Cellular Automata for Imaging, Art, and Video

The techniques known as Cellular Automata (CA) can be used to create a variety of visual effects. As the state space for each cell, 24-bit photo realistic color was used. Several new state transition rules were created to produce unusual and beautiful results, which can be used in an interactive program or for special effects for images or videos. This chapter presents a technique for applying CA rules to an image at several different levels of resolution and recombining the results. A “soft” artistic look can result. The concept of “targeted” CAs is introduced. A targeted CA changes the value of a cell only if it approaches a desired value using some distance metric. This technique is used to transform one image into another, to transform an image to a distorted version of itself, and to generate fractals. The author believes that the techniques presented can form the basis for a new artistic medium that is partially directed by the artist and partially emergent. Images and animations from this work are posted on the World Wide Web at (http://www.scruznet.com/~hughes/CA.html). All cellular automata (CA) operate on a space of discrete states. The simplest CAs, such as the Game of Life, use a 1-bit state space. Most modern personal computers represent color as a 24-bit value, allowing for approximately 16 million possible colors. The work presented in this chapter uses a 24-bit color space that is represented in a 32-bit-long integer. This color space can be conceptualized as a three-dimensional bounded continuous vector space. Often, it is desirable to work with in the HSV (Hue, Saturation, Value) color space. Some of the rules encode the value (luminance) of a cell in the otherwise unused 8 high-order bits of a 32-bit word. The hue and saturation can be estimated “on the fly” with simple, fast algorithms. The hue is represented as an angle on the color wheel. For some rules, it is necessary to know the “distance” between two colors. Estimating the distance in perceptual space would be a difficult problem, as it would be dependent on the monitor used and the gamma exponent applied for a particular setup.

Constructive Molecular Dynamics Lattice Gases: Three-Dimensional Molecular Self-Assembly

Realistic molecular dynamics and self-assembly is represented in a lattice simulation where water, water-hydrocarbons, and water-amphiphilic systems are investigated. The details of the phase separation dynamics and the constructive self-assembly dynamics are discussed and compared to the corresponding experimental systems. The method used to represent the different molecular types can easily be expended to include additional molecules and thus allow the assembly of more complex structures. This molecular dynamics (MD) lattice gas fills a modeling gap between traditional MD and lattice gas methods. Both molecular objects and force fields are represented by propagating information particles and all microscopic interactions are reversible. Living systems, perhaps the ultimate constructive dynamical systems, is the motivation for this work and our focus is a study of the dynamics of molecular self-assembly and self-organization. In living systems, matter is organized such that it spontaneously constructs intricate functionalities at all levels from the molecules up to the organism and beyond. At the lower levels of description, chemical reactions, molecular selfassembly and self-organization are the drivers of this complexity. We shall, in this chapter, demonstrate how molecular self-assembly and selforganization processes can be represented in formal systems. The formal systems are to be denned as a special kind of lattice gas and they are in a form where an obvious correspondence exists between the observables in the lattice gases and the experimentally observed properties in the molecular self-assembly systems. This has the clear advantage that by using these formal systems, theory, simulation, and experiment can be conducted in concert and can mutually support each other. However, a disadvantage also exists because analytical results are difficult to obtain for these formal systems due to their inherent complexity dictated by their necessary realism. The key to novelt simpler molecules (from lower levels), dynamical hierarchies are formed [2, 3]. Dynamical hierarchies are characterized by distinct observable functionalities at multiple levels of description. Since these higher-order structures are generated spontaneously due to the physico-chemical properties of their building blocks, complexity can come for free in molecular self-assembly systems. Through such processes, matter apparently can program itself into structures that constitute living systems [11, 27, 30].

Still Life Theory

In Conway’s Game of Life [2], if one starts with a large array of randomly set cells, then after around twenty thousand generations one will see that all motion has died down, and only stationary objects of low period remain, providing a final density of about .0287. No methods are known for proving rigorously that this behavior should occur, but it is reliably observed in simulations. This brings up several interesting related questions. Why does this “freezing” occur? After everything has frozen, what is the remaining debris composed of? Is there some construction that can “eat through” the debris? If we start with an infinitely large random grid, so that all constructions appear somewhere, what will the long term behavior be? It seems clear that knowing the composition of typical debris is central to many such questions. Much effort has gone into analyzing the objects that occur in such stationary debris, as well as into determining what stationary objects can exist at all in Life [4, 8], Both of these endeavors depend on having some notion of what an “object” is in the first place. One simple notion is that of an island, a maximal set of live cells connected to each other by paths of purely live cells. But many common objects, such as the “aircraft carrier,” are not connected so strongly. They are composed of more than one island, but we think of them as a single object anyway, since their constituent islands are not separately stable. Any pattern that is stable (has period one, i.e., does not change over time) is called a still life. Since a collection of stable objects can satisfy this definition, the term strict still life is used to refer to a single indivisible stable object, and pseudo still life is used to refer to a stable pattern that is composed of distinct strict still lifes. For example, the bi-block is a pseudo still life, since it is composed of two blocks, but the aircraft carrier is a strict still life, since its islands are not stable on their own.

A Two-Dimensional Cellular Automaton Crystal with Irrational Density

We solve a problem posed recently by Gravner and Griffeath [4]: to find a finite seed A0 of 1s for a simple {0, l}-valued cellular automaton growth model on Z2 such that the occupied crystal An after n updates spreads with a two-dimensional asymptotic shape and a provably irrational density. Our solution exhibits an initial A0 of 2,392 cells for Conway’s Game Of Life from which An cover nT with asymptotic density (3 - √5/90, where T is the triangle with vertices (0,0), (-1/4,-1/4), and (1/6,0). In “Cellular Automaton Growth on Z2: Theorems, Examples, and Problems” [4], Gravner and Griffeath recently presented a mathematical framework for the study of Cellular Automata (CA) crystal growth and asymptotic shape, focusing on two-dimensional dynamics. For simplicity, at any discrete time n, each lattice site is assumed to be either empty (0) or occupied (1). Occupied sites after n updates grows linearly in each dimension, attaining an asymptotic density p within a limit shape L: . . . n-1 A → p • 1L • (1) . . . This phenomenology is developed rigorously in Gravner and Griffeath [4] for Threshold Growth, a class of monotone solidification automata (in which case p = 1), and for various nonmonotone CA which evolve recursively. The coarse-grain crystal geometry of models which do not fill the lattice completely is captured by their asymptotic density, as precisely formulated in Gravner and Griffeath [4]. It may happen that a varying “hydrodynamic” profile p(x) emerges over the normalized support L of the crystal. The most common scenario, however, would appear to be eq. (1), with some constant density p throughout L. All the asymptotic densities identified by Gravner and Griffeath are rational, corresponding to growth which is either exactly periodic in space and time, or nearly so. For instance, it is shown that Exactly 1 Solidification, in which an empty cell permanently joins the crystal if exactly one of its eight nearest (Moore) neighbors is occupied, fills the plane with density 4/9 starting from a singleton.

Universal Cellular Automata Based on the Collisions of Soft Spheres

Fredkin’s Billiard Ball Model (BBM) is a continuous classical mechanical model of computation based on the elastic collisions of identical finite-diameter hard spheres. When the BBM is initialized appropriately, the sequence of states that appear at successive integer time steps is equivalent to a discrete digital dynamics. Here we discuss some models of computation that are based on the elastic collisions of identical finite-diameter soft spheres: spheres which are very compressible and hence take an appreciable amount of time to bounce off each other. Because of this extended impact period, these Soft Sphere Models (SSMs) correspond directly to simple lattice gas automata—unlike the fast-impact BBM. Successive time steps of an SSM lattice gas dynamics can be viewed as integer-time snapshots of a continuous physical dynamics with a finite-range soft-potential interaction. We present both two-dimensional and three-dimensional models of universal CAs of this type, and then discuss spatially efficient computation using momentum conserving versions of these models (i.e., without fixed mirrors). Finally, we discuss the interpretation of these models as relativistic and as semiclassical systems, and extensions of these models motivated by these interpretations. Cellular automata (CA) are spatial computations. They imitate the locality and uniformity of physical law in a stylized digital format. The finiteness of the information density and processing rate in a CA dynamics is also physically realistic. These connections with physics have been exploited to construct CA models of spatial processes in Nature and to explore artificial “toy” universes. The discrete and uniform spatial structure of CA computations also makes it possible to “crystallize” them into efficient hardware [17, 21]. Here we will focus on CAs as realistic spatial models of ordinary (nonquantum- coherent) computation. As Fredkin and Banks pointed out [2], we can demonstrate the computing capability of a CA dynamics by showing that certain patterns of bits act like logic gates, like signals, and like wires, and that we can put these pieces together into an initial state that, under the dynamics, exactly simulates the logic circuitry of an ordinary computer.

Phase Transition via Cellular Automata

The dynamics of unit-charged graphs under iterated local majority rule observed in Moran [2] strongly suggested to me a phase-transition phenomenon. In a correspondence with D. Ruelle on this matter in late 1993, he expressed his feelings that the connection was too vague and that temperature was absent in it. This note is a reproduction of my 1993 response, where I try to force my suggestive feelings into a bit more formal frame. A recent work of Yuval Ginosar and Ron Holzman [1], which extends Moran [2], allows us to replace the definition of a solid, given in section 4, by a sharper one, namely that of a “puppet” in their terminology. This means that in section 4 we may define a G ∈ Y to be a solid if every initial charge upon it decays under these dynamics—possibly in infinite time—into a time-periodic charging of a time period not longer than two. This note suggests an approach to the phenomenon of phase transition based on the behaviour of some cellular automata on infinitely countable nets, as noted recently in Moran [2]. Specifically, we use a majority automaton operating simultaneously on a countably infinite graph as a test device determining its “phase.” Results in Moran [2] suggest some sharp partition of a configuration space made up of the totality of such graphs into “solids,” where the only periods allowed for the automaton are 1 or 2, versus the others. Results in Moran [2] allow also the introduction of a “temperature” functional—a numerical parameter defined for each configuration, with the property that a configuration is “solid” whenever its “temperature” is negative. We first describe a possible physical interpretation of such a model, taking the nodes of a graph to be “particles” (stars, electrons, ions, atoms, molecules, radicals—as the case may be) in some Riemannian manifold. Our interpretation is obviously open to a wide diversity of modifications. It is hoped that in spite of its admittedly speculative nature, it may invoke a novel approach to the theoretical treatment of phase transition.

Continuous-Valued Cellular Automata in Two Dimensions

We explore a variety of two-dimensional continuous-valued cellular automata (CAs). We discuss how to derive CA schemes from differential equations and look at CAs based on several kinds of nonlinear wave equations. In addition we cast some of Hans Meinhardt’s activator-inhibitor reaction-diffusion rules into two dimensions. Some illustrative runs of CAPOW, a. CA simulator, are presented. A cellular automaton, or CA, is a computation made up of finite elements called cells. Each cell contains the same type of state. The cells are updated in parallel, using a rule which is homogeneous, and local. In slightly different words, a CA is a computation based upon a grid of cells, with each cell containing an object called a state. The states are updated in discrete steps, with all the cells being effectively updated at the same time. Each cell uses the same algorithm for its update rule. The update algorithm computes a cell’s new state by using information about the states of the cell’s nearby space-time neighbors, that is, using the state of the cell itself, using the states of the cell’s nearby neighbors, and using the recent prior states of the cell and its neighbors. The states do not necessarily need to be single numbers, they can also be data structures built up from numbers. A CA is said to be discrete valued if its states are built from integers, and a CA is continuous valued if its states are built from real numbers. As Norman Margolus and Tommaso Toffoli have pointed out, CAs are well suited for modeling nature [7]. The parallelism of the CA update process mirrors the uniform flow of time. The homogeneity of the CA update rule across all the cells corresponds to the universality of natural law. And the locality of CAs reflect the fact that nature seems to forbid action at a distance. The use of finite space-time elements for CAs are a necessary evil so that we can compute at all. But one might argue that the use of discrete states is an unnecessary evil.

Emerging Markets and Persistent Inequality in a Nonlinear Voting Model

Since it is one of the few spin systems that can be studied analytically, the Voter Model has been extensively discussed in the interacting particle systems literature. In the original interpretation of this model, voters choose their political positions with probabilities equal to the voting frequency of their friends. One of the main results is that, in one and two dimensions, the system clusters—i.e., converges to a homogeneous steady state—while heterogeneity can persist only in dimensions higher than two. This chapter develops an economic model that is similar to the Voter Model, in that agents decide between economic positions, conditional on the economic choices of their trade partners. The choices considered here are market production and nonmarket production, where the payoffs associated with market production for a given agent are a function of the amount of market goods produced by others. Intuitively, the more people are producing for the market, the more potential trade partners exist, hence the higher the expected payoff associated with market production. Similarly, the smaller the extent of the market, the lower the expected gains from trade, hence the smaller the incentive to produce for the market. When each agent is assumed to have an equal probability of trading with any other agent in his or.her trade network, the payoffs associated with market production are linearly increasing in the network’s total market output. However, this linearity in payoffs does not necessarily imply that the conditional probability of working for the market is linearly increasing in total market production, as the Voter Model would have it. As it turns out, this follows only if agents believe, mistakenly, that their trade partners will decide to work for the market with a probability that is exactly proportional to their current market output. Clearly, a more general approach is obtained by allowing for different types of expectations agents may have about the production decisions of their trade partners. A model that allows for such an approach is the so-called Nonlinear Voter Model (NLVM), studied by Molofsky et al.

Replicators and Larger-than-Life Examples

After watching a substantial number of cellular automaton dynamics generated by rules containing suitable ingredients, eventually a particular time-dependent pattern catches the eye. A configuration of occupied sites makes copies of itself, then the copies make copies of themselves, and these copies move toward one another and also toward the boundaries of the evolution. This continues as long as there is room for the evolution. When the innermost copies collide, they annihilate one another. Meanwhile, the outermost copies continue to reproduce, provided that no occupied sites from the outside impede. The pattern repeats, ad infinitum. We first saw this kind of evolution, which we call a replicator, in our studies of the Larger-than-Life (LtL) family of cellular automaton (CA) rules. The first replicators we found were all in the same region of LtL space. We thought an intrinsic property of this specific region was necessary for the existence of a replicator. However, we began seeing similar configurations, with slight variations, in many different subregions of LtL space. Then we learned of the range 1 HighLife cator. However, we began seeing similar configurations, with slight variations, in become quite famous. We saw more examples on Christopher Langton’s computer at the Santa Fe Institute in 1995; this convinced us that the behavior was not exclusive to LtL-like rules. Since then, new replicators have been discovered for a variety of CA rules. In this chapter, we define a replicator using an axiomatic approach and prove various theorems that follow from the axioms. We also present a collection of Larger-than-Life replicator examples, HighLife’s famous example, and propositions that generalize several of the LtL examples. We will begin by presenting a collection of Larger-than-Life replicator examples, but first let us define the family of Larger-than-Life update rules. Larger-than-Life (LtL) is a four-parameter family of two-state cellular automaton rules. The four parameters are the upper and lower bounds of the birth and survival intervals. At each time t, each site x∊ Zd is either live or dead. We think of a live site as being in state 1 and a dead site as being in state 0.

Synthesis of Complex Life Objects from Glider

Life, like many other cellular automata, contains many interesting objects, such as still lifes, oscillators, spaceships, spaceship guns, puffer trains, breeders, and the like. While many of these, like blocks, blinkers, and gliders, occur naturally with great frequency, there are many others that occur infrequently, and countless others that have never yet been observed in any natural context. This chapter deals with methods for synthesizing such complex objects from simple building blocks, such as gliders or other easy-to-synthesize objects. Once an object can be shown to be built in this manner, the object may be used as a building block in larger relocatable structures, such as Turing machines or universal constructors. In addition, the existence of a natural synthesis of an object from a bounded number of gliders implies that the object will form naturally in a sufficiently large, sufficiently sparse field [2]. Inasmuch as this chapter deals mainly with practical aspects of object synthesis, rather than theoretical ones, it may resemble a talk on chemical engineering, rather than abstract mathematics. All figures shown here, unless otherwise specified, show “before” and “after” images of collision sequences; the “before” sequences are shown on the left with dark cells, and the “after” sequences to the right of them in lighter cells. In some cases, unwanted debris is also generated and must be removed later; this debris is shown in the lightest color. There are several basic ways in which objects can be synthesized. The most common objects occur in great abundance in nature, so there are many random collisions of a small number of gliders that will produce them. There have been many random broth experiments conducted in Life, in which fields initialized to random initial configurations have been run until they became periodic, and then the resulting ash analyzed. The results of two such series of experiments, performed by Achim Flammenkamp [1] and Heinrich Koenig [3], are available on the Web. If the objects are sorted in order of decreasing frequency of natural occurrence, the list is also in order of increasing synthesis cost in gliders (with a few rare objects out of place).