Curiouser and curiouser

"A scientist," say the authors of this book, "when confronted with a complex problem feels a sensation of distress that is often not attributable to a definite cause ... the behaviour is so involved that any specifically designed finite model eventually departs from the observation". Laymen confronted with a tax form will sympathise. It is important to distinguish this characteristic of complexity from the "predictability horizon" of chaotic systems, in which small perturbations in the initial conditions grow exponentially even if the model is exact. (The predictability horizon of the motion of the planets is about ten million years: the cosmos is chaotic.) Of course, chaos is an important subset of complexity, but the literature on it is extensive and it is not discussed in detail in this book. The authors do not define "complexity" exactly, and it means slightly different things to different workers in the field.

The book begins with three chapters on phenomenology and models: here and later, an impressively wide range of phenomena is covered and unified. The description of my own speciality, turbulence, is apparently based on the very special case of buoyant convection between a pair of horizontal plates, and also suggests, incorrectly, that the cascade of turbulent energy to smaller wavelengths takes place in one-octave steps and stops exactly at the Kolmogorov length scale. Similar minor inaccuracies may occur in other physical examples about which I am ignorant, but, at least in the case of turbulence, the description of the essential features of complexity is entirely satisfactory - and anyway this is not a book from which to learn about individual phenomena.

The authors elect to treat all the phenomena as discrete rather than continuous, which perhaps detracts from physical understanding but permits the use of digital and symbolic (and statistical) concepts in a unifying way. For instance, there are frequent references to "cellular automata" that change the states of points on a lattice according to rules involving the present states of neighbouring points: they can be set up to reproduce, more or less exactly, discretised versions of various field equations. The classical "relaxation" solution of the discretised Laplace equation is an example. An automaton with a more complex output is the Game of Life, in which patterns grow, die or in some cases move unchanged across the lattice. The rules of cellular automata have standard mnemonic numbers and are classified into "regular" and "chaotic" according to the behaviour of the solutions: apparently by delightful chance, the first chaotic example given by the authors is rule 42. (Rule 42: "All persons more than one-mile high to leave the court ..." "Well I shan't go, at any rate," said Alice, "besides, that's not a regular rule: you invented it just now.") Chapters four to six discuss mathematical tools - symbolic representations and languages, probability, and the unhappily named "thermodynamics" of information theory. Various definitions of entropy appear, but at least the authors are careful in their discussion of the relevance to true thermodynamic entropy (itself a battlefield). A wide range of mathematical tools is used, but most of them should be in the toolkit of physicists or mathematically minded engineers and biologists, and there are short appendices on the less common concepts (measurable sets, Lyapunov exponents).

The next chapters deal with the formal characterisation of complexity, mainly by using symbolic languages, and slide into discussion of computability in the Turing sense. Many physicists may feel these are unfamiliar matters best left to computer scientists, but the book serves to warn that this attitude may have to change, because symbolic-logic concepts are needed in defining quantitative measures of complexity (the most familiar measure, the attractor dimension of a chaotic process, is a very special case).

This is a book for professional scientists, but it is by no means solely for specialists: its admirable breadth of outlook should provide something for every scientist.

Peter Bradshaw is emeritus professor of engineering,Stanford University,California.