cause evolution has sampled a huge num- his approach has yielded no new successful 61 tures of behavior that are of interest and the ber of simple programs and these often predictions, and none of his interesting 60 available perceptual and analytical tools. give rise to complex behavior. Wolfram of- speculations on physics propose any experi- 59 Wolfram examines these connections in fers no convincing evidence for this claim, ments to support this view. 58 considerable detail, and he devotes an entire nor does he discuss where these programs 57 chapter to a discussion of perception and are implemented—at the level of the Computational Universality 56 analysis. Nonetheless, there is little support genome? of cells? Although cellular au- A centerpiece of the book is Wolfram’s 55 for his claim that because simple rules tomata provide plausible models of biologi- sketch of a proof done by Caltech graduate 54 commonly produce random and complex- cal pigmentation and some aspects of mor- student (and former Wolfram research as- 53 looking behavior, simple rules must be the phology, there is as yet no compelling link sistant) Matthew Cook that one of the ele- 52 source of most such behavior in nature. between simple programs and complex bi- mentary cellular automaton rules can sup- 51 In addition, Wolfram distances his results ological systems such as the brain, the im- port universal computation. In describing 50 on cellular automata from earlier results on mune system, or cellular metabolism. On this result, Wolfram gives an excellent re- 49 chaotic systems. He contends that whereas view of some central ideas 48 some cellular automata can generate ran- Pigmentation patterns in theoretical computer sci- 47 domness “intrinsically” (starting from a on mollusc shells. ence. The notion of univer- 46 very simple initial condition), a chaotic sys- Wolfram interprets sal computation was first 45 tem can do so only when its initial condition their striking resem- developed by Alan Turing 44 is random. This is incorrect; there are many blance to the patterns in the 1930s. Roughly, a 43 chaotic systems in which even a very simple Image not produced by simple device is said to be “uni- 42 initial condition produces a random-looking one-dimensional cellu- versal” or “can support 41 output—the Lorenz system of equations (4) available for lar automata as evi- universal computation” if it 40 is one well-known example. In general, cel- online use. dence that they are can run any program on 39 lular automata are a type of discrete dynam- generated by processes any input. Nowadays, ap- whose basic rules are 38 ical system and can exhibit behavior analo- proximations to a universal chosen at random from 37 gous to chaos. Thus, many of the results a set of the simplest devices are commonplace; 36 from the theory of dynamical systems apply possibilities. they are known as pro- 35 to cellular automata. grammable computers.