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Home , A New Kind of Science, Matthew Cook, Stephen Wolfram

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 a Universal Computer? 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 , 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 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 . $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 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 “.” 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 , 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 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 , 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 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 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 as- 53 looking behavior, simple rules must be the phology, there is as yet no compelling link sistant) 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 , 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 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. The 34 the contrary, it is increasingly clear that no- computer on your desk can (given enough 33 Cellular Automata as Models tions of selection and adaptation are crucial memory) calculate any function, as long as 32 The author traces the origins of his new for understanding such systems. the function is “computable.” (One of Tur- 31 kind of science to his frustration with ana- ing’s greatest contributions was to demon- 30 lytical approaches. He claims that, in con- Framework for Understanding Nature strate that noncomputable functions exist.) 29 trast to traditional , the re- Since the beginning of the computer age, the In the early 1980s, Wolfram had found 28 search program he develops in the book “is process of computation has been proposed as that of the 256 possible elementary cellu- 27 for the first time able to make meaningful an explanatory framework for many natural lar automaton rules (i.e., rules for one- 26 statements about even immensely complex systems. Artificial-intelligence practition- dimensional cellular automata with two 25 behavior.” This promise is not borne out. ers have suggested that the brain is actually states and two neighbors per cell), a small 24 One chapter of the book presents pat- a computer and that thinking is equivalent to subset, including the rule he numbered 23 terns formed by cellular automata and sim- processing information. In the earliest use 110, exhibited particularly interesting be- 22 ilar systems that model crystal growth (par- of cellular automata, von Neumann de- havior. Cook showed that, for any program 21 ticularly snowflakes), material fracture, flu- scribed biological self-reproduction in com- and any input, one can specially design an 20 id turbulence, plant morphologies, and pig- putational terms (5, 6). More recently, all initial condition that encodes the program 19 mentation patterns of shells and animals. In kinds of behavior (including the immune re- and input and then iterate (of the 18 each case, the behavior of the model visu- sponse, the regulatory networks formed 256 possible rules for a one-dimensional 17 ally resembles in some way the behavior of among genes, and the collective behavior of cellular automaton with two states and two 16 the actual system; this is particularly strik- ants in a colony) have been cast as “natural neighbors per cell) on the initial condition 15 ing for snowflakes and sea shells. Wolfram computation.” Wolfram takes this notion of to, in effect, run the given program on the 14 makes a compelling visual case that simple “computation as a framework for nature” to given input. Thus, rule 110 is a universal 13 cellular automata–like rules might underlie an extreme. He believes that nearly every- computer. This is an impressive result, and 12 such behavior in nature, and he gives a thing in the universe can be explained not Wolfram claims it is counterintuitive: 11 concise and readable review of some ideas just as computation but specifically in terms 10 in developmental morphology. But the re- of simple programs such as cellular automa- [I]t has almost always been assumed… 9 sults in this chapter do not advance beyond ta. In a long, rather technical chapter, he dis- that in order to get something as sophisti- 8 work done twenty or more years ago, and cusses how fundamental physics (thermody- cated as universality there must be no 7 there are no new “meaningful statements” namics, quantum mechanics, relativity, and choice but to set up rules that are them- 6 about any of these phenomena. the like) can be cast in terms of cellular au- selves special and sophisticated. One of 5 Less compelling are Wolfram’s specula- tomata, a research program that was previ- the dramatic discoveries of this book, 4 tions on natural selection. He contends that ously pursued by , Tomasso however, is that this is not the case. 3 because the generation of complexity is so Toffoli, , , 2 common in simple cellular automata, it and others [see (7)]. Wolfram claims that Wolfram views this accomplishment as

1 must be common in nature as well and, “remarkably simple programs are often able extremely significant for science; he believes CREDIT: STEPHEN WOLFRAM

66 4 OCTOBER 2002 VOL 298 SCIENCE www.sciencemag.org S CIENCE’ S C OMPASS 65 that a logical consequence is that universality is that it is easy to find universal comput- Wiener. (To be sure, Wolfram’s own con- 64 is ubiquitous throughout nature. Rule 110’s ers even among simple cellular automata. tribution of the Mathematica software 63 universality will not, however, be very sur- The second claim is also plausible, though package has been of great value for such 62 prising to computer scientists; Cook’s proof supported by less evidence than the first. efforts.) I also agree that ideas from the 61 is only the latest in a series of demonstra- There are, in principle, processes that can field of computation will be increasingly 60 tions that relatively simple devices (Turing compute things universal computers can- useful for understanding natural phenome- 59 machines, neural networks, iterative maps) not, but these “super-universal” processes na, particularly in the study of living sys- 58 can be universal computers. Von Neumann require continuous-valued numbers (11). tems. Nonetheless, analytical approaches 57 was the first to show that a cellular au- Whether continuous values actually exist to illuminating complexity in nature thus 56 tomaton can be universal. He constructed in nature and can be harnessed by natural far have been much more successful than 55 a two-dimensional example with 29 states processes to surpass universal computa- cellular automata and related computation- 54 and four neighbors per cell, al methods. A clear example is the suc-

53 and others eventually reduced Image not cessful use of the renormalization group to 52 the complexity to four states available for explain complex behavior in a wide range 51 per cell (8). In the 1970s, online use. of dynamical systems (12). 50 Conway sketched a proof that A universal computer. Rule 110 uses these eight functions to Given its length and content, A New 49 his Game of Life (with two specify a cell’s new color (lower row) for each possible combina- Kind of Science is surprisingly readable. 48 states and eight neighbors per tion of previous colors of the cell and its two neighbors (upper row). Wolfram’s use of pictures to illustrate diffi- 47 cell) is universal (2). Before cult concepts works superbly well, and 46 Cook’s work with rule 110, the simplest tion is unknown. Wolfram strongly be- non-scientists will find it possible to un- 45 known universal cellular automaton was lieves they cannot. The third claim does derstand much of what he covers. The prin- 44 one-dimensional with seven states and two not make sense to me. I find it quite plau- cipal obstacle readers face is the plethora 43 neighbors per cell (9). Rule 110 is now the sible that my brain is a universal computer of self-aggrandizement, some statements 42 simplest, and it is hard to see how a univer- (or would be, if I had infinite memory) of which seem like they could not possibly 41 sal cellular automaton could get any sim- and that the brain of the worm Caenorhab- be serious. For example, the author claims 40 pler. Although it is interesting that such a ditis elegans is also (approximately) uni- his principle of computational equivalence 39 simple rule (not specifically constructed to versal, but I don’t accept that the computa- “has vastly richer implications...than essen- 38 perform computation) turns out to be uni- tions we engage in are equivalent in so- tially any single collection of laws in sci- 37 versal, the result is an incremental step over phistication. ence.” Even more disturbing are the sug- 36 what had been done before. gestions that Wolfram himself invented ev- 35 The significance of universality is also Summary erything of interest here: 34 tempered by practicality. Whereas rule 110 I think Wolfram is on the right track in 33 (and other cellular automata) can be shown proposing that simple computer models But to develop the new kind of science 32 to be universal in principle, in practice it is and experiments can lead to much that I describe in this book I have had no 31 almost impossible to design the initial progress. This approach may even come to choice but to take several large steps at 30 condition necessary to perform a desired be seen as , though it once, and in doing so I have mostly ended 29 computation. And even if such an initial will be the result of the contributions of a up having to start from scratch—with new 28 condition were known, the time needed to very large number of people—beginning ideas and new methods that ultimately de- 27 perform the computation might be ex- with pioneers of the computer age such as pend very little on what has gone before. 26 tremely long compared with that on a tradi- von Neumann, Turing, and Norbert 25 tional computer. Many people have In fact, most of what 24 claimed that the concepts of universal com- Wolfram describes is the 23 putation and uncomputability are relevant work of many people (in- 22 to science; in a notable example, Roger cluding himself), and 21 Penrose claimed these notions preclude the most of it was done at 20 possibility of machine intelligence (10). least ten to twenty years 19 But I know of no generally accepted scien- ago. Nearly no credits to 18 tific explanations or predictions in which the contributions of oth- 17 these concepts play any role. ers appear in the book’s 16 main text. Some credits 15 Computational Equivalence Image not can be found in the long 14 The final chapter discusses Wolfram’s notes section at the 13 “principle of computational equivalence,” available for book’s end, but many are 12 which is based on the idea that the best online use. not given at all. For ex- 11 way to understand processes in nature is to ample, the snowflake 10 view them as performing computations. It models Wolfram discuss- 9 consists of three claims: (i) The ability to es are based on the work 8 support universal computation is very of Packard (13), but 7 common in nature. (ii) Universal computa- Packard is not mentioned 6 tion is an upper limit on the sophistication in connection with them. 5 of computations in nature. (iii) This is only one example 4 processes in nature are almost always of such inexcusable omis- 3 equivalent in sophistication. sions. Moreover, the book 2 The first claim is plausible, though by Artistic output. Igor Bakshee created this image using rule 110 does not contain a single

1 CREDIT:WOLFRAM; STEPHEN IGOR BAKSHEE (TOP) (BOTTOM) no means established. Wolfram’s argument and Mathematica. bibliographic citation—

www.sciencemag.org SCIENCE VOL 298 4 OCTOBER 2002 67 S CIENCE’ S C OMPASS 65 an astounding lapse that will put off serious rillas mate only with the group’s silverback evolutionary models. Through the device 64 scientific readers. Wolfram’s Web site (14) who, in the absence of sperm competition of having organisms describe their situa- 63 includes “relevant books,” but this list is no from other males, needs to provide just tions (and predicaments) to her, she is able 62 substitute. enough sperm to ensure that fertilization is to enter a dialog that uses individual case 61 To benefit from the book, one must get successful. The promiscuous female chim- studies to illustrate general principles. This 60 past these issues without becoming too an- panzee, on the other hand, has the sperm technique draws in the reader to a witty, 59 gry and take most of the claims with a from different males competing for access racy, informed, entertaining, and instructive 58 large grain of salt. Wolfram’s discussions to her eggs, so those males have evolved read. 57 and speculations will interest many people the capacity to produce inordinate quanti- There will be opposition to Judson’s ap- 56 in a wide variety of fields, but they do not ties of the stuff. proach. Some will argue that anthropomor- 55 constitute a new kind of science. Dr. Tatiana is the brainchild of Olivia phism on this level is unjustified and leads 54 Judson, whose doctoral studies were su- inevitably to inaccuracies. Who cannot feel 53 References and Notes pervised by the late W. D. Hamilton. She for the plight of the green spoon worm 52 1. N. Metropolis, M. L. Stein, P. R. Stein, J. Comb. Theor. wanted to describe to 15A, 25 (1973). 51 2. E. Berlekamp, J. H. Conway, R. Guy, Winning Ways for her audience our current 50 Your Mathematical Plays,vol. 2. (Academic Press, understanding of the 49 New York, 1982). evolutionary biology of 3. P. Prusinkiewicz, A. Lindenmayer, The Algorithmic 48 Beauty of Plants (Springer-Verlag, New York, 1990). sex. The topic is mani- 47 4. E. N. Lorenz, The Essence of Chaos (Univ. of Washing- fold, wondrous, and Image not 46 ton Press, Seattle,WA, 1993). characterized by diversi- 45 5. J. von Neumann, A. W. Burks, Theory of Self- ty: Why do some organ- available for Reproducing Automata (Univ. of Illinois Press, 44 Urbana, IL, 1966). isms have sexual repro- online use. 43 6. A.W. Burks, Ed., Essays on Cellular Automata (Univ. of duction whereas others 42 Illinois Press , Urbana, IL, 1970). do not? Why do differ- 7. E. Fredkin, R. Landauer, T. Toffoli, Eds., Int. J. Theor. 41 Phys. 21 (nos. 3–4, 6–7, 12), (1982). ent species have differ- 40 8. E. R. Banks, dissertation, Massachusetts Institute of ent numbers of sexes? 39 Technology (1971). What determines whether 9. K. Lindgren, M. G. Nordahl, Complex Syst. 4, 299 38 (1990). individuals are single-sexed or herma- (Bonellia viridis) that just inhaled her 37 10. R. Penrose, The Emperor’s New Mind (Oxford Univ. phrodite? Why do some species usually “husband”? But, then again, in what sense 36 Press, 1989). have imbalanced sex ratios while others was the male a husband before being in- 11. C. Moore, Theor. Comp. Sci. 162, 23 (1996). 35 12. M. J. Feigenbaum, Los Alamos Sci. 1 (no. 1), 4 (1980). do not? Why is sex sometimes determined haled? Only after being inhaled does the 34 13. N. H. Packard, in Theory and Applications of Cellular genetically and sometimes environmen- male start to fertilize the female’s eggs. 33 Automata,S.Wolfram, Ed. (World Scientific, Singa- tally? What are the causes and conse- This example leads Dr. Tatiana to a descrip- pore, 1986), pp. 305–310. 32 14. www.wolframscience.com quences of the different mating systems tion of environmental sex determination in 31 15. I am grateful to R. Axelrod, D. Farmer, J. Holland, G. seen in the natural world? Over the years the spoon worm: lone larvae mature into 30 Huber, and C. Moore for discussions and insightful the variety has been described and the large females and larvae that subsequently 29 comments on an earlier version of the manuscript problems of explaining it have been develop near a female become male. In a and to the Santa Fe Institute for support during the 28 writing of this review. solved, to varying degrees. Many of the carefully crafted discourse that follows, she 27 major contributions came from biologists explains why and when sex is environmen- 26 BOOKS: EVOLUTION like Darwin who became familiar with tally versus genetically determined. 25 the natural history of many, many species It would be wrong to think about Sex 24 Explanations for the and were then able to make comparisons Advice to All Creation as merely a collec- 23 to explain the differences. tion of anecdotes followed by descriptions 22 Birds and the Bees Familiarity with natural history is of general principles. Instead, the book is a 21 equivalent to becoming intimate with pri- developing text, meaning that it should be 20 Paul Harvey vate lives, except that the for- read from the beginning be- 19 mer lacks the taboo of anthro- Dr.Tatiana’s cause answers to some ques- 18 f I were an intellectually challenged pomorphism. Some of the best Sex Advice tions require familiarity with 17 adult male gorilla who stumbled across evolutionary biologists work by to All Creation earlier chapters. For those who 16 Ian adult male chimpanzee, I should in attempting to identify them- by Olivia Judson want to check the facts for 15 all likelihood be at a loss to explain my selves with the species they themselves or to delve more Chatto and Windus, 14 comparatively tiny testicles. Fortunately, study: “What would I do if?” is London, 2002. 317 pp. deeply into the problems that 13 my angst might be eased by consulting Dr. often useful shorthand for £16.99. ISBN 0-7011- Judson tackles, notes at the 12 Tatiana, the agony aunt, who would point “What would natural selection 6925-7. Metropolitan, end of the book cleverly refer- 11 out that large testicles are characteristic of produce under particular cir- New York, 2002. 319 pp. ence the original research pa- 10 those primates and other mammalian cumstances?” Of course there $24, C$34.95. ISBN 0- pers used in its construction. 9 species in which the female often mates can be dangers in this way of 8050-6331-5. The bottom line is that the 8 with more than one male during a given thinking, which is why formal book actually works. Like 7 estrus. Large testicles produce more models often reveal logical pit- Richard Dawkins’s Selfish 6 sperm, thereby providing more tickets in falls. But, even then, the results of a logical Gene (Oxford University Press, Oxford, 5 the sperm competition lottery. Female go- modeling process need to be described ver- 1976), it uses unabashed anthropomor- 4 bally. Judson has gone the whole hog by phism to create scenarios with which the 3 employing anthropomorphism to its ex- open-minded reader can identify. Also like The author is in the Department of Zoology, Univer- 2 sity of Oxford, South Parks Road, Oxford OX1 3PS, tremes, in the assurance that most of the Dawkins, Judson is a gifted writer, and her

1 UK. E-mail: [email protected] work she describes has been backed by book helps further understanding. CREDIT: JOE SUTLIFF

68 4 OCTOBER 2002 VOL 298 SCIENCE www.sciencemag.org