% -*- Mode:Text -*- %\def\mail{margolus nhm@mit.edu } \documentclass[10pt]{feynman} \usepackage{epsfig} \usepackage{psfig} \usepackage{graphics} %\usepackage{mybib} \begin{document} %\setcounter{chapter}{17} %\setcounter{page}{281} \newcommand{\chapterheader}{CRYSTALLINE COMPUTATION} \newcommand{\chapterauthor}{NORMAN H. MARGOLUS} %\newcommand{\eqlabel}[1]{\label{eq.#1}} %\newcommand{\eq}[1]{(\ref{eq.#1})} % Ref. to equation %\newcommand{\bra}[1]{\mbox{$\left\langle #1 \right|$}} %\newcommand{\ket}[1]{\mbox{$\left| #1 \right\rangle$}} %\newcommand{\braket}[2]{\mbox{$\left\langle #1 | #2 \right\rangle$}} %\newcommand{\av}[1]{\mbox{$\left| #1 \right|$}} %\newcommand{\ve}{\varepsilon} %\newcommand{\osc}{{\mbox{\rm \scriptsize osc}}} %\newcommand{\tot}{{\mbox{\rm \scriptsize tot}}} %\newcommand{\ground}{{\mbox{\rm \scriptsize ground}}} %\renewcommand{\max}{{\mbox{\rm \scriptsize max}}} %\renewcommand{\min}{{\mbox{\rm \scriptsize min}}} \newcommand{\fref}[1]{Figure~\ref{fig.#1}} \newcommand{\byby}[0]{$\times$} \newcommand{\figspace}[3]{% name height caption \begin{figure} \begin{tt} % figure name \vspace*{#2} % figure height \end{tt} \caption{#3} % figure caption \label{fig.#1} \end{figure}} \newcommand{\figfig}[3]{% label, picture, caption \begin{figure} $$#2$$\relax \caption{#3} \label{fig.#1} \end{figure} } \setlength{\fboxsep}{.1pt} \setlength{\fboxrule}{.1pt} \chapter{Crystalline Computation} \begin{center} %\begin{tabular}{c} {\bf Norman Margolus} \input where.tex %\\ %{\it Boston University Center for Computational Science and}\\ %{\it Massachusetts Institute of Technology Artificial Intelligence Laboratory} %\end{tabular} \end{center} \section*{Abstract} Discrete lattice systems have had a long and productive history in physics. Examples range from exact theoretical models studied in statistical mechanics to approximate numerical treatments of continuum models. There has, however, been relatively little attention paid to exact lattice models which obey an {\em invertible dynamics}: from any state of the dynamical system you can infer the previous state. This kind of microscopic reversibility is an important property of all microscopic physical dynamics. Invertible lattice systems become even more physically realistic if we impose locality of interaction and exact conservation laws. In fact, some invertible and momentum conserving lattice dynamics---in which discrete particles hop between neighboring lattice sites at discrete times---accurately reproduce hydrodynamics in the macroscopic limit. These kinds of discrete systems not only provide an intriguing information-dynamics approach to modeling macroscopic physics, but they may also be supremely practical. Exactly the same properties that make these models physically realistic also make them efficiently realizable. Algorithms that incorporate constraints such as locality of interaction and invertibility can be run on microscopic physical hardware that shares these constraints. Such hardware can, in principle, achieve a higher density and rate of computation than any other kind of computer. Thus it is interesting to construct discrete lattice dynamics which are more physics-like both in order to capture more of the richness of physical dynamics in informational models, and in order to improve our ability to harness physics for computation. In this chapter, we discuss techniques for bringing discrete lattice dynamics closer to physics, and some of the interesting consequences of doing so. % add comp u? \section{Introduction}\label{sec.intro} In 1981, Richard Feynman gave a talk at a conference hosted by the MIT Information Mechanics Group. This talk was entitled ``Simulating Physics with Computers,'' and is reproduced in this volume. In this talk Feynman asked whether it is possible that, at some extremely microscopic scale, nature may operate exactly like discrete computer-logic. In particular, he discussed whether crystalline arrays of logic called {\em Cellular Automata} (CA) might be able to simulate our known laws of physics in a direct fashion. This question had been the subject of long and heated debates between him and his good friend Edward Fredkin (the head of the MIT Group) who has long maintained that some sort of discrete classical-information model will eventually replace continuous differential equations as the mathematical machinery used for describing fundamental physical dynamics\cite{fredkin-dm,fredkin-nature}. For classical physics, Feynman could see no fundamental impediment to a very direct CA simulation. For quantum physics, he saw serious difficulties. In addition to discussing well known issues having to do with hidden variables and non-separability, Feynman brought up a new issue: simulation efficiency. He pointed out that, as far as we know, the only general way to simulate a lattice of quantum spins on an ordinary computer takes an exponentially greater number of bits than the number of spins. This kind of inefficiency, if unavoidable, would make it impossible to have a CA simulation of quantum physics in a very direct manner. Of course the enormous calculation needed to simulate a spin system on an ordinary computer gives us the result of not just a single experiment on the system, but instead approximates the complete statistical distribution of results for an {\em infinite} number of repetitions of the experiment. Feynman made the suggestion that it might be more efficient to use one quantum system to simulate another. One could imagine building a new kind of computer, a {\em quantum spin computer}, that was able to mimic the quantum dynamics of any spin system using about the same number of spins as the original system. Each simulation on the quantum computer would then act statistically like a {\em single} experiment on the original spin system. This observation that a quantum computer could do some things easily that we don't know how to do efficiently classically, stimulated others to look for and find algorithms for quantum computers that are much faster than any currently known classical equivalents\cite{shor,grover}. In fact, if we restrict our classical hardware to perform the ``same kind'' of computation as the quantum hardware---rather than to just solve the same problem---then we can actually {\em prove} that some quantum computations are faster. These fast quantum computations present further challenges to hypothetical classical-information models of quantum physics\cite{joe-thesis}. Despite such difficulties, Feynman did not rule out the possibility that some more subtle approach to the efficient classical computational modeling of physics might yet succeed. He found something very tantalizing about the relationship between classical information and quantum mechanics, and about the fact that in some ways quantum mechanics seems much {\em more} suited to being economically simulated with bits than classical mechanics: unlike a continuous classical system, the entropy of a quantum system is finite. The informational economy of quantum systems that Feynman alluded to has of course long been exploited in statistical mechanics, where classical bits are sometimes used to provide finite combinatorial models that reproduce some of the macroscopic {\em equilibrium} properties of quantum systems\cite{ising,huang}. It is natural then to ask how much of the macroscopic {\em dynamical} behavior of physical systems can also be captured with simple classical information models. This is an interesting question even if your objective is not to revolutionize quantum physics: we can improve our understanding of nature by making simple discrete models of phenomena. % even classical not easy -- this is what we'll discuss My own interest in CA modeling of physics stems from exactly this desire to try to understand nature better by capturing aspects of it in exact informational models. This kind of modeling in some ways resembles numerical computation of differential equation models, where at each site in a spatial lattice we perform computations that involve data coming from neighboring lattice sites. In CA modeling, however, the conceptual model is not a continuous dynamics which can only be approximated on a computer, but is instead a finite logical dynamics that can be simulated {\em exactly} on a digital computer, without roundoff or truncation errors. Every CA simulation is an exact digital integration of the discrete equations of motion, over whatever length of time is desired. Conservations can be exact, invertibility of the dynamics can be exact, and discrete symmetries can be exact. Continuous behavior, on the other hand, can only emerge in a large-scale average sense---in the {\em macroscopic limit}. CA models have been developed in which realistic classical physics behavior is recovered in this limit\cite{rothman-book,chopard-book}. Physics-like CA models are of more than conceptual and pedagogical interest. Exactly the same general constraints that we impose on our CA systems to make them more like physics also make them more efficiently realizable as physical devices. CA hardware that matches the structure and constraints of microscopic physical dynamics can in principle be more efficient than any other kind of computer: it can perform more logic operations in less space and less time and with less energy dissipation\cite{toffoli-nature,frank-thesis}. It is also scalable: a crystalline array of processing elements can be indefinitely extended. Finally, this kind of uniform computer is simpler to design, control, build and test than a more randomly structured machine. The prospect of efficient large-scale CA hardware provides a practical impetus for studying CA models. \section{Modeling dynamics with classical spins}\label{sec.ising} {From} the point of view of a physicist, a CA model is a fully discrete classical field theory. Space is discrete, time is discrete, and the state at each discrete lattice point has only a finite number of possible discrete values. The most essential property of CA's is that they emulate the spatial locality of physical law: the state at a given lattice site depends only upon the previous state at nearby neighboring sites. You can think of a CA computation as a regular spacetime crystal of processing events: a regular pattern of communication and logic events that is repeated in space and in time. Of course it is only the structure of the computer that is regular, not the patterns of data that evolve within it! These patterns can become arbitrarily complicated. Discrete lattice models have been used in statistical mechanics since the 1920's\cite{ising,huang}. In such models, a finite set of distinct quantum states is replaced by a finite set of distinct classical states. Consider, for example, a hypothetical quantum system consisting of $n$ spin-${1 \over 2}$ particles arranged on a lattice, interacting locally. The spin behavior of such a system can be fully described in terms of $2^n$ distinct (mutually orthogonal) quantum states. The {\em Ising model} accurately reproduces essential aspects of phase-change behavior in such a system using $n$ classical bits---which give us $2^n$ distinct classical states. In the Ising model, at each site in our lattice we put a {\em classical spin:} a particle that can be in one of two classical states. We define bond energies between neighboring spins: we might say, for example, that two adjacent spins that are parallel (i.e., are in the same state) have a bond energy of $\epsilon_=$, while two antiparallel (not same) neighbors have energy $\epsilon_{\neq}$. This gives us a classical system which has many possible states, each of which has an energy associated with it. In calculating the equilibrium properties of this system, we simply assume that the dynamics is complicated enough that all states with the same energy as we started with will appear with equal probability. Thus we ignore the actual quantum dynamics of the original spin system, and instead substitute an energy-conserving random pseudo-dynamics. % mention different universality classes % look at Creutz's papers! We could equally well substitute any classical dynamics that has a sufficiently complicated evolution. We will consider one simple CA model that has been successfully used in this manner\cite{vichniac,pomeau-invariant,herrmann,creutz}. We assume that we are dealing with an isolated spin system, not in contact with any heat bath, so that total energy must be exactly conserved. We will also impose the realistic constraint that the microscopic dynamics of an isolated physical system must be exactly invertible: there must always be enough information in the current state to recover any previous state. This constraint helps to ensure that a deterministic dynamics explores its available state-space thoroughly, and so can be analyzed statistically---this issue is discussed in Section~\ref{sec.rev}. We can construct a simple CA that has these properties, in which the next value of the spin at each site on a 2D square lattice only depends upon the current values of its four nearest neighbors. The rule is very simple: a given spin changes state if and only if this doesn't change the total energy associated with its bonds to its four nearest neighbors. Equivalently, a given spin (bit) is flipped (complemented) if exactly two of its four neighbors are zero's, and two are one's. This doesn't change its total bond energy: both before and after the flip, it will be parallel to half of its neighbors (contributing $2\epsilon_=$ to the total), and antiparallel to the rest (contributing $2\epsilon_{\neq}$). The rule as stated above would be fine if we updated just one spin on the lattice at a time, but we would like to update the lattice in parallel. To make this work, we will adopt a checkerboard updating scheme: we imagine that our lattice is a giant black and white checkerboard, and we alternately hold the bits at all of the black sites fixed while we update all of the white ones, and then hold the white sublattice fixed while updating the black. In this way, the neighbors of a spin that is changed are not also simultaneously changed, and so our logic about conserving energy remains valid. Now how can we add invertibility? We already have! If we apply our rule to the same checkerboard sublattice twice in a row, then each spin is either flipped twice or not at all---the net effect is no change. Thus the most recent step in our time evolution can always be undone simply by applying the rule a second time to the appropriate sublattice, and we can recover any earlier state by undoing enough steps. This example demonstrates that we can simultaneously capture several basic aspects of physics in an exact digital model. First of all, the finite-state character of a quantum spin system is captured by using classical bits. Next, spatial locality is captured by making the rule at each lattice site depend only on nearby neighbors. Finally, both energy conservation and invertibility are captured by splitting the updating process into two phases, and alternately looking at one half of the bits while changing the other half. By making only changes that conserve a bond energy locally, we conserve energy globally. By making the change at each site be a permutation operation that depends only upon unchanged neighbor information, we can always go backwards by taking the same neighbor information and performing the inverse permutation. \figfig{creutz}{ \hfill \fbox{% \epsfig{figure=margolus/2d-ising-10.ps,height=1.5in}}\hskip .2in \fbox{% \epsfig{figure=margolus/2d-ising-wave.ps,height=1.5in}}\hskip .2in \fbox{% \epsfig{figure=margolus/2d-ising-close.ps,height=1.5in}}\hfill}% {An Ising CA (a)~A state that evolved from a random pattern of 90\% 0's and 10\% 1's. (b)~Wave on the boundary between a domain of all 1's, and a domain of all 0's. (c)~Closeup of a portion of the boundary.} Figure~\ref{fig.creutz}a shows the state of the Ising CA after 100,000 steps of time evolution on a 512\byby512 lattice, when started from a randomly generated configuration of site values that consisted of 10\% 1's and 90\% 0's. This is an {\em equilibrium} configuration: if we compare this to the configuration after 100,000,000 steps, the picture looks qualitatively the same. This equilibrium configuration is divided about equally between large domains that are mostly 0's, and large domains that are mostly 1's. Since bond energy is conserved, the total length of boundary between regions of 0's and regions of 1's must be unchanged from that in the initial configuration---the numbers of 0's and 1's are not themselves conserved. This CA has some surprising behavior when started from a more ordered initial state: it supports the continuum wave equation in an exact fashion. Figure~\ref{fig.creutz}b illustrates a wave on the boundary between two {\em pure} domains (all 0's, or all 1's). If we hold the values at the edges of the lattice fixed, then we find that the boundary shown behaves like a standing wave, oscillating in a harmonic fashion that repeats forever without any damping. In fact, it is easy to show that any waveform that we set up along this diagonal boundary---as long as it isn't too steep---exactly obeys the wave equation ({\em cf.} \cite{hrgovcic}). To see this, notice (Figure~\ref{fig.creutz}c) that the boundary {\em between} the two domains consists of a sequence of vertical and horizontal line-segments each the height or width of one site. If we number these segments sequentially along the boundary, then it is easy to verify that, at each update of the lattice, all of the even-numbered segments move one position along the boundary in one direction, while all of the odd-numbered segments move one position in the opposite direction. Thus the shape of the boundary is exactly the superposition of two discrete waveforms moving in opposite directions. \figfig{other-ising}{ \hfill \hbox{% \epsfig{figure=margolus/scarves.ps,height=2in}}\hskip .6in \hbox{% \epsfig{figure=margolus/3d-ising.ps,height=2in}}\hfill}% {Ising-like CA's in 1D and 3D. (a)~Time history of Bennett's 1D CA. (b)~A 3D Ising CA cooled with a heatbath.} Similar techniques to those used in the Ising CA give a variety of related CA models\cite{cambook,margolus-thesis,creutz,raissa}. For example, in Figure~\ref{fig.other-ising}a we show the time-history of a 1D rule invented by Charles Bennett that has exactly the same bond-energy conservation that we've just seen\cite{pomeau-invariant}. In Bennett's CA, instead of 1-bit at each site we put 2-bits, which we'll call $A_i$ and $B_i$. The $A$'s and $B$'s will play the roles of the two sublattices in the Ising CA. We first update all of the $A$'s in parallel, holding the $B$'s fixed, and then vice versa. For each $A_i$, it's neighbors along the 1D chain will be the two $B$'s on either side of it: $B_{i-2}$, $B_{i-1}$, $B_{i+1}$ and $B_{i+2}$. Our rule is the same as before: we complement an $A_i$ if exactly half of its four neighbors are 1's, and half are 0's. Once we have updated all of the $A$'s, then we update the $B$'s in the same manner, using the $A$'s as neighbors. If we consider that there is a bond between each ``spin'' and its four neighbors, then we are again flipping the spin only if it doesn't change the total bond energy. If we update the same sublattice twice in a row, the net effect is no change: the rule is invertible, exactly like the Ising CA. In the figure, our 1D lattice is 512 sites wide, with periodic boundaries (joined at the edges). We started the system with all sites empty except for a patch of randomly set bits in sites near the center. Time advances upward in the figure, and we show a segment of the evolution after about 100,000 steps. Rather than show the domains directly, we show all bonds that join antiparallel spins---the number of such ``domain boundary'' bonds is not changed by the dynamics. Note the variety of ``particle'' sizes and speeds. In Figure~\ref{fig.other-ising}b, we show a 3D Ising dynamics with a heat bath. Here the rule is an invertible 3D checkerboard Ising CA similar to our 2D version, except that at every site in our 3D lattice we have added a few extra {\em heatbath} bits. The heatbath bits at each site record a binary number that is interpreted as an energy. Now our invertible rule is again ``flip whenever it is energetically allowed.'' As long as the heatbath energy at a given site is not too near its maximum, then a spin flip that would lower the bond energy is allowed, because we can put the energy difference into the heatbath. Similarly with transitions that would raise the bond energy. This heatbath-CA technique is due to Michael Creutz\cite{creutz}. He thought of the bond energy as being potential energy, and the heatbath energy as being the associated kinetic energy. This heatbath CA is perfectly invertible, since applying the dynamics twice to the same sublattice leaves both the spins and the heatbath unchanged. By adjusting the energy in the heatbath portion of this 3D CA, we can directly control the {\em temperature} of our system. We simply stop the simulation for a moment while we reach into our system and reset the heatbath values---without changing the spin values. As we {\em cool} the system in this way, energy will be extracted from bonds, and so if (for example) $\epsilon_{\neq} > \epsilon_=$, then there will be fewer domain boundaries---the domains will grow larger. The system shown has been cooled in this manner, and we render the interface between the up and down spins. \figfig{same}{ \hfill \fbox{% \epsfig{figure=margolus/same-out.ps,height=2in}}\hskip .6in \fbox{% \epsfig{figure=margolus/same-in.ps,height=2in}}\hfill}% {An Ising-like 3D CA. (a)~A macroscopic equilibrium configuration. (b)~The same configuration, with the front half of the ball removed.} Figure~\ref{fig.same} shows another Ising-like CA defined on a 3D cubic lattice. As in our 2D Ising CA, we have only one bit of state at each lattice site. Each site has a bond with each of its six nearest neighbors, and we perform a 3D checkerboard updating. This time our rule is, ``flip a given spin if its six neighbors all have the same value: six 0's or six 1's.'' We'll call this the ``Same'' rule. If we label half of the bonds attached to each site as ``antiferromagnetic'' (i.e., the energy values associated with parallel and antiparallel spins are interchanged for these labeled bonds), then this rule again conserves the total bond energy. Notice, though, that there are many different ways of labeling half of the bonds, and each way corresponds to a different additively conserved energy. We need to use several of these energies simultaneously if we want to express the Same rule as ``flip whenever permitted by energy conservation.'' The system in Figure~\ref{fig.same}a is 512$\times$512$\times$512 and was started from an empty space (all 0's) with a small random block of spin values in the center. After about 5000 steps of time evolution, this invertible system settles into the ball shown, which then doesn't change further macroscopically. Microscopically it must keep changing---otherwise if we ran the system backwards it couldn't tell when to start changing again and {\em unform} the ball. The local density of 1's defines the surface that is being rendered. In Figure~\ref{fig.same}b we remove the front half of the ball to show its interior structure. The analogous rule in 2D does not form stable ``balls.'' % We've seen a variety of behaviors % using Ising-like CA's. This raises the question, ``How complicated % can the behavior of our CA's be It is easy to define other energy-conserving invertible Ising-like CA's. We could, for example, take any model that has a bond energy defined, find a sublattice of sites that aren't directly connected to each other by bonds, and update those sites in an invertible and energy conserving manner, holding their neighbors fixed. By running through a sequence of such sublattices, we would eventually update all sites. We could also make CA models with the same energy conservations with just two sublattices, by using the technique illustrated in Bennett's CA. Simply duplicate the state at each site in the original model, calling one copy $A_i$, and the other $B_i$. If the $A$'s are only bonded to the $B$'s and vice versa, then we can update half of our system in parallel, while holding all neighbors that they depend on fixed. Of course we can construct additional invertible energy-conserving rules by taking any of these examples and forbidding some changes that are energetically allowed. \section{Simple CA's with arbitrarily complex behavior}\label{sec.complex} When the Ising model was first conceived in the 1920's, it was not thought of as a computer model: there were no electronic computers yet! It was only decades later that it and other discrete lattice models could begin to be investigated on computers. One of the first to think about such models was John von Neumann\cite{burks,ulam}. He was particularly interested in using computer ideas to construct a mechanical model that would capture certain aspects of biology that are essential for reproduction and evolution. What he constructed was a discrete world in which one could arrange patterns of signals that act much like the logic circuitry in a computer. Just as computer programs can be arbitrarily complex, so too could the animated patterns in his CA world. Digital ``creatures'' in his digital universe reproduced themselves by following a digital program. This work anticipated the discovery that biological life also uses a digital program (DNA) in order to reproduce itself. As we will see below, the level of complexity needed in a CA rule in order to simulate arbitrary patterns of logic and hence {\em universal computation} is quite low. In physics, this same possibility of building arbitrarily complicated mechanisms out of a fixed set of components seems to be an essential property of the evolution that built us. Is this the {\em only} essential property of evolution? In a paradoxical sense, computation universality gives us so much that it really gives us very little. Once we have computation universality, we have a CA that is just as powerful as any conventional computer! By being able to simulate the logic circuitry of any computer, given enough time and space, any universal CA can compute exactly the same things as any ordinary computer. It can play chess. It can simulate quantum mechanics. It can even perform a simulation of a Pentium Processor running Tom Ray's Tierra evolutionary-simulation program\cite{ray1}, and thus we know that it is capable of exhibiting Darwinian evolution. But if we don't put in such an unlikely initial state by hand, is evolution of interesting complexity something that we are ever likely to see? Is it a {\em robust} property of the dynamics? Nature has computation universality along with locality, exact conservations and many other constraints. Which of these constraints are important for promoting evolution, and whether it is possible to capture all of the important constraints simultaneously in a CA model, are both interesting questions. Here we will examine a well-known CA rule that is universal, and then discuss some physical constraints that it lacks that might make it a better candidate as a model for Darwinian evolution. \figfig{life}{ \hfill \fbox{% \epsfig{figure=margolus/life-1k.ps,height=1.5in}}\hskip .2in \fbox{% \epsfig{figure=margolus/life-16k.ps,height=1.5in}}\hskip .2in \fbox{% \epsfig{figure=margolus/life-guns.ps,height=1.5in}}\hfill}% {Conway's non-invertible ``Game of Life'' CA (128\byby128 closeups taken {from} a 2K$\times$2K space). (a)~One thousand steps into an evolution started from a random configuration of 1's and 0's. (b)~The same region after 16 thousand steps---the evolution has settled down into small uncoupled repeating patterns. (c)~A configuration that started with two ``glider guns.''} In Figure~\ref{fig.life} we illustrate Conway's {\em Game of Life}, probably the most widely known CA rule\cite{life}. This is a CA that involves a 2D square lattice with one bit at each lattice site, and a rule for updating each site that depends on the total number of 1's present in its eight nearest neighboring sites. If the total of its neighbors is 3, a given site becomes a 1, if the total is 2, the site remains unchanged; in all other cases the site becomes a 0. This rule is applied to all sites simultaneously. The Life rule is clearly non-invertible since, for example, an isolated 1 surrounded by 0's turns into a 0: you cannot then tell from the resulting configuration of site values whether that site was a 0 or a 1 in the previous configuration. If you fill your computer screen with a random pattern of 0 and 1 pixels, and run the Life dynamics on it at video rates, then you see a lively churning and boiling pattern of activity, dying down in places to a scattering of small-period oscillating structures, and then being reignited from adjacent areas (Figure~\ref{fig.life}a). If you speed up your simulation to a few hundred frames per second, then typically after less than a minute for a 2K$\times$2K system all of the interesting activity has died out, and the pattern has settled down into a set of isolated small-period oscillators (Figure~\ref{fig.life}b). If you watch the initial activity closely, however, and pick out some of the interesting dynamical structures that arise, you can ``build'' configurations containing constructs such as the ones in Figure~\ref{fig.life}c. These are called {\em glider guns}. When the Life dynamics is applied to a glider gun, at regular intervals the gun spits out a small pattern that then goes through a short cycle of shapes, with the same shape reappearing every few steps in a shifted position. These {\em gliders} are the smallest moving objects in the Life universe. By putting together such constructs, one can show how to build arbitrary logic circuits, using sequences of gliders as the signals that travel around and interact\cite{life,lwod}. This then is our first example of a universal CA rule. Many other non-invertible universal CA rules are known---the simplest is due to Roger Banks\cite{cambook}. All of these can support arbitrary complexity if you rig up a special enough initial state. Life is notable because it spontaneously develops interesting complexity starting from {\em most} initial states. Small structures that do something recognizable occasionally appear briefly, before being sucked back into the digital froth. \section{Invertible CA's are more interesting}\label{sec.rev} One problem with Conway's Life as a model of evolution is that it lasts for such a short time when started from generic initial conditions. For a space of 2K$\times$2K bits, there are $2^{4,194,304}$ distinct possible configurations, and this rule typically goes through fewer than $2^{14}$ of them before repeating a configuration and entering a cycle. This doesn't allow much time for the evolution of complexity! Furthermore, useful computing structures in Life are very fragile: gliders typically vanish as soon as they touch anything. The short Life-time problem can be attributed largely to the non-invertible nature of the Life rule---invertible rules do not behave like this. We typically have no idea just how long the cycle times of our invertible CA's actually are, because we have never seen them cycle, except from very special initial states or on very tiny spaces. The reason that invertible CA's have such long cycle-times is actually the same as the reason that essentially all invertible information dynamics have long cycles: {\em an invertible dynamics cannot repeat any state it has gone through until it first repeats the state it started in.} In other words, if we run an invertible rule for a while, we know what the unique predecessor of every state we have already seen is, except for the first state---its predecessor is still coming up! Thus an invertible system is forced to keep sampling distinct states until it stumbles onto its initial state. Since there is nothing forcing it toward that state as it explores its state space, the cycle of distinct states is typically enormously long: if our invertible CA really did sample states at random without repetition, it would typically have to go through about half of all possible states before chancing upon its initial state\cite{toffoli-four}. A non-invertible system doesn't have this constraint, and can re-enter its past trajectory at any point\cite{kauffman}. The moral here is that if you want to make a discrete world that lasts long enough to do interesting things, it is a good idea to make it invertible. As a bonus, a more thorough exploration of the available state-space tends to make a system more amenable to statistical mechanical analysis. To make it easier to capture physical properties in CA's, we will use a technique called {\em partitioning}, which was developed specifically for this purpose\cite{margolus-bbm,cambook}. This technique is closely related to the sublattice technique introduced in Section~\ref{sec.ising}\cite{ica,margolus-thesis}. The idea of partitioning is to divide up all of the bits in our CA system into disjoint local groupings---each bit is part of only one group. Then we update all of the groups independently, before changing the groupings and repeating the process---changing the groupings allows information to propagate between groups. If the updating of each group conserves the number of 1's, for example, then so does the global dynamics. If the updating applied independently to each group is an invertible function, then the global dynamics is also invertible. Since all invertible CA's can be reexpressed isomorphically in a partitioning format---where conservations and invertibility are manifest---this is a particularly convenient format to use for our models\cite{ica,kari-uncomputable,kari-partitioning}. \figfig{critters-rule}{ \hfill \hbox{% \epsfig{figure=margolus/marg-grid.ps,height=2in}}\hskip 1in \hbox{% \epsfig{figure=margolus/crit-rule.ps,height=2in}}\hfill}% {The invertible ``Critters'' CA. (a)~The solid and dotted blockings are used alternately. (b)~The Critters rule.} Our first example of a partitioned CA is called ``Critters.'' This is a universal invertible CA that evolves interesting complexity. The Critters rule uses a 2$\times$2 block partition on a 2D square lattice. In Figure~\ref{fig.critters-rule}a we show an 8$\times$8 region of the lattice---each square represents a lattice site that can hold a 0 or a 1. The solid lines show the grouping of the bits into 2$\times$2 blocks that is used on the even time-steps, the dotted lines show the odd-time grouping. The Critters rule is shown in Figure~\ref{fig.critters-rule}b. This same rule is used for both the even-time grouping and the odd-time grouping. All possible sets of initial values for the four bits in a 2$\times$2 block are shown on the left, the corresponding results are shown on the right. The rule is rotationally symmetric, so not all cases need to be shown explicitly: each of the four discrete rotations of a block that is shown on the left turns into the same rotation of the corresponding result-block shown on the right. Notice that each of the 16 possible initial states of a block is turned into a distinct result state. Thus the Critters rule is invertible. Notice also that the number of 1's in the initial state of each block is, in all cases, equal to the number of 0's in the result. Thus this property is true for each update of the entire lattice. If we call 1's {\em particles} on even steps, and call 0's {\em particles} on odd steps, then particles are conserved by this dynamics. Notice that the Critters rule also conserves the parity (sum mod 2) along each diagonal of each block, which leads to conservation of parity along every second diagonal line running across the entire space. It is not interesting to run an invertible rule such as Critters starting from a completely random initial state, as we did in the case of Life. This is because the vast majority of all possible states are random-looking and so, by a simple counting argument, almost all of them have to turn into other random-looking states. To see this, note that any given number of steps of an invertible dynamics must turn each distinct initial state into a distinct final state. Since the set of states with recognizable structure is such a tiny subset of the set of all possible states, almost every random-looking state must turn into another random-looking state. \figfig{critters}{ \hfill \fbox{% \epsfig{figure=margolus/critters-a.ps,height=1.5in}}\hskip .2in \fbox{% \epsfig{figure=margolus/critters-m.ps,height=1.5in}}\hskip .2in \fbox{% \epsfig{figure=margolus/critters-close.ps,height=1.5in}}\hfill}% {A Critters simulation. (a)~The initial state of the full 2K$\times$2K lattice. (b)~The state after 1,000,000 steps. (c)~A closeup of a region on the right.} Thus instead of a random state, a ``generic'' initial state for an invertible CA will be some easily generated ``low-entropy'' state---we saw several examples of invertible evolutions from such states in Section~\ref{sec.ising}. For the Critters CA, we show a sample simulation started from an empty 2K$\times$2K lattice with a randomly filled 512$\times$512 block of 0's and 1's in the middle (Figure~\ref{fig.critters}a). In Figure~\ref{fig.critters}b we see the state after 1,000,000 updates of the entire space. In this simulation, opposite edges of the space have been connected together (periodic boundaries). Figure~\ref{fig.critters}c shows a closeup of a region near the right edge of the space: all of the structure present has arisen from collisions of small moving objects that emerged from the central region. In analogy to Life, we will call these small moving objects {\em gliders}. You can see several of these gliders in various phases of their motion in the closeup: they are the compact symmetrical structures composed of four particles, with two particles adjacent to each other, and two slightly separated (see also Figure~\ref{fig.crit-universal}). In the Critters dynamics, a glider goes through a cycle of four configurations each time it shifts by two positions. Unlike the gliders in Life, Critters gliders are quite robust. Consider, for example, what happens when two of these gliders collide in an empty region. At first they form a blob of eight particles that goes through some pattern of contortions. If nothing hits this blob for a while, we always see at least one of the gliders emerge. This property arises from the combination of conservation and invertibility: we can prove, from invertibility, that the blob must break up, but since the only moving objects we can make with eight particles are one or two gliders, then that's what must come out. To see that the blob must break up, we can suppose the opposite. The particles that make up the blob can only get so far apart without the blob breaking up, and so there are only a finite number of possible configurations of the blob. The blob cannot repeat any configuration (and hence enter a cycle of states) because of invertibility: a local cycle of states would be stable going back in time as well as forward, but we know that the blob has to break up if we run backwards past the collision that formed it. Since the blob runs out of distinct configurations and cannot repeat, it must break up. At least one glider must come out. If the collision that formed the blob was rotationally symmetric, then both gliders must come out, since the dynamics is also rotationally symmetric. The robustness of particles that we saw in Figure~\ref{fig.other-ising}a arises in a similar manner. The Critters rule is fascinating to watch because of the complicated structures that form, with swarms of gliders bouncing around within them and slowly altering them. Sometimes, for example, a glider will hit a little flap left from a previous glider collision, flipping it {from} one diagonal orientation to another. This will affect what another glider does when it subsequently hits that flap. Gliders will hit obstacles and turn corners, sometimes going one way, sometimes another, depending on the details of the collisions. The pattern must gradually change, because the system as a whole cannot repeat. After a little observation it is clear that there are many ways to build arbitrary logic out of the Critters rule---one simple way is sketched in the next section in order to demonstrate this rule's universality. Started from an ordered state, the Critters CA accumulates disorder for the same reason that a neat room accumulates disorder: most changes make it messier. As we have already noted, in an invertible dynamics, a simple counting argument tells us that most messy states don't get neater. Localized patterns of structured activity that may arise within this CA must deal with an increasingly messy environment. No such structure in an invertible world can take in inputs that have statistical properties that are unpredictable by it, and produce outputs that are less messy, again because of our counting argument. Thus its fair to call a measure of the messiness of an invertible CA world the total {\em entropy}. We can think of this total entropy as being approximated by the size of the file we would get if we took the whole array of bits that fills our lattice and used some standard compression algorithm on it. It is possible to construct invertible CA's in which a simple initial state turns into a completely random looking mess very quickly. While it is still true that this invertible CA will probably take forever to cycle, it has found another way to end its interesting activity quickly---what we might call a rapid {\em heat death}. Of course heat death is the inevitable fate of any CA evolution that has a long enough cycle: since the vast majority of states are random-looking, very long cycles must consist mostly of such states. We can, however, try to put off the inevitable. In the Critters CA, symmetries and conservation laws act as constraints on the rate of increase of entropy, and so make the interesting low-entropy phase of the dynamics last much longer. It would be interesting to try to capture within CA dynamics other mechanisms that occur in real physics that contribute to metastability and hence delay the heat death. \section{A bridge to the continuum} % already saw this in Ising waves, will see again in HPP % BBM is a bridge with Comp Universality! % can't reduce entropy on purpose Historically, the partitioning technique used in the previous section was first de\-vel\-op\-ed\cite{margolus-bbm} for use in the construction of a very simple universal invertible CA modeled after Fredkin's Billiard Ball Model (BBM) of computation\cite{fredkin-bbm}.\footnote{The first universal invertible CA was actually constructed by Toffoli\cite{toffoli-universal}, who showed how to take a universal 2D CA that was non-invertible, and add a third dimension that would keep a complete time history of the dynamics, thus rendering it invertible.} The BBM is a beautiful example of a continuum physical system that is turned into a digital system simply by constraining its initial conditions and the times at which we observe it. This makes it a wonderful bridge between the tools and concepts of continuum mechanics, and the world of exact discrete information dynamics. This model is discussed in \cite{feynman-comp}, but we will review it very briefly here. \figfig{fred-coll}{ \hfill \hbox{% \epsfig{figure=margolus/bbm-a.ps,height=2in}}\hskip .6in \hbox{% \epsfig{figure=margolus/bbm-coll.ps,height=2in}}\hfill}% {Fredkin's Billiard Ball Model. (a)~Balls heading toward a collision. (b)~Paths taken in collision are displaced from straight paths.} In Figure~\ref{fig.fred-coll}a we show two finite-diameter billiard balls heading toward a collision, both moving at the same speed. Their centers are initially located at integer coordinates on a Cartesian lattice---we will refer to these as {\em lattice points}. At regular intervals, the balls will be found at consecutive lattice points, until they collide. In Figure~\ref{fig.fred-coll}b we show a collision. The outer paths show the actual course that the balls take after a collision; the inner paths illustrate where each of the two balls would have gone if the other one wasn't there to collide with it. Thus we see that a locus at which a collision might or might not happen performs logic: if the presence of a ball at a lattice point at an integer time is taken to represent a 1, and the absence a 0, then we get 1's coming out on the outer paths only if balls at $A$ AND $B$ came in at the same time. The other output paths correspond to other logical functions of the inputs. It is easy to verify that this collision is a universal and invertible logic element (just reverse all velocities to run BBM circuits backwards). We also allow fixed mirrors in our model to help route ball-signals around the system---these are carefully placed so that the centers of balls are still always found at lattice points at integer times. In order to make a simple CA model of the BBM, we will represent finite diameter balls in our CA by spatially separated {\em pairs} of particles, one following the other---the leading edge of the ball followed by the trailing edge. When such a ball collides with something, the front-edge particle collides first, and then the influence is communicated to the trailing edge. This kind of ``no rigid bodies'' approach to collisions is more consonant with the locality of interaction that we are trying to capture in CA's than a larger-neighborhood model in which distant parts of a fixed-size ball can see what's going on right away. \figfig{bbmca}{ \hfill \hbox{% \epsfig{figure=margolus/bbm-rule.ps,height=2in}}\hskip 1in \fbox{% \epsfig{figure=margolus/bbm-circuit.ps,height=2in}}\hfill}% {A Billiard Ball Model CA. (a)~The BBMCA rule. (b)~A BBMCA circuit.} Figure~\ref{fig.bbmca}a shows the BBMCA rule. Like the Critters rule, this rule is rotationally symmetric and so, again, only one case out of every rotationally equivalent set is shown. Note that the rule conserves 1's (particles), and that only two cases change. This is the complete rule that is applied alternately to the even and odd 2$\times$2 blockings. Note that, much like the Ising CA, this rule is its own inverse: if we simply apply the update rule to the same blocking twice in a row, the net effect is no change.\footnote{The Ising CA is actually very closely related. It can be put into the same 2$\times$2 block-partitioned format if we model bonds instead of sites\cite{cambook}.} \figfig{bbmca-coll}{ \hfill \hbox{% \epsfig{figure=margolus/bbmca-a.ps,height=.7in}}\hskip .1in \hbox{% \epsfig{figure=margolus/bbmca-b.ps,height=.7in}}\hskip .1in \hbox{% \epsfig{figure=margolus/bbmca-c.ps,height=.7in}}\hskip .1in \hbox{% \epsfig{figure=margolus/bbmca-d.ps,height=.7in}}\hskip .1in \hbox{% \epsfig{figure=margolus/bbmca-e.ps,height=.7in}}\hskip .1in \hbox{% \epsfig{figure=margolus/bbmca-f.ps,height=.7in}}\hfill}% {A BBMCA collision. We show succesive snapshots of a small area where a collision is happening. In the first image, the solid-blocks are about to be updated. The blocking alternates in successive images.} Figure~\ref{fig.bbmca-coll} shows a BBMCA collision between two minimum-size balls---the gap between the two particles that make up the front and back of the ball can be any odd number of empty sites. Until the balls get close together the particles that form them all propagate independently: a single 1 in one corner of a block moves to the opposite corner. When we change the blocking, the particle again finds itself alone in a block in the same corner it started in, and again moves in the same direction. When two leading-edge particles find themselves in the same block, the collision begins. These particles are stuck for one step---this case doesn't change. Meanwhile the trailing edge particles catch up, each colliding head-on with a leading-edge particle which was about to head back to meet it (if the gap had been wider). New particles come out at right angles to the original directions, due to the ``two-on-a-diagonal'' case of the rule, which switches diagonal. Now one of the particles from each head-on collision becomes the new leading edge particle; these are done with the collision and head away from the collision locus, once again propagating independently of the trailing particles. Meanwhile the two new trailing-edge particles are headed toward each other. They collide and are stuck for one step before reflecting back the way they came, each following along the path already taken by a leading edge particle. Each two-particle ball has been displaced from its original path. If the other two-particle ball hadn't been there, it would have gone straight. Mirrors are built by placing square patterns of four particles straddling two adjacent blocks of the partition. It is easy to verify that such squares don't change under this rule, even if you put them right next to each other. Single particles just bounce back from such mirrors. The collision of a two-particle ball with such a mirror looks just like the collision of two balls that we have already seen; we just replace one of the balls with a mirror whose surface lies along the axis of symmetry of the two-ball collision. The remaining ball can't tell the difference. For more details about the BBMCA, see \cite{cambook,margolus-thesis}. \figfig{crit-universal}{ \hfill \fbox{% \epsfig{figure=margolus/crit-a.ps,height=.7in}}\hskip .1in \fbox{% \epsfig{figure=margolus/crit-b.ps,height=.7in}}\hskip .1in \fbox{% \epsfig{figure=margolus/crit-c.ps,height=.7in}}\hskip .1in \fbox{% \epsfig{figure=margolus/crit-d.ps,height=.7in}}\hskip .1in \fbox{% \epsfig{figure=margolus/crit-e.ps,height=.7in}}\hskip .1in \fbox{% \epsfig{figure=margolus/crit-f.ps,height=.7in}}\hfill}% {A BBMCA-style collision of pairs of gliders in the Critters CA. The images shown are not consecutive states of the lattice, but are instead spaced in time to correspond (with a $45^\circ$ rotation) to the images in the previous figure.} Figure~\ref{fig.bbmca}b shows a BBMCA circuit, computing a permutation sequence. Because of their long cycle times, invertible circuits tend to make good pseudo-random number generators. In fact, a perfect random number generator would go through all of its internal states before cycling, and so it would be perfectly invertible. It is also interesting to use the BBMCA to construct circuits that are more directly analogous to thermodynamic systems, since the constraint of invertibility means that {\em it is impossible to design a BBMCA circuit that, acting on unpredictable statistical inputs that it receives, can reduce the entropy of those data}---for the reasons discussed in the previous section\cite{margolus-thesis,bennett-thermo,bennett-demons}. The BBMCA is simple enough that it provides a good theoretical model to use for other inquiries about connections between physics and computation. For example, one can use its dynamics as the basis of quantum spin models\cite{margolus-pqc}. % async? mention rel? Using Figure~\ref{fig.crit-universal}, we sketch a simple demonstration that the Critters rule of the previous section is universal---we show that pairs of Critters-gliders suitably arranged can act just like the ``balls'' in the BBMCA, which is universal. Figure~\ref{fig.crit-universal} shows a collision that is equivalent to that shown in Figure~\ref{fig.bbmca-coll}. We don't show every step of the Critters time-evolution; instead we show the pairs of gliders at points corresponding to those in the collision of Figure~\ref{fig.bbmca-coll}. Mirrors can be implemented by two single Critters particles, one representing each end of the mirror. \section{Discrete molecular dynamics}\label{sec.md} Having constructed a CA version of billiard ball dynamics, it seems natural to try to construct CA's that act more like real gases\cite{toffoli-pde,margolus-bbm}. With enough particle directions and particle speeds, our discrete Molecular Dynamics (MD) should approximate a real gas. The BBMCA has just four particle directions and a single particle speed. We will make our first MD model by modifying the BBMCA rule. For simplicity, we won't worry about modeling finite-diameter balls: single 1's will be our gas molecules. The simplest such rule would be, ``During each 2$\times$2 block update, each molecule ignores whatever else is in the block and simply moves to the opposite corner of its block.'' Then, when we switch partitions, that molecule would again be back in the same kind of corner that it started in and so it would again move in the same direction---moving exactly like an isolated particle in the BBMCA. This simple rule gives us a non-interacting gas, with four directions and one speed. \figfig{hpp}{ \hfill \hbox{% \epsfig{figure=margolus/hpp-rule.ps,height=1.85in}}\hskip .35in \hbox{% \epsfig{figure=margolus/stone-a.ps,height=1.85in}}\hskip .2in \hbox{% \epsfig{figure=margolus/stone-c.ps,height=1.85in}}\hfill}% {A simple four-direction lattice gas. (a)~A momentum conserving invertible rule. (b)~A 512$\times$512 lattice filled randomly with particles, with a square block of ones's in the center. (c)~A round pressure wave spreads out from the center.} We would like to add a {\em momentum conserving} collision to this non-interacting gas rule. We can begin by defining what we mean by momentum. If we imagine that our discrete lattice dynamics is simply a succession of snapshots of a continuum system, as it was in the case of the BBM, then we automatically inherit definitions of energy and momentum from the continuous system. To add a momentum conserving collision to our simple four-direction gas, we should have two molecules that collide head-on come out at right angles. Figure~\ref{fig.hpp}a shows the non-interacting gas rule with one case modified to add this collision: when exactly two molecules appear on one diagonal of a block, they come out on the other diagonal. We would not expect such a simple model to behave very much like a real gas. In Figure~\ref{fig.hpp}b, we show a 512$\times$512 2D space filled with a random pattern of 1's and 0's, with a square block in the center of the pattern that is all 1's. Figure~\ref{fig.hpp}c shows this system after about 200 updates of the space: we see a round pressure wave. We were amazed when we first ran this simulation in the early 1980's\cite{toffoli-pde}. How could we get such continuous looking behavior from such a simple model? This is the point at which we began to think that perhaps CA MD might be immediately practical for fluid modeling\cite{super,cambook}. Discrete lattice models are well adapted to meshes of locally interconnected digital hardware, which can be very large and very fast if the models are simple. It turns out, though, that this particular model is too simple to simulate fluid flow---though it is useful for other purposes. This four-direction {\em lattice gas automaton} (LGA) is now commonly known as the HPP gas after its originators\cite{hpp}, who analyzed it about a decade before we rediscovered it. Their analysis showed that this four-velocity model doesn't give rise to normal isotropic hydrodynamics. Notice that the HPP gas is perfectly invertible---like the BBMCA, its rule is its own inverse. Thus we can run our pressure wave backwards, getting back exactly to the square block we started from. This doesn't contradict what we said earlier about entropy in invertible CA's, since a messy state can always be cleaned up if you undo {\em exactly} the sequence of actions that produced the mess. Normal forward time evolution doesn't do this. \figfig{refract}{ \hfill \hbox{% \epsfig{figure=margolus/refract-a.ps,height=1.5in}}\hskip .2in \hbox{% \epsfig{figure=margolus/refract-b.ps,height=1.5in}}\hskip .2in \hbox{% \epsfig{figure=margolus/refract-c.ps,height=1.5in}}\hfill}% {A slightly modified gas CA. The vertical bar is a density wave traveling to the right, the circle is a region in which waves are slower. We see the wave reflect and refract.} Once we are able to model one macroscopic phenomenon, it is often obvious how to model other phenomena. Starting from a model with sound waves, we can make a model with reflection and refraction of such waves. In Figure~\ref{fig.refract} we show a simulation using a 2-bit variant of the HPP CA. Here we have added a bit to each site, and used it to mark a circular region of the space: one bit at each site is a {\em gas} bit, and the other bit is a {\em mark} bit. We now alternate the rule with time so that, for one complete even-time/odd-time update of the lattice we apply the HPP rule to the gas bits at all sites; then we do a complete even/odd update only of unmarked blocks, with all gas particles in blocks containing non-zero mark-bits left unchanged. This gives us two connected HPP systems, one running half as fast as the other. In particular, waves travel half as fast in the marked region, and so we get refraction of the wave that is incident from the left. Notice that the dynamics is still perfectly invertible, and that we can make our ``lens'' any shape we desire---it is a general feature of CA MD that we can simply ``draw'' whatever shaped obstacles and potentials we need in a simulation\cite{smith-topol}. Related LGA techniques have been used for complex antenna simulations\cite{simons}. We can model many other phenomena by coupling 2$\times$2 block rules. We can, for example, use the HPP gas or a finite-impact-parameter variant of it (the TM gas\cite{super,cambook}, which has better momentum-mixing behavior) as a source of pseudo-randomness in a diffusion model. We start by again putting two bits at each lattice site---one bit will belong to the {\em diffusing system}, while the other belongs to the {\em randomizing system}. Let the four bits of the randomizing system in each block simply follow the HPP dynamics described above. Let the four bits of the diffusing system be rotated $90^\circ$ clockwise or counterclockwise, depending on the parity of the number of 1's in the four ``random'' bits. This results in a perfectly invertible diffusion in which no more than one diffusing particle can ever land at the same site\cite{cambook}. Using this approach with enough bits at each site we can, for example, model the diffusion and interaction of different chemical species. The HPP CA was originally presented in a different format than we have used above\cite{hpp}. Since this other format is in many cases very natural for discussing MD models, we will describe it here, and then relate it to a partitioned description. We start by putting four bits of state at each site of a square lattice. We will call the bits at the $i^{\rm th}$ site $N_i$, $S_i$, $E_i$ and $W_i$. The dynamics consists of alternating two steps: (1) {\em move} the data, and then (2) let the groups of bits that land at each site {\em interact} separately. The first step moves the data in sheets: if we think of the directions on our lattice as being North, South, East and West, then all of the $N$ bits are moved one position North, all the $S$ bits one position South, etc. Our interaction rule at each site combines the bits that came from different directions and sends them out in new directions. A state consisting of two 1's (particles) that came in from opposite directions and two 0's from the other directions, is changed into a state in which the 1's and 0's are interchanged---the particles come out at right angles to their original directions. In all other cases, particles come out in the same directions they came in. We can think of this as a particular kind of partitioning rule, where the four bits at each site are the groups, and we use the data-movement step to rearrange the bits into new groups. Although in some ways this {\em site-partitioned} description of the HPP gas is simpler, it also suffers from a slight defect. If we imagine, as we did in our discussion of the Ising model, that our lattice is a giant black and white checkerboard, then we notice that in one data movement step all of the bits that land at black squares came from white squares, and vice versa. No data that is currently on a black square will ever interact with data that is currently on a white square: we have two completely independent subsystems. The 2$\times$2 block version of the HPP rule is isomorphic to just one of these subsystems, and so lets us avoid simulating two non-interacting systems. Of course we can also avoid this problem in the site-partitioned version with more complicated time-dependent shifts: we can always reexpress any partitioned CA as a site-partitioned CA. This fact has been important in the design of our latest CA machines. The HPP lattice gas produces a nice round-looking sound wave, but doesn't reproduce 2D hydrodynamics in the large-scale limit. We can clearly make a CA model with more speeds and directions by having molecules travel several lattice positions horizontally and/or vertically at each step---just add more particles at each site, and shift the different momentum fields appropriately during the movement step. With enough speeds and directions, it seems obvious that we can get the right macroscopic limit---this should be very much like a hard-sphere gas, which acts like a fluid. The fact that so many different fluids obey the same hydrodynamic equations also suggests that the details of the dynamics can't matter very much, just the constraints such as momentum and particle conservation. \figfig{fhp2}{ \hfill \hbox{% \epsfig{figure=margolus/fhp2.ps,height=2in}}\hfill}% {Flow past a half-cylinder using a six-direction lattice gas on a triangular lattice. We simulate ``smoke'' streamers to visualize the flow.} So how simple a model can work? It was found\cite{fhp} that we can recover macroscopic 2D hydrodynamics from a model that is only slightly more complicated than the HPP gas. A single-speed model with six particles per site, moving in six directions on a triangular lattice, will do. If all zero-net-momentum collisions cause the molecules at the collision site to scatter into a rotated configuration, and otherwise particles go straight, then in the low speed limit we recover isotropic macroscopic fluid dynamics. Figure~\ref{fig.fhp2} shows a simulation of a slightly more complicated six-direction lattice gas\cite{filga}. The simulation shown is 2K$\times$1K, and we see vortex shedding in flow past a half-cylinder. The white streamers are actually a second gas (more bits per site!), inserted into the hydrodynamic gas as a kind of smoke used to visualize flows in this CA wind tunnel. This is an invertible CA rule, except at the boundaries which are irreversibly being forced (additional bits per site mark the boundaries). Simple single-speed CA's have also been used to simulate 3D hydrodynamics\cite{doolen,rothman-cam}. When it was discovered that lattice gases could simulate hydrodynamic behavior, there was a great deal of excitement in some circles and skepticism in others. The exciting prospect was that by simplifying MD simulations to the point where only the essence of hydrodynamic behavior remained, one could extend the scale of these simulations to the point where interesting hydrodynamics could be done directly with an MD method. This spawned an entire new field of research\cite{doolen,doolen91,torino,nice,waterloo,princeton,boston-lga}. This optimistic scenario has not yet been realized. One problem is that simple single speed models aren't well suited for simulating high-speed flows. As in a photon gas, the sound speed in a single-speed LGA is almost the same as the maximum particle speed, making supersonic flows impossible to simulate. You need to add more particle speeds to fix this. The biggest problem, though, is that you need truly enormous CA systems to get the resolution needed for hydrodynamic simulations with high Reynold's numbers\cite{yakhot}. % Lattice boltzmann % porous flow, complex fluids % lga are higher level of agg than md, lower than cfd \figfig{chem-xtal}{ \hfill \hbox{% \epsfig{figure=margolus/porous.ps,height=1.5in}}\hskip .2in \fbox{% \epsfig{figure=margolus/knot.ps,height=1.5in}}\hskip .2in \hbox{% \epsfig{figure=margolus/xtal.ps,height=1.5in}}\hfill}% {Some CA MD simulations. (a)~Flow through a porous medium. (b)~A topologically complicated structure within a chemical reaction simulation. (c)~Crystallization using irreversible discrete forces.} For the near term, for those interested in practical modeling, it makes sense to avoid high-Reynold's numbers and fast fluid flows, and to use MD CA models to simulate other kinds of systems that are hard to simulate by more conventional numerical techniques. Suitable candidates would include systems for which the best current simulation techniques are in fact some form of molecular dynamics, as well as systems for which there are at present no good simulation techniques because traditional MD cannot reach the hydrodynamic regime. An example would be systems with very complicated flows. Figure~\ref{fig.chem-xtal}a shows a simulated flow through a piece of sandstone. The shape of the sandstone was obtained from MRI imaging of an actual rock, taking advantage of the ability of CA MD simulations to handle arbitrarily shaped obstacles. Shading in the figure indicates flow velocity. Simulations were compared against experiments on the same rock that was imaged, and agreement was excellent\cite{rothman-book,rothman-cam}. More complicated flows, involving immiscible liquids, have been simulated with this same technique. CA models of complex systems can be built up by combining simpler models. We simply pile up as many bits as we need at each lattice site, representing as many fluids, random variables, heat baths, and other fields as we desire. Then we update them in some repeated sequence, including extra steps that make different groups of subsystems interact, much as we did in our diffusion and our refraction examples. For practical purposes we will often dispense with invertibility, and be satisfied with irreversible rules coupled to pseudo random subsystems. Figure~\ref{fig.chem-xtal}b shows an example of a 3D chemical reaction simulation of this sort, which simulates the FitzHugh-Nagumo reaction-diffusion dynamics\cite{kapral-pre}. The knot and its surroundings are composed of two different chemical phases. The connectivity of the knot in conjunction with domain repulsion keeps the knot from shrinking away. Many kinds of multiphase fluids, microemulsions, and other complex fluids have been simulated using related techniques\cite{reaction-diffusion,microemulsion,rothman-book}. We can easily add discrete forces by having particles at a discrete set of vector separations interact. If two such particles are heading away from each other we can point them toward each other and otherwise leave them unchanged---this results in an attraction. This kind of rule isn't invertible, but it is energy and momentum conserving. Figure~\ref{fig.chem-xtal}c shows a 3D crystallization simulation using a potential built up out of such interactions\cite{yepez-crystal}. This is not currently a very practical way to simulate crystals, but this kind of technique is generally useful\cite{apert,yepez-blobs}. For example, the ``smoke'' in Figure~\ref{fig.fhp2} has a weak cohesive force of this kind, which makes the smoke streams thinner. There are many other ways to build new CA MD models. We often appeal to microdynamical analogy, or to simulating ``snapshots'' of a hypothetical continuous dynamics. We can take aspects of existing atomistic models and model them statistically at a higher level of aggregation using exact integer counts and conservations, to avoid any possibility of numerical instability\cite{filga}. We can combine CA's with more traditional numerical mesh techniques, using discrete particles to handle difficult interface regions\cite{interface}. We can adapt various energy-based techniques from statistical mechanics\cite{gunn,microemulsion}. We can also build useful models in a less systematic manner, justifying their use by simulation and careful measurements\cite{rothman-book}. Combining well understood CA MD components to build up simulations of more complex systems is a kind of iterative programming exercise that involves testing components in various combinations, and adjusting interactions. Although there is already a role for CA MD models even on conventional computers, there is a serious mismatch on such machines between hardware and algorithms. If we are going to design MD simulations to fit into a CA format, we should take advantage of the uniformity and locality of CA systems, which are ideally suited to efficient and large-scale hardware realization. \section{Crystalline computers} Computer algorithms and computer hardware evolve together. What we mean by a good algorithm is that we have found some sort of efficient mapping between the computation and the hardware. For example, CA and other lattice algorithms are sometimes ``efficiently'' coded on conventional machines by mapping the parallelism and uniformity of the lattice model onto the limited parallelism and uniformity of word-wide logical operations---so-called ``multi-spin coding.'' This is a rather grotesque physical realization for models that directly mimic the structure and locality of physics: we first build a computer that hides the underlying spatial structure of nature, and then we try to find the best way to contort our spatial computation to fit into that mold! Ultimately all physical computations have to fit into a spatial mold, and so our most efficient computers and algorithms will eventually have to evolve toward the underlying spatial ``hardware'' of nature\cite{toffoli-nature}. Because physical information can travel at only a finite velocity, portions of our computation that need to communicate quickly must be physically located close together. Computer architects can only hide this fact-of-life from us for so long. At some point, if we want our computations to run faster, our algorithms must take on the responsibility of dealing with this constraint. Computer engineers are not unaware of this spatial constraint. Various locally-interconnected parallel computers have been built and studied\cite{ftl}. Mesh architectures are organized like a kind of CA, but usually with a rather powerful computer with a large memory at each lattice site. Unlike CA's, they normally don't have the same operation occurring everywhere in the lattice at the same time. {\em SIMD} or {\em data parallel} mesh machines are more CA-like, since they typically have a simpler processor at each site, and they do normally have the operation of all processors synchronized in perfect lockstep. % This approach of course just mimics what we do in % physics: the laws are uniform, and the fact that different things are % happening in different places we ascribe to the {\em state} of the % system. Another important spatial computing device is the {\em gate array}. These regular arrays of logic elements are very much like a universal CA. Initially, we build these chips with arrays of logic elements, but we leave out the wiring. Later, when we need a chip with a specific functionality, we can quickly ``program'' the gate array by simply adding wires to connect together these elements into an appropriate logic circuit. FPGA's (field programmable gate arrays) make programming the interconnections even easier. What should be connected to what is specified by some bits that we communicate directly to the chip: this rapid transition from bits to circuitry eliminates much of the distinction that is normally made between hardware and software\cite{fccm}. As general purpose computing devices, none of these CA-like machines are significant mainstream technologies---the evolutionary forces that are pushing us in the CA direction haven't yet pushed hard enough. This was even more true when Tom Toffoli and I first started playing with CA's together, almost two decades ago. There were no machines available to us then that would let us run and display CA's quickly enough that we could experience them as dynamical worlds. We became increasingly frustrated as we tried to explore the new and exciting realm of invertible CA's on available graphical workstations: each successive view of our space took minutes to compute. \figfig{cams}{ \hfill \fbox{% \epsfig{figure=margolus/cam6.ps,height=1.9in}}\hskip .4in \fbox{% \scalebox{.864}{\includegraphics*[1.65in,7.15in][3.85in,9.35in]{margolus/tight-grid.ps}}}\hfill}% {CA machines. (a)~Our earliest CA machines scanned their memory like a framebuffer, applying discrete logic to a sequence of neighborhood windows. (b)~CAM-8 uses a 3D mesh array of SIMD processors. Each processor handles part of the lattice, alternating data movement with sequential lookup-table updating of the groups of bits that land at each lattice site.} Tom designed and built our first CA simulation hardware. This CA machine was a glorified frame-buffer that kept the CA space in memory chips. As it scanned a 2D array of pixels out of the memory, instead of just showing them on a screen, it first applied some neighborhood logic to the data, then sent each resulting pixel to the screen while also putting it back into memory. At first the neighborhood logic was on a little prototyping board (this is shown schematically in Figure~\ref{fig.cams}a). I learned about digital circuitry by building site-update rules out of TTL chips---each small logic circuit I built became the dynamical law for a different video-rate universe! Eventually, we switched to lookup tables, which largely replaced the prototype circuits. The first few generations of machines, however, all had wires that you could move around to change things---whenever you wanted to demonstrate something, you would invariably find that someone else had rewired your machine\cite{cam5}. We went through several generations of hardware, playing with models that no one else had ever seen\cite{cambook}. At a certain point we rediscovered the HPP lattice gas, and our simulations rekindled interest in this kind of model. At this point, our machines became inadequate. They had been designed for interacting and experimenting with small 2D systems that you could watch on a monitor. Real CA MD was going to require large-scale 3D machines designed for serious scientific simulation, with provisions for extensive data analysis and visualization. A new dedicated CA MD machine could be about 1000 times as cost effective as existing supercomputers for this task, and would provide the interactivity and flexibility of a personal CA workstation. % virtual % what was original? I designed the new architecture (CAM-8) based on the experience that Tom and I had with our earlier machines\cite{margolus-waterloo}. As shown in Figure~\ref{fig.cams}b, this machine uses a 3D mesh array of SIMD processors running in perfect lockstep. The lattice being simulated (which can be $n$ dimensional) is divided up evenly among the processors, each processor handling an equal {\em sector} of the overall lattice. As in our earlier machines, the state of the lattice is kept in ordinary memory chips, and the updating is done by lookup tables---also memory chips. Data cycles around and around between these two sets of memory chips. Unlike our previous machines, which provided a fixed set of traditional CA neighborhoods, the only neighborhood format supported by CAM-8 is {\em site-partitioning} (discussed in Section~\ref{sec.md}). Any {\em bit field} (i.e., set of corresponding bits, one from every site) can be shifted uniformly across the lattice in any direction. Whatever data land at a given lattice site are updated together as a group. This is a particularly convenient neighborhood format from a modeling point of view, since any CA dynamics on any neighborhood can be accomplished by performing an appropriate sequence of data shifts and site updates, each acting on a limited number of site bits at a time. Also, as we've seen, partitioning is a particularly good format for constructing models that incorporate desired physical properties. Site partitioning is also a very convenient neighborhood format from a hardware standpoint. Since the pattern of data movement is very simple and regular, mesh communication between processors is also very simple. Since the updating is done on each site independently, it doesn't matter what order the sites are updated in, or how many different processors are involved in the updating. All of this can be organized in the manner most efficient for the hardware. CAM-8 machines were built and performed as expected. All of the simulations depicted in this chapter were done on CAM-8, except for that of Figure~\ref{fig.chem-xtal}a (which is similar to \cite{rothman-cam}). CAM-8 has not, unfortunately, had the impact that we hoped it might. First of all, during the time that we were building the machine, it was found that lattice gases weren't as well suited for high Reynold's number hydrodynamic flow simulations as people had hoped. In addition, in the absence of any good CA machines, interest in lattice dynamics calculations had shifted to techniques that make better use of the floating point capabilities of conventional computers. Also, most researchers interested in developing practical applications already had good access to conventional supercomputers, which were 1000 times less cost effective than CAM-8, but had familiar and high-quality software and system support. Finally, the evolutionary forces favoring CA-like machines were temporarily on the wane at the time when CAM-8 was completed, as multiprocessor funding dried up and fine-grained parallel computing companies folded. We didn't build any versions of our indefinitely scalable CAM-8 machine that were large enough to make previously unreachable realms of CA modeling accessible---as our early machines first opened the CA world to us. In the near term, prospects again look good for CA machines. Although our small personal-computer-scale CAM-8 machines are still about as good as any supercomputer for LGA computations, advances in technology make radical improvements possible. By putting logic directly on DRAM memory chips, which is now routinely done, and by exploiting the enormous memory bandwidth that can be made available on-chip, it is possible today to make a SIMD machine similar to the CAM-8 that is over 10,000 times faster {\em per memory chip} than the current CAM-8 hardware\cite{margolus-hsc}. Putting together arrays of such chips, qualitatively new physical simulations will become possible. Other SIMD applications such as logic simulation, image processing and 3D bit-map manipulation/rendering will also run at about a trillion bit-operations per second per chip. Whether we manage to make our next dream machine, the time is ripe for commercial SIMD-based CA machines. What of the more distant future? In the preceding sections, we have emphasized invertible CA's. Aside from their intrinsic interest, they have the virtue that they mimic the microscopic invertibility of physical dynamics. From a macroscopic point of view, this means that these CA's can in principle be implemented using frictionless reversible mechanisms---they don't depend on dissipative thermodynamically irreversible processes in order to operate\cite{frank-thesis,vieri-thesis,younis-knight,younis-thesis}. Thus 3D machines based on invertible CA's won't have the same problem getting rid of dissipated heat that irreversible machines do\cite{bennett-thermo,bennett-fund,feynman-comp}. From a more microscopic point of view, we can see the match between invertible computation and invertible quantum physics as making possible direct use of quantum scale elements and processes to do our computations. We can make use of discrete properties of these quantum elements to represent our digital quantities and perform our digital computations. Thus in the distant future I expect that our most powerful large-scale {\em general purpose} computers will be built out of macroscopic crystalline arrays of identical invertible computing elements. We would make such large arrays out of identical elements because they will then be easier to control, to design, to build and to test. These will be the distant descendants of todays SIMD and FPGA computing devices: when we need to perform inhomogeneous computations, we will put the irregularities into the program, not the hardware. The problem of arranging the pieces of a computation in space will be part of the programming effort: architectural ideas that are used today in physical hardware may reappear as data structures within this new digital medium. With molecular scale computing elements, a small chunk of this {\em computronium}\cite{margolus-fund} would have more memory and processing power than all of the computers in the world today combined, and high Reynold's number CA MD calculations of fluid flow would be practical on such machines. Note that I don't expect our highest performance general purpose computers to be quantum spin computers of the sort discussed in Section~\ref{sec.intro}. In such a machine, the whole computer operates on a superposition of distinct computations simultaneously. This kind of {\em quantum parallelism} is very delicate, and the overhead associated with the difficult task of maintaining a superposition of computations over a large spatial scale will be such that it will only be worth doing in very specialized situations---if it is possible at all\cite{preskill}. This won't be something that we will do in our general purpose computers. \section{What makes a CA world interesting?} Future CA machines will make extensive large-scale CA simulations possible---we will be able to study the macroscopic properties of CA worlds that we design. Aside from issues of size and speed, there doesn't seem to be any obvious reason why exact classical information models cannot achieve as high a level of rich macroscopic complexity as we see in our universe. This is a very different modeling challenge than trying to simulate quantum mechanics with CA's. We would like to simulate an interesting macroscopic world which is built out of classical information. It is instructive to try to see what the difficulties might be. The most important thing in our universe that makes it interesting to us is of course {\em us}. Or more generally, the existence of complex organisms. Thus let's begin by seeing what it might take to simulate a world in which Darwinian evolution is {\em likely} to take place. Since no one has yet made a CA that does this, our discussion will be quite speculative. One of the most successful computer models of evolution is Tom Ray's Tierra\cite{ray1,ray2}, which was designed to capture---in an exact digital model---an essential set of physical constraints abstracted {from} biology. His model did not include spatial locality or invertibility, but we could try to add these features. % mention conservations and metastability Modeling evolution in a robust spatial fashion may, however, entail incorporating some physical properties into our CA systems that are not so obvious\cite{bridge}. For example, in nature we have the property that we can take complicated objects and set them in motion. This property seems to be essential for robust evolution: it is hard to imagine the evolution of complex organisms if simpler pieces of them can't move toward each other! No known universal CA has this property (but see \cite{chopard-book,apert,yepez-blobs}). There is nothing in the Life CA, for example, that corresponds to a glider-gun in motion. The general property of physics that allows us to speak about an object at rest and then identify {\em the same object} in motion is {\em relativistic invariance}. The fact that the laws of physics look the same in different relativistic frames means that we can have the same complex macroscopic objects in all states of motion: an organism's chemistry doesn't stop working if it moves, or if the place it lives moves! In a spatial CA dynamics, some sort of spatial macroscopic motion invariance would clearly make evolution more likely. Since our CA's have a maximum speed at which information can travel---a finite speed of light---relativistic invariance is a possible candidate. Full relativistic invariance may be more than we need, but it is interesting to ask, ``Can we incorporate relativistic invariance into a universal CA model?'' We have already seen that we can have macroscopic rotational invariance in our lattice gas models, and we know that numerical mesh calculations of relativistic dynamics are possible. Thus achieving a good approximation of relativistic invariance in the macroscopic limit for an exact CA model seems possible\cite{hrgovcic}. Such a system would, at least in the limit, have the conservations associated with the continuous Lorentz group of symmetries. Although it is not possible to put a continuous symmetry directly into a discrete CA rule, it is certainly possible to put these conservations into the rule, along with a discrete version of the symmetries---just as we did in our lattice gas models.\footnote{In continuum physics, continuous symmetries are regarded as fundamental and conservations arise as a consequence of symmetry. Fredkin has pointed out that in discrete systems, it must be the conservation that is fundamental.} Thus we might imagine our relativistically invariant CA to be a refinement of lattice gases---we would also like to make it invertible for the reasons discussed in Section~\ref{sec.rev}. But we also demand that this CA incorporate computation universality. This may not be easy: since a relativistically invariant system must have momentum conservation, we will need to worry about how to hold complex interacting structures together. Thus we may need to incorporate some kind of relativistically invariant model of forces into our system. Simulating forces in an exact and invertible manner is not so easy, particularly if we want the forces to last for a long time\cite{yepez-thesis}. Models in which forces are communicated by having all force-sources continuously broadcast field-particles have the problem that the space soon fills up with these field-particles---which cannot be erased because of local invertibility---and then the force can no longer be communicated. Directly representing field gradients works better, but making this work in a relativistic context may be hard. At this point, we might also begin to question our basic CA assumptions. We introduced crystalline CA's to try to emulate the spatial locality of physics in our informational models, but we are now discussing modeling phenomena in a realm of physics in which modern theories talk about extra dimensions and variable topology. Perhaps whatever is essential fits nicely into a simple crystalline framework, but perhaps we need to consider alternatives. We could easily be led to informational models in which the space and time of the microscopic computation becomes rather divorced from the space and time of the macroscopic phenomena. \smallskip We started this section with the (seemingly) modest goal of using a CA to try to capture aspects of physics necessary for a robust evolution of interesting complexity, and we have been led to discuss incorporating larger and larger chunks of physics into our model. Perhaps our vision is too limited, and there are radically different ways in which we can have robust evolution in a spatial CA model. Or perhaps we can imitate nature, but cheat in various ways. We may not need full relativistic invariance. We may not need exact invertibility. On the other hand, it is also perfectly possible that we can't cheat very much and still get a system that's nearly as interesting as our world. \section{Conclusion} I was in high school when I first encountered cellular automata---I read an article about Conway's ``Game of Life.'' At that time I was intensely interested in both Physics and Computers, and this game seemed to combine the two of them. I immediately wondered if our universe might be a giant cellular automaton. The feeling that physics and computation are intimately linked has remained with me over the years. Trying to understand the difficulties of modeling nature using information has provided an interesting viewpoint from which to learn about physics, and also to learn about the ultimate possibilities of computer hardware. I have learned that many properties of macroscopic physics can be mirrored in simple informational models. I have learned that quantum mechanics makes both the amount and the rate-of-change of information in physical systems finite---all physical systems do finite information processing\cite{margolus-speed}. I have learned that the non-separability of quantum systems makes it hard to model them efficiently using classical information---it is much easier to construct quantum spin computer models\cite{lloyd-sim}. I have learned that physics-like CA models can be the best possible algorithms when the computer hardware is also adapted to the constraints and structure of physical dynamics. I have learned that developing computer hardware that promotes this viewpoint can consume an enormous amount of time! Since classical information is much easier to understand than quantum information I have mostly studied classical CA models. In these systems, a macroscopic dynamical world arises from {\em classical} combinatorics. Continuous classical-physics behavior can emerge in the large-scale limit. We can try to model and understand (and perhaps teach our students about) all sorts of physical phenomena without getting involved in quantum complications. We can also try to clarify our understanding of the fundamental quantities, concepts and principles of classical mechanics {\em and of classical computation} by studying such systems\cite{fredkin-comp}. The principle of stationary action, for example, must arise in such systems solely from combinatorics---there is no underlying quantum substratum in a CA model. Conversely, we should remember that information (in the guise of entropy) was an important concept in physics long before it was discovered by computer scientists. Just as Ising-like systems have provided intuitive classical models that have helped clarify issues in statistical mechanics, CA's could play a similar role in dynamics. Since we have focused so much on discrete classical CA models of physics, it might be appropriate to comment briefly on their relationship to discrete quantum models---Feynman's quantum spin computer of Section~\ref{sec.intro}. Exactly the same kinds of grouping and sublattice techniques that we have used to construct invertible CA's also allow us to construct quantum CA's---QCA's\cite{margolus-qc}. We simply replace invertible transformations on groups of bits with unitary transformations on groups of spins. Just as it is an interesting problem to try to recover classical physics from ordinary CA's, it is also interesting to try to find QCA's that recover the dynamics of known quantum field theories in the macroscopic limit\cite{boghosian-qc,meyer-qlga,meyer-boston,yepez-thesis}. Following our Ising CA example, it might be instructive to investigate classical CA's that are closely related to such QCA's. Although people have often studied CA's as abstract mathematical systems completely divorced from nature, ultimately it is their connections to physics that make them so interesting. We can use them to try to understand our world better, to try to do computations better---or we can simply delight in the creation of our own toy universes. As we sit in front of our computer screens, watching to see what happens next, we never really know what new tricks our CA's may come up with. It is really an exploration of new worlds---live television from other universes. Working with CA's, anyone can experience the joy of building simple models and the thrill of discovering something new about the dynamics of information. We can all be theoretical physicists. \section{Acknowledgments} Much of what I know of this subject I learned from Edward Fredkin, Tommaso Toffoli and Charles Bennett. They were my close collaborators in the MIT Information Mechanics Group and my dear friends. They provided me with the support and encouragement I needed to begin work in this field. It was a privilege to be a part of their group. Richard Feynman has been a constant source of inspiration and insight in my physics career. His {\em Lectures on Physics}\cite{feynman-lec} sustained me through my undergraduate years, and I spent some wonderful months as a graduate student visiting with him at CalTech and giving lectures on CA's and on reversible computing in his course on computation\cite{feynman-comp}. I have many fond memories of his warmth, charm, humor and {\em joie de vivre}. I learned much from conversations with Mark Smith, Mike Biafore, Gerard Vichniac and Hrvoje Hrgov\u{c}i\'{c}---colleagues in the MIT IM Group. I have also learned much from my colleagues in the lattice gas field who have helped me in my research, particularly Jeff Yepez, who has been a close collaborator on CA research with the CAM-8 machine. The construction of CAM-8 itself was a large effort that involved many people---including Tom Toffoli, Tom Durgavich, Doug Faust, Ken Streeter and Mike Biafore---each of whom contributed in a vital way. My current physics/computation related research has been made possible by the support, collaboration and encouragement of Tom Knight. This manuscript has benefitted greatly from discussions with and comments by Raissa D'Souza, Lov Grover, David Meyer, and Ilona Lappo. Thanks to Dan Rothman for providing Figure~\ref{fig.chem-xtal}a, Ray Kapral for providing Figure~\ref{fig.chem-xtal}b, and Jeff Yepez for providing Figure~\ref{fig.chem-xtal}c. Figure~\ref{fig.other-ising}b comes from a simulation written by Tom Toffoli and Figure~\ref{fig.same} comes from an investigation that I'm working on jointly with Raissa D'Souza and Mark Smith. Support for this work comes from DARPA contract DABT63-95-C-0130, the MIT AI Lab's Reversible Computing Project, and Boston University's Center for Computational Science. \section*{Notes on the references} Much of the material in this chapter is discussed at greater length in \cite{margolus-thesis} and \cite{cambook}. These documents were both strongly influenced by ideas and suggestions from Edward Fredkin, some of which are also discussed in \cite{fredkin-bbm,fredkin-dm,fredkin-comp,fredkin-nature,fredkin-cosmogony,fredkin-perspective}. Some recent books on CA modeling of physics are \cite{rothman-book} and \cite{chopard-book}. Many of the early lattice-gas and quantum computing papers are reproduced in \cite{doolen} and \cite{lloyd} respectively. Recent related papers can be found online in the {\tt comp-gas} and {\tt quant-ph} archives at {\tt http://xxx.lanl.gov}, and cross-listed there from other archives at LANL such as {\tt chao-dyn}. 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Feynman, {\em Feynman Lectures on Computation,} edited by J. G. {\sc Hey} and R. W. {\sc Allen} (Addison-Wesley, 1996). \bibitem{frank-thesis} M.~P. Frank, ``Reversibility for efficient computing,'' MIT Ph.D. Thesis (1999). \bibitem{fhp} U. Frisch, B. Hasslacher and Y. Pomeau, ``Lattice-gas automata for the navier-stokes equation,'' {\em Phys.\ Rev.\ Lett.} {\bf 56}, 1505--1508 (1986). \bibitem{endicott} E. Fredkin, R. Landauer and T. Toffoli (eds.), {\em Proceedings of the Physics of Computation Conference}, in {\em Int.\ J. Theor.\ Phys.}, issues {\bf 21}:3/4, {\bf 21}:6/7, and {\bf 21}:12 (1982). \bibitem{fredkin-bbm} E. Fredkin and T. Toffoli, ``Conservative logic,'' in \cite[p.\ 219--253]{endicott}. \bibitem{fredkin-dm} E. Fredkin, ``Digital mechanics: an informational process based on reversible universal CA,'' in \cite[p.\ 254--270]{gutowitz}. \bibitem{fredkin-comp} E. 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