CAM-8: a computer architecture based on cellular

automata∗

Norman MargolusMIT Laboratory For Computer Science

Cambridge Massachusetts 02139

December 15, 1993

Abstract

The maximum computational density allowedby the laws of physics is available only in aformat that mimics the basic spatial localityof physical law. Fine-grained uniform compu-tations with this kind of local interconnectiv-ity (Cellular Automata) are particularly goodcandidates for efficient and massive micro-physical implementation.Conventional computers are ill suited to run

CA models, and so discourage their develop-ment. Nevertheless, we have recently seenexamples of interesting physical systems forwhich the best computational models are cel-lular automata running on ordinary comput-ers. By simply rearranging the same quantityand quality of hardware as one might find ina low-end workstation today, we have made alow-cost CA multiprocessor that is about asgood at large CA calculations as any existingsupercomputer. This machine’s architecture isscalable in size (and performance) by orders ofmagnitude, since its 3D spatial mesh organi-zation is indefinitely extendable.Using a relatively small degree of paral-

lelism, such machines make possible a level

∗This research was supported by the Advanced Re-search Projects Agency, grant N0014-89-J-1988.

of performance at CA calculations much su-perior to that of existing supercomputers, butvastly inferior to what a fully parallel CA ma-chine could achieve. By creating an interme-diate hardware platform that makes a broadrange of new CA algorithms practical for realapplications, we hope to whet the appetiteof researchers for the astronomical computingpower that can be harnessed in microphysicsin a CA format.

1 Introduction

Within the Information Mechanics Group atthe MIT Laboratory for Computer Science,a primary focus of our research has been onthe question, “How can computations andcomputers best be adapted to the constraintsand opportunities afforded by microscopicphysics?” This has led us to study spatiallyorganized computations, since the maximumcomputational density allowed by the lawsof physics is available only in a format thatmimics the basic spatial locality of physicallaw. Fine-grained uniform computations withthis kind of local interconnectivity (CellularAutomata) are particularly good candidatesfor efficient and massive micro-physical imple-mentation.

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We have been involved for over a decadein the design and use of machines optimizedfor studying Cellular Automata (CA) compu-tations ([42, 23, 46, 30, 28]). This involvementbegan in response to our need for more power-ful CA simulation tools—suitable for investi-gating the large-scale behavior of CA systems.Using our early CA machines (cams) we de-veloped a number of new CA models and mod-eling techniques for physics and for spatially-structured computation ([46]). Eventually we“published” a commercial version of our CAmachines, along with a collection of models assoftware examples ([46, 24]).Some of our earliest models were reversible

lattice gases that simulated a billiard-ball com-puter ([23]). It was a natural step to usethese and related lattice gases to try to sim-ulate fluid flow ([43]). Although only the lin-ear hydrodynamics worked correctly (see Fig-ure 8), our cam simulations made Pomeau andothers realize that lattice gases were not justconceptual models, but might be turned intopowerful computational tools (cf. the seminal“FHP” lattice-gas paper[12], and our compan-ion paper[30]).The design of our latest CA machine, cam-

8, builds upon our accumulated experiencewith previous cellular automata machine de-signs, and represents both a conceptual andpractical breakthrough in our understandingof how to efficiently simulate CA systems([28, 27]). This new machine is an indefinitelyscalable three-dimensional mesh-network mul-tiprocessor optimized for large inexpensivesimulations, rather than for ultimate perfor-mance. Our small-scale prototype—with anamount and kind of hardware comparable tothat in a low-end workstation—already per-forms a wide range of CA simulations atspeeds comparable to the best numbers re-ported for any supercomputer ([3, 18, 31, 53]).Machines orders of magnitude bigger and pro-portionately faster can be built immediately.

Most of the current exploration of cellularautomata as computational models for scienceis being done using machines which were de-signed for very different purposes. Such exper-imentation doesn’t make apparent the tremen-dous computational power that is potentiallyavailable to models tailored for uniform arraysof simple processors. Nevertheless, we alreadyhave seen examples of interesting physical sys-tems for which the best computational mod-els are cellular automata running on ordinarycomputers (cf. [17, 35, 36]). Cam-8—using arelatively small degree of parallelism—makespossible a level of performance at CA calcula-tions much superior to that of existing super-computers, but vastly inferior to what a fullyparallel CA machine could achieve. By cre-ating an intermediate hardware platform thatmakes a broad range of new CA algorithmspractical for real applications, we hope to whetthe appetite of researchers for the astronomi-cal computing power that can be harnessed inmicrophysics in a CA format.

2 An architecture based on

cellular automata

In nature, we have a uniform and local lawin the world that is operating everywhere inparallel. A CA model is a synchronous digitalanalog of such a law. As a basis for a com-puter architecture, CA’s have the advantagethat there can be a direct mapping betweenthe computation and its physical implementa-tion: a small region of the computer can im-plement a small region of the CA space, andadjacent regions of physical space can imple-ment adjacent regions of the CA space. Thuslocality is preserved, and very efficient realiza-tions are in principle possible. This efficiency,however, comes at the cost of requiring that allmodels run on the machine must be spatially

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lookup

SRAM

table

DRAM

cell

array

Figure 1: Cam-8 system diagram. (a) A single processing node, with dram site data flowingthrough an sram lookup table and back into dram. (b) Spatial array of cam-8 nodes, withnearest-neighbor (mesh) interconnect (1 wire/bit-slice in each direction).

organized. Thus the unavoidable problem ofultimately making your computation fit intoa uniform and local physical world is shiftedinto the software domain: you must directlyembed your software problems into a uniformand local spatial matrix.

Cam-8 is a parallel computer built on thisspatial paradigm. For technological conve-nience, it time-shares individual processorsover “chunks” of space—and also time-sharesthe wires connecting each processor with itsneighboring processors. The time-sharing ofcommunication resources reduces the numberof interprocessor wires dramatically and thusallows the scalability that is inherent in theCA paradigm to be practically achieved usingcurrent technology, even in three dimensions.The time-sharing of processors allows a highlyefficient “assembly-line” processing of spatialdata, in which exactly the same operations arerepeated for every spatial site in a predeter-mined order.

From the viewpoint of the programmer, thisvirtualization of the spatial sites is not appar-ent: you simply program the local dynamics

in a uniform CA space.

2.1 System overview

Figure 1 is a schematic diagram of a cam-8 system. On the left is a single hardwaremodule—the elementary “chunk” of the archi-tecture. On the right is an indefinitely ex-tendable array of modules (drawn for conve-nience as two-dimensional, the array is nor-mally three-dimensional). A uniform spatialcalculation is divided up evenly among thesemodules, with each module simulating a vol-ume of up to millions of fine-grained spatialsites in a sequential fashion.In the diagram, the solid lines between mod-

ules indicate a local mesh interconnection.These wires are used for spatial data move-ment. There is also a tree network (not shown)connecting all modules to the front-end work-station that controls the CA machine. Theworkstation uses this tree to broadcast sim-ulation parameters to some or all modules,and to read back data from selected modules.Normally, the parameters of the next updat-

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ing scan of the space are broadcast while thecurrent scan is in progress, and analysis datafrom the modules are also read back while thecurrent scan runs.Each module contains a separate copy of

the current program for updating the space(data transformation parameters, data move-ment parameters, etc.), and all modules oper-ate in lockstep. This allows both the compu-tation within modules and communication be-tween modules to be pipelined, so that one vir-tual processor within each module completesits update (including all communication) ateach machine clock.Spatial site data is kept in conventional

dram chips which are all accessed continu-ously in a predictable and optimized scan or-der, achieving 100% utilization of the availablememory bandwidth. Within a module, eachdram chip belongs to a separate bit-slice, andeach dram chip has its address controlled sep-arately from the rest. The group of bits thatare scanned simultaneously (one bit from eachbit-slice) constitute a hardware cell. Data isreshuffled between hardware cells by control-ling the relative scan order of the dram bit-slices.Updating is by table lookup. Data comes

out of the cell-array, is passed through alookup table, and put back exactly where itcame from (Figure 1a). The lookup tables aredouble buffered, so that the front-end work-station can send a new table while the cam-modules are busy using the previous table toupdate the space. There are also hardwareprovisions for replacing the lookup tables withpipelined logic (to allow versions of cam-8with a large number of bits in the hardwarecell—too many to update by table lookup),and for connecting external data sources oranalysis hardware.There are only a handful of connections be-

tween modules—one per bit-slice to each ofthe six adjacent modules. Uniform data shifts

across the entire three-dimensional space areachieved by combining dram address manip-ulation with static routing ([25, 19])—data aresent over the intermodule wires at preordainedtimes, exactly when they are needed by adja-cent modules.

2.1.1 A sample implementation

For comparison purposes, here is a descriptionof the amount and kind of technology used inone of our prototype 8-module cam-8 units:

• System clock: 25 MHz

• DRAM: 64Megabytes (4 Megabit chips,70ns)

• SRAM: 2 Megabytes (256 Kilobit chips,20ns)

• Logic: about 2 Million gates total

• Logic technology: 1.2 micron CMOS

This level of technology is comparable towhat is used in a low-end workstation—a smallcam-8 unit is really a CA personal computer.1

For CA rules with one bit per site, this 8 mod-ule machine runs simulations at a rate of about3 billion site updates per second on spaces ofup to half a billion sites; with 16 bits per site,simulations run at about 200 million site up-dates per second on spaces of up to 32 mil-lion sites. Several of our 8-module prototypescan be connected together to construct biggermachines—repackaging the modules would bedesirable for constructing substantially largermachines.

1In comparing the performance of this unit againstnumbers reported for simulations on supercomputers(which have a similar performance) one should alsotake availability into account: a personal computercan be run on a single problem for a very long pe-riod of time. Economies of scale (mass production) arealso potentially available to personal-computer levelhardware.

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Our cam-8 prototype can directly accumu-late and format data for a real-time video dis-play; provision is also made to accept data di-rectly from a video camera, in order to allowcam-8 to perform real-time video processingwith CA rules. For a detailed description ofthe prototype cam-8 implementation, includ-ing the cam-8 register model, the workstationinterface, and system configuration and initial-ization, see “STEP: a Space Time Event Pro-cessor ([29]).”

2.2 Programmer’s model

In addition to more specialized resources hav-ing to do with display, analysis, and I/O, themain programmable resources in cam-8 are:

• Number of dimensions.• Size and shape of the space.• Number of bits at a site.• Initial state of the space.• Directions and distances of datamovement.

• Rules for data interaction.

All of these parameters are normally spec-ified as part of a cam-8 experiment. Oftenthe data movement and data interaction willchange with time, either cyclically or progres-sively as the simulation runs: the overhead as-sociated with changing these parameters be-fore every update of the space is negligible.

2.2.1 The space

Our earlier cam machines were all 2-dimensional, with severe restrictions on theoverall size of the space and the number ofbits at each spatial site. In cam-8, these pa-rameters may be freely specified.

The overall space-array is configured asa multi-dimensional Cartesian lattice with achosen size, shape, number of bits per site, andnumber of dimensions. The boundaries areperiodic—if you move from site to site alongany dimension, you eventually get back to yourstarting point. Three of the dimensions canbe arbitrarily extended by adding “chunks” ofhardware (modules). The maximum numberof bits in the array is of course governed by thetotal amount of storage in all of the modules(64 Megabytes in our prototype): each mod-ule processes an equal fraction of the overallspace-array. There is no architectural limit onhowmany modules a cam-8 machine can have.

2.2.2 Data movement

In earlier cam machines, there were severeconstraints on neighborhoods: restrictions onwhich data from sites near a given site couldbe seen by the CA update rule acting at thatsite. In cam-8, we have eliminated these con-straints. This was accomplished by abandon-ing the use of traditional CA neighborhoods,and basing our machine on the kind of datapartitioning characteristic of lattice gas mod-els. Instead of having a fixed set of neighbor-hood data visible from each site, we shift dataaround in our space in order to communicateinformation from one place to another.In traditional CA rules, each bit at a given

site is visible to all neighbors. In contrast,the pure data-movement used in cam-8 sendseach bit in only one direction. Informationfields move uniformly in various directions,each carrying corresponding bits from everyspatial site along with them—in two dimen-sions think of uniformly shifting bit-planes, inhigher dimensions bit-hyperplanes.2 Interac-

2The term information field is a bit of a pun, sincewe intend by this both the computer science meaning,namely a fixed set of bits in every “record” (spatialsite), and also the physics meaning of a field, which is

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tions act separately on the data that land ateach lattice point, transforming this block ofdata into a new set of bits. If some piece ofdata needs to be sent in two directions, theinteraction must make two copies.There is a constraint on how far bit-fields

can move in one updating step, but it is quitemild. Each bit-field can independently shiftby a large amount in any direction—the max-imum shift-component along each dimensionis one that would transfer the entire sector ofa bit-field contained in one hardware moduleinto an adjacent module. For a two dimen-sional simulation on our prototype, for exam-ple, the x and y offsets for each bit-field thatcan be incorporated as part of a single updat-ing step can be any pair of signed integers withmagnitudes of up to a few thousand. In general(for any number of dimensions), each updatingevent brings together a selection of bits chosenfrom the few million neighboring sites.

2.2.3 Data interaction

Data movement and data interaction alter-nate: once we have all the data in the rightplace, we update each site using only the infor-mation present at that site.3 In our prototype,there is a constraint that only 16 bit-fields canbe moved in independent directions simultane-ously, and only 16 bits at a time can interactand be updated arbitrarily (by table lookup).4

a number attached to each site in space.3Actually, the hardware does both movement and

updating in a single pipelined operation.4Alternative implementations (using the same cam

data-movement chips) would allow many more simul-taneously moving bit-fields, but would use pipelinedlogic in place of lookup tables, since tables grow insize exponentially with the number of inputs. Suffi-ciently wide programmable logic can perform any de-sired many-input function if there are enough levels oflogic; an arbitrary number of levels can be simulatedby changing the program for the logic from one scanof the space to the next. An interesting application ofthis would be for efficiently running lattice gases with

Thus a program for this machine consists ofa sequence of specifications of (wide ranging)particle-like data movements and (arbitrary)16-bit interaction events. Simulations involv-ing the interaction of large numbers of bits ateach site have to be broken down into a se-quence of 16-bit events—a space-time eventprogram.

3 Applications

Cam-8 is good at spatially moving data, andat making the data interact at lattice sites.This makes it well suited for simulating phys-ical systems using lattice-gas-like dynamics.This also makes it appropriate for a wide rangeof other spatially organized calculations in-volving localized interactions.We are actively collaborating with several

groups to develop sample applications whichillustrate the use of this CA machine for phys-ical simulations (e.g., fluid flow, chemical re-actions, polymer dynamics), two and three di-mensional image processing (e.g., documentreading, medical imaging from 3D data), andlarge logic simulations (including the simu-lation of highly parallel CA machines). Ofcourse all of the models developed for our ear-lier cam machines run well on cam-8 ([46]),and can now be extended far beyond the ca-pabilities of these earlier machines. Many spa-tial algorithms (pipelined, systolic, simd, etc.)designed for other machines ([19, 21]) can alsobe adapted to this architecture.As illustrations of the use of cam-8, some

sample applications and simulation techniquesare discussed below. All of these exampleshave been developed on the prototype machinediscussed in Section 2.1.1, and performancefigures are for this workstation-scale device.

large numbers of bits per site (cf. [20]).

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Figure 2: Flows for two simulations using the FHP lattice gas. (a) Von Karman streets,(b) Kelvin-Helmholtz shear instability.

3.1 Lattice gases

Cam-8 is at heart a lattice gas machine. Parti-cle streaming is an efficient, low-level hardwareoperation, and the large spatial data shiftsavailable make it convenient to investigatemodels with widely varying particle speeds.Multi-dimensional shifts are useful for investi-gating models with shallow extra dimensions.Our most advanced lattice gas collaboration

is with Jeff Yepez and his group at the U.S. AirForce’s Phillips Laboratory. He and PhillipsLabs have started a new initiative on geophys-ical simulation that involves the constructionof a large cam-8 machine.Geophysical phenomena are good candi-

dates for lattice dynamics modeling since thereis so much distributed complexity involved,and since many of these phenomena are sohard to model using traditional differentialequation techniques. With lattice gases, thesimulation runs just as fast with the most com-plex boundary condition as with the simplest.One can use a great deal of physical intuitionin incorporating desired properties into modelsby constructing simplified discrete versions ofthe actual physical dynamics. The process ofmaking these models is closely akin to that of

making models in statistical mechanics, whereone strives to include only the essence of thephenomenon ([52]).Figures 2 and 3 illustrate some simple

“warmup” calculations done in collaborationwith Yepez that illustrate the use of cam-8’sstatistics gathering hardware. Here, we splitthe system up into bins of a chosen size and uselookup tables to count a function of the stateof the sites in each bin. These event counts arecontinuously reported back to the workstationthat is controlling the simulation.Figure 2a shows momentum flow in a two-

dimensional 2K×1K lattice, illustrating vortexshedding in lattice-gas flow past a flat plate.Here we use a 7-bit “FHP” model, whichruns on our prototype at a rate of 382 mil-lion site updates per second (for pure simula-tion). Both the time averages (over 100 steps)and the space averages (over 32×32 sites) wereaccumulated by cam; the workstation simplydrew the arrows.Similarly, Figure 2b uses the same model to

illustrate a Kelvin-Helmholtz shear instabilityon a 4K×2K lattice. Most of the fluid wasinitially set in motion at Mach 0.4 to the right,except for a narrow strip in the middle which

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was started with the opposite velocity. TheFigure shows the situation after 40,000 timesteps (about 15 minutes of simulation). Theaveraging is over regions of 128×128 sites, andover 50 time steps.

Cold Plate

Gra

vity

Hot Plate

Figure 3: Rayleigh-Benard convection.

Finally, Figure 3 illustrates Rayleigh-Benard convection, following Shi-Yi Chen([4]). The simulation uses a 13-bit hexago-nal lattice-gas, with 3 particle speeds, heating(at the bottom), cooling (at the top), wallsaround the box, and gravity. The simulationsize is 1024×512, and the prototype runs thisat a rate of 191 million site updates per sec-ond. The time and space averaging was doneby cam as in the previous figures.

We have also been working with BruceBoghosian and Dan Rothman on three dimen-sional lattice gas models. Since the CM-2 alsohas 16-bit lookup tables, the “random isom-etry” techniques that were used to partitionlattice-gas updates into a composition of 16-bit lookups on the Connection Machine carryover directly to cam-8 ([8, 2]): a 24-particleFCHC lattice gas with solid boundaries runsat about 7 million site-updates per second. Weare using these techniques as the basis for im-plementing simulations of the flow of immisci-ble fluids through porous media ([36]).

3.2 Statistical mechanics

Physicists have long used discrete models instatistical mechanics to model material sys-tems. In simulating such systems it is of-ten important to have available large quanti-ties of precisely controllable random variables.By independently applying large random spa-tial shifts to each of a few randomly filled bitfields (and other related techniques), it is pos-sible to avoid local correlations and continu-ously generate high quality random variableswithout slowing the simulation down. Usingsuch random variables, we have run three di-mensional thermalized annealing models ([49])on our 8-module prototype at about 200 mil-lion site-updates per second on a space of 16million 16-bit sites (about 12 updates of the3-dimensional space per second), with simul-taneous rendering (by discrete ray tracing aspart of the CA dynamics) and display. Fig-ure 4a shows one rendered image from thecam-8 display for a 512× 512× 64 simulation.Figure 4b shows a two-dimensional diffusion

limited aggregation simulation on a 1024 ×1024 space, driven by random variables. Thesystem shown is started with a single fixedparticle in the center of about 100,000 ran-domly diffusing particles. Whenever a diffusertouches a fixed particle, it becomes fixed atthat position. This is a large version of acam-6 experiment ([46]), but run more thantwo orders of magnitude faster than cam-6could have run it. The simulation performsabout 800 million site updates per second, andthe Figure shows the state of the system afterabout two minutes of evolution.Figure 4c shows another statistical simula-

tion: a CA polymer model due to Yaneer Bar-Yam ([31, 39]; the cam-8 program was writ-ten by Michael Biafore). This discrete modelcaptures certain essential features of polymers:conservation of the total number of monomers,preservation of connectivity, monomers can’t

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Figure 4: Cam-8 used as “programmable matter.” (a) Spongy three-dimensional structureobtained by “majority” annealing. (b) Diffusion limited aggregation. (c) Polymers diffusingfrom an initial concentrated region. (d) Typical texture produced by one of Griffeath’s large-neighborhood voting rules.

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overlap (excluded volume), etc. It employsa statistical dynamics (controlled by cam-8random variables) that uses space-partitioningto maintain these constraints ([23]). Thesimulation discussed in [31] ran at a rate ofabout 30 million site updates per second ona space 512 × 512. Problems that are beingaddressed with these models include dynam-ics in polymer melts, gelation and phase sep-aration, polymer collapse, and pulsed field gelelectrophoresis ([40]).Figure 4d shows a deterministic simulation

of a model due to David Griffeath at the Uni-versity of Wisconsin. He and some of his col-laborators are engaged in the analysis of com-binatorial mathematics problems that havespatial locality. They have been using our ear-lier, much more limited cam-6 machine in thiscapacity for a number of years ([13]). The sim-ulation shown is a kind of annealing rule: eachsite in the space (512×512) takes on whatevervalue is in the majority in its neighborhood.The neighborhoods are quite large—they in-volve the 121 neighbors in an 11 × 11 regionsurrounding each site. Since there are 5 dif-ferent species (3 bits of state), the updatingrule must deal with 363 bits of state in eachneighborhood. This is done as a compositionof about 70 distinct updating steps, and so weget about 10 complete updates of the spaceper second (about 2.5 million site updates persecond). A better algorithm, that doesn’t re-calculate the species-counts for overlapping re-gions of adjacent neighborhoods, would run anorder of magnitude faster. In either case, thisexample serves to illustrate how rules that in-volve the interaction of large numbers of bitsat each site are handled by composing updat-ing steps.5

5At the opposite extreme of few bits per site, the“Bonds Only” ([46]) version of Michael Creutz’s dy-namical Ising model ([6]) is a 1-bit per site partitioningrule that runs at a rate of about 3 billion site updatesper second on our prototype.

Cam-8 is designed to numerically analyzethe models run on it—largely through the useof the event counters ([46]). By appropriatelyaugmenting the system dynamics with extradegrees of freedom, we can make essentiallyany desired property of the system quanti-tatively visible. For example, localized spa-tial averages (such as density, pressure, en-ergy density, temperature, magnetization den-sity) can be gathered as we did to producethe momentum flows in the previous section;global correlation statistics can be accumu-lated quickly for occurrences of given spatialpatterns; autocorrelations can be computedby comparing the system to a copy of it-self shifted in time and/or space ([30]); andblock-spin transformations can be quickly per-formed, simplifying renormalization group cal-culations of critical exponents.

3.3 Data visualization and imageprocessing

Another area we’ve been exploring is two- andthree-dimensional image processing. We wereled into this area initially by the display needsof our physical simulations (e.g., see Figure 4a,discussed above), but this activity has takenon a life of its own.Our cam-8 machine simulates a kind of

raster-scan universe, in which each hardwaremodule sequentially scans its chunk of theoverall simulation space. This raster scancan in fact be programmed to be two di-mensional, and synchronized and interfacedwith an external video source. The neces-sary hardware is included as part of our pro-totype, and allows us to perform realtimeimage processing. Generic bit-map process-ing/smoothing/improving techniques are sup-ported through a combination of local (CA)operations and global statistics gathering viathe hardware event counters ([33]). Well-known CA image-processing algorithms, such

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as those used commercially for locating andcounting objects in images, can also be runefficiently ([34, 41, 50]).Many novel algorithms are also directly sup-

ported by the architecture. For example, the8-node prototype can rotate a 512 × 512 bit-map image through an arbitrary angle in lessthan 10 milliseconds by permuting the ar-rangement of the pixels to move every pixelto within one pixel-width of its best possiblerotated position. Figure 5a shows camera in-put of a closeup of the cam-8 chip (the semi-custom chip that knits memory chips togetherinto a CA machine). Figure 5b shows the sameimage rotated by cam-8 through an angle of35 degrees ([32]).6

In three dimensions, local CA techniquescan be used to find and to smooth two-dimensional surfaces to be visualized. For ex-ample, magnetic resonance imaging can pro-duce three-dimensional arrays of spatial den-sity data that subsequently need to be visu-alized. Interesting features might be the sur-face of the brain, the surfaces of lesions, bloodvessels, etc. Local rules can be used to tracefeatures (e.g., blood vessels are regions con-nected to segments that have already beenidentified as blood vessels) and to smooth sur-faces (e.g., using annealing rules that have sur-face tension). Once a surface has been dis-tinguished, many bit-map oriented renderingtechniques are available. The simplest is prob-ably the same one used in Figure 4a: just sim-ulate “photons” of light moving from site tosite, entering the system from one direction,and being observed from another. Figure 6

6This same kind of technique is applicable in othercontexts. For example, a matrix transpose can be ac-complished by a 90 degree rotation and a flip—thiscombined operation on cam-8 takes the same time asthe rotation alone. Some of our collaborators (BryantYork and Leilei Bao at Northeastern University) areperforming combinatorial searches on cam-8 by apply-ing these kinds of techniques to large multidimensionalmatrices.

shows the surface of the brain generated fromMRI data, and rendered by such a technique.The two images are rotated versions of thesame data—we can actually do an arbitrarythree-dimensional rotation of site data usingthe same technique used in Figure 5 in justthree updating scans of the space ([45]).If you render a surface twice, once from

each of two slightly separated vantage points,you can quickly produce stereo pairs. Wehave tested this technique7 in some of ourphysical simulations: we have run a versionof the three-dimensional annealing simulationpictured in Figure 4a while continuously gen-erating such stereo pairs, without slowing thesimulation down at all. Using this techniqueto generate images from many vantage points,one can quickly generate data needed for pro-ducing holograms from computer volumetricdata.8

3.4 Spacetime circuitry

Cam-8 can rapidly perform not only arbitraryrotations, but also affine transformations onits data—the hardware can skip or repeatedlyscan sites during updating in order to rescalean image. Actually, we can do far betterthan this: cam-8 can perform arbitrary rear-rangements of bits, with any set of local, non-uniform operations along the way. To get anarbitrary transformation, you simply simulatethe right logic circuit!A digital logic circuit is a physical system

that (not surprisingly) can be simulated ef-ficiently by a (digital) CA space. Figure 7ashows a straightforward simulation of logic us-ing cam-89. Here we have a CA space that

7Mike Biafore led this effort8Cam-8 should also be useful for reconstructing

three-dimensional surfaces from holographic data. Thealgorithm implemented by the horn machine ([54])should run faster on our cam-8 prototype than on thespecial-purpose horn hardware.

9The circuit shown is due to Ruben Agin.

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Figure 5: Continuous rotations of realtime video data.

Figure 6: Continuous 3D rotations of MRI data.

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0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

t

x

Figure 7: Logic simulation. (a) Gate-array-like CA simulation of a random number generator.(b) Spacetime wires reverse the bits in a 1-dimensional space.

simulates a kind of sea-of-gates gate-array,with one gate at each spatial site. Local rout-ing information recorded at each site deter-mines how data hops between bit-fields thatshift in various directions, in order to im-plement the wires that connect the gates to-gether. Large three dimensional logic simu-lations can be performed by cam-8 in thismanner: just as with other spatially orga-nized computations, the kind of virtualizationof spatial sites (gates here) that cam-8 doesmakes such simulations practical.10 The in-vestigation of CA rules that permit efficientlogic simulation is also important for highly-parallel fixed-rule CA hardware: if the fixedrule supports logic simulation, then the ma-chine can simulate any other CA rule by tiling

10Since cam-8 shares each processor over up to afew million spatial sites, much higher performance spe-cialized machines with a lower virtualization ratio canbe made to implement specific cam-8 rules—such as alogic simulation rule, or an image processing rule. Youtrade flexibility (large spatial shifts and large lookuptables) and simulation size for speed. Notice that evenif fpga’s are used for implementing these specializedmachines, very high silicon utilization ratios can beachieved, since the regular structure of a CA mapsnaturally onto the regular structure of an fpga.

the simulation space with appropriate blocksof logic.11

Now consider the problem of producingrather general transformations of the data inour CA space. One approach would be to di-rectly simulate a gate-array-like rule that op-erates on the original data, and eventually pro-duces the transformed data. An efficient tech-nique for doing this on cam-8 is called space-time circuitry. This involves adding an extradimension to your system to hold the transfor-mation circuitry, laid out as a pipeline in whicheach stage is evaluated only once, as the datapasses through it ([19, 1]).As a simple example, consider a 1-

dimensional space where the desired transfor-mation is to reverse the order of the data bitsacross the width of the space. We add a di-mension (labeled t in Figure 7b) and draw acircuit that accomplishes the reversal—in thissimple case, we only need wires. The circuitshown is a data pipeline that copies informa-

11The idea of using CA’s to do logic is quite old. Infact, much of the present work on field programmablegate arrays carries forward ideas that originated inearly work on CA’s (cf. [48, 16, 22, 38].)

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tion up one row at each stage, and possiblyover by one position right or left: the infor-mation about which way the data should go isstored locally. The cam-8 rule that achievesthis transformation only involves 5-bit sites—two bits of stationary routing information, andthree shifting bit-fields to transport the sig-nals. If we continually add new informationat the bottom of the picture, reversed datacontinually appears, with a 10-stage propaga-tion delay, at the top. But if we only wantto accomplish the transformation once, thenwe only need to update each consecutive rowof the circuit once, performing the data move-ments between row updates. In this case, in-stead of one update of the space moving thewhole pipeline forward by one stage, the rowby row update will move one set of data allthe way through the pipeline!12 We still getone result per update of the space (exactly asbefore), but the propagation delay has beenreduced to a single scan of the space! Thusgiven a CA space, by adding a dimension con-taining a sufficiently complicated pipelined cir-cuit, any desired transformation of the orig-inal space can be achieved in one scan ofthe augmented space—limited only by the to-tal amount of space available for the extra-dimensional circuitry.If the problem we’re interested in is the sim-

ulation of a clocked logic circuit, this techniquecan be used to greatly speed up the simulation.Instead of updating our CA space over andover again while signals propagate around thesystem, passing through gates and eventuallybeing latched in preparation for the next clockcycle, we can pipeline this calculation using anextra dimension, and perform the entire clockcycle in a single update of the space. Sincethe total volume of space (number of sites)

12The rendering algorithm of Figure 4a uses essen-tially this technique to propagate the light all the waythrough the material system in a single scan of thespace.

needed to represent all of the gates and wiresshould be comparable to the volume withoutthe pipeline dimension,13 this represents anenormous speedup. If we think of the rout-ing and gate information that is spread out inthe pipeline dimension as being spread out intime, then we greatly reduce the space neededfor the calculation by making what happensat each spot time dependent—hence the termspacetime circuitry.An additional benefit of spacetime circuitry

on cam-8 is that it allows us to take goodadvantage of the large spatial shifts that areavailable in this architecture. In the logic ex-ample, we could use big spatial shifts at somestages of the pipeline, and smaller ones atother stages, in order to route all signals in asfew stages as possible—this provides a furtherspeedup of the simulation. Of course thesesorts of techniques (extra dimensions and bigshifts) will not be applicable to fully paral-lel CA machines built at the most microscopicscale, but they add greatly to the power andflexibility of our virtual-processor implemen-tation.

4 Software

During the design of the cam-8 asic, wedecided to implement a version of the soft-ware that would drive the real hardware, anduse that to drive complete system simulationsof the cam-8 hardware, including the work-station interface hardware. Thus when thehardware arrived, we immediately had soft-ware that would drive it, and could run thesame tests that we used to validate the design.This initial software was intentionally rather

low level, since it was necessary to have lowlevel access and control to thoroughly and ef-

13Since routing signals in a higher dimension is gen-erally much easier than in a lower dimension, the cir-cuit should actually be more compact.

14

new-experiment 512 by 512 space

0 0 == north 1 1 == south

2 2 == east 3 3 == west

define-rule hpp-rule north south = east west = and

if east <-> north west <-> south then

end-rule

define-step hpp-step lut-data hpp-rule

site-src lut

lut-src site

kick north field -1 y

south field 1 y

east field 1 x

west field -1 x

run new-table

end-step

Figure 8: Sample experiment.

ficiently drive gate-level simulations that raneight orders of magnitude slower than a singlemodule of the actual hardware. The present(still rather rudimentary) cam-8 systems soft-ware has been built as several layers on top ofthis initial work. It provides a prototypicalprogramming environment for cam-8 whichdemonstrates how to access and control allfacets of the hardware.

4.1 A high level machine lan-guage

For simple CA models running on regular crys-tal lattices, the mapping between the modeland the cam-8 architecture is quite direct.14

To illustrate this direct mapping for thesimplest lattice gas model, Figure 8 shows acam-8 assembly language program for run-ning the HPP lattice gas ([15]). This programtranslates into about a dozen cam-8 machine-language instructions to be broadcast to cam.It has two main parts: a rule definition, anda definition of what constitutes an updatingstep. The updating step broadcasts the rule,adjusts some cam data paths, specifies someuniform data movements of the four bit-fieldsused to transport particles, and initiates a scan

14Embedding any regular lattice into cam’s Carte-sian lattice generally involves combining several adja-cent sites of the original lattice into one cam site.

of the space. Despite being at such a low level,this program runs without change on a ma-chine with any number of modules.15 Issuessuch as making the data move smoothly acrossmodule boundaries are handled directly by thehardware.Figure 8 also shows a “snapshot” from the

cam-8 display of a sound pulse resulting whenthis exact code is run from an initial pattern ofrandom particle data with a cavity (a 64×64particle vacuum) in the center.

4.2 Zero-module scalability

The cam-8 machine language is directly inter-preted by the hardware interface that residesin the workstation that controls cam. Thislanguage forms a sharp and simple boundarybetween the software and the hardware—allinteraction gets funneled through this inter-face. A software simulator of cam-8 has onlyto correctly interpret this machine languagein order to be compatible with all higher levelsoftware written for cam.Since the cam-8 architecture depends so

heavily on data movement by pointer manip-ulation and updating by table lookup, it is in

15Utilities that download initial patterns and thatmanage the video display are not shown here—the ac-tions of these routines depend explicitly on the numberof modules.

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fact very well suited to direct software simu-lation on serial machines. A functionally ac-curate software simulator of cam-8 has beenconstructed for the Sun SPARCstation whichruns CA models about as fast as the best ex-isting CA simulators for that machine—as fastas simulators that are not burdened with theconstraint of also simulating cam-8 function-ality ([37]).This property that cam-8 simulations have

of running well even in a pure software contextwe sometimes refer to as zero-module scalabil-ity. Efficient simulability on a variety of par-allel and serial architectures should encouragethe use of the cam-8 machine model as a stan-dard for CA work—which would make othercam-8 software efforts much more widely use-ful. Applications developed on faithful soft-ware simulators (and on small cam-8 instal-lations) will be directly transferable to largecam-8 machines when faster or more massivesimulations are needed.

4.3 Programming environment

For specific applications, it will be the sim-ulation context that defines the “high level”programming environment. For a logic simula-tion, the high level environment might includehardware description languages, logic synthe-sizers, chip-model libraries, etc. For a fluidsimulation, the high level environment mightallow one to “design” a wind-tunnel, obstacles,probes, etc. In general, one needs facilities forconveniently producing interesting initial con-ditions, for visualizing the state of the system,for monitoring and analyzing the progress ofthe simulation, etc. Our task here is to pro-vide examples, utilities, and “hooks” to facil-itate the construction and integration of suchenvironments.For developing models, one great simplifi-

cation has been the sharing of code that ispossible between models that employ a sim-

ilar spatial format. For example, we have con-structed a set of libraries that specialize thecam-8 machine to run cam-6 style neighbor-hoods on fixed-sized two-dimensional spaces.This allows generic mechanisms for display,analysis, etc. to be shared, allowing the pro-grammer to concentrate on developing models.These libraries serve both to allow the exper-iments and experience of cam-6 to be appliedrapidly to this new domain and to allow usersto develop applications in a simplified and welldocumented context. The library routines alsoserve as examples of how to directly programcam-8 itself.The task of providing high-level tools for

model development has barely begun. Someof the work involves only software engineer-ing: for example, writing good compilers thatcan automatically partition a rule on sites withmany bits into a composition of 16-bit opera-tions would be a valuable aid (cf. [11]). Com-pilers that can perform specified transforma-tions on a space by constructing spacetime cir-cuitry would be similarly valuable. Access toarithmetic array operations directly on cam-8 would be useful not only for model build-ing, but for model analysis. High level de-bugging tools that let one quickly compare amodel’s behavior against expectations are es-sential. Where adequate models exist, workneeds to be done on parameterizing knownmodeling techniques and ways of combiningmodels.

4.4 Theoretical challenges

Ideally, one would like to be able to specify avery high level description of a physical sys-tem, and have software use some set of corre-spondence rules to generate an efficient, fine-grained CA model of that system. In gen-eral, we don’t know how to do this. Presentmodeling techniques are rather ad hoc, andthe best progress has been made by “dressing

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up” lattice gases by adding additional particlespecies and interactions, resulting in complexmodels with large numbers of bits at each site.Such models are ill suited to an ultimate goalof harnessing fine-grained, high-density micro-physical systems for CA computations. Fur-thermore, there are at present no fine-grainedCA models of many basic physical phenomena,such as motion of an elastic-solid, long-rangeforces, or relativistic effects.We know that more general methods of con-

structing models are possible. For example,the numerical integration of a finite differenceequation is actually a type of CA computation,and it can reproduce a differential equation.This correspondence, however, yields a ratherrestricted class of CA rules, constrained to useonly arithmetic operations and large numbersof bits at each site. Without these constraints,other general methods may be possible whichyield much simpler CA rules that also repro-duce a desired macroscopic dynamics—rulesbetter suited to high-density microphysical im-plementation. Finding such general methodsis an open problem.Many basic questions remain in the develop-

ment and analysis of CA models, and progresson their resolution will both facilitate, and befacilitated by, the use of CA machines.

5 Conclusions

By exploiting the uniformity of a virtual pro-cessor simulation of fully parallel CA hard-ware, we were able to make workstation-class hardware outperform supercomputers formany CA simulation tasks. Using the sametechnology, a new generation of largescale CAmachines becomes possible that will make en-tirely new classes of spatially organized com-putations practicable. Our aim in all of thishas been to promote the development of CAmodels that can begin to harness the astro-

nomical computing power that is available, ina CA format, in microphysics.As stated, this goal is directed toward bring-

ing computational models closer to physics inorder to improve computation, not physics.But computational models that match wellwith microphysics also tell us something aboutthe structure of information dynamics inphysics. Since a finite physical system has afinite entropy, not only computer science butalso physics itself must deal with the dynam-ics of finite-information systems at increas-ingly microscopic scales ([26]). Thus it seemspossible that promoting the development ofphysics-like computational models will one daycontribute to the conceptual development ofphysics itself.

Acknowledgments

Many people contributed to the successful con-clusion of the cam-8 hardware project. First Iwould like to thank our funding agency, arpa,which has strongly supported our CA machineresearch for over a decade. I would also liketo acknowledge the intellectual debt that thismachine owes to Tom Toffoli, who built thefirst cam machine and had the basic idea oftime-sharing a lookup-table processor over alarge array of cells. He was also a close collab-orator in helping realize this design for a newCA machine, along with Michael Biafore (in-tensive functional testing), Tom Cloney (CM-2 simulator), Tom Durgavich (CAM chip de-sign), Doug Faust and David Harnanan (in-terface design), Nate Osgood (backplane), Mi-lan Shah (SPARC simulator), Mark Smith(early circuit prototyping), and Ken Streeter(project programmer, and also prototypedpart of our SBus interface). Harris Gilliam(project programmer), Frank Honore and KenMacKenzie (board design), and David Zaig(technician) have been helping get copies of

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our prototypes into shape for our collabora-tors. In addition, I would like to thank BillDally and Tom Knight (MIT AI Lab) for ad-vice and design reviews; John Gage, BruceReichlen and Emil Sarpa (Sun Microsystems)for help and encouragement; and Mike Der-touzos and Al Vezza (MIT LCS) for their sup-port. Also, I’d like to thank Jonathan Babband Russ Tessier (LCS) for pointing out to methe applicability of ideas about static routingto logic simulations on cam-8, and Gill Pratt,John Pezaris, and Steve Ward (LCS) for ideasabout how to actually build a large 3D mesh.Finally, I’d like to thank Mark Smith and JeffYepez for comments on this manuscript.

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