Universal Cellular Automata Based on the Collisions of Soft Spheres

Abstract

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.