Collision-based computing in Belousov–Zhabotinsky medium

Abstract

A photosensitive sub-excitable Belousov–Zhabotinsky medium exhibits propagating wave fragments that preserve their shapes during substantial periods of time. In numerical studies we show that the medium is a computational universal architectureless system, if presence and absence of wave fragments are interpreted as truth values of Boolean variable. When two or more wave fragments collide they may annihilate, fuse, split or deviate from their original paths, thus values of the logical variables are changed and certain logical gates are realized in result of the collision. We demonstrate exact implementation of basic operations with signals and logical gates in Belousov–Zhabotinsky dynamic circuits. The findings provide a theoretical background for subsequent experimental implementation of collision-based, architectureless, dynamical computing devices in homogeneous active chemical media.

Introduction

Certain families of thin-layer reaction-diffusion (RD) chemical media can implement sensible transformation of initial (data) spatial distribution of chemical species concentrations to final (result) concentration profile [1], [32]. In these RD computers a computation is realized via spreading and interaction of diffusive or phase waves. Specialized, intended to solve a particular problem, experimental RD processors implement basic operations of image processing [2], [18], [24], [25], computation of optimal paths [3], [7], [26], [33] and control of mobile robots [4]. A number of computationally universal RD chemical devices were implemented, the findings include logical gates [31], [34] and diodes [12], [19], [21] in Belousov–Zhabotinsky (BZ) medium, and XOR gate in palladium processor [5]. All the known so far experimental prototypes of RD processors exploit interaction of wave fronts in a geometrically constrained chemical medium, i.e. the computation is based on a stationary architecture of medium's inhomogeneities. Constrained by a stationary wires and gates RD chemical universal processors pose a little computational novelty and none dynamical reconfiguration ability because they simply imitate architectures of silicon computing devices.

To appreciate in full massive-parallelism of thin-layer chemical media and to free the chemical processors from limitations of fixed computing architectures we adopt an unconventional paradigm of dynamical, architectureless, or collision-based, computing. The paradigm originates from computational universality of Game of Life [9], conservative logic and billiard-ball model [14] and their cellular-automaton implementations [20]. A collision-based (CB) computation employs mobile compact patterns, in our particular case they are self-localized excitations in active non-linear medium. The localizations travel in space and perform computation (implement logical gates) when they collide with each other. There are no predetermined stationary wires––a trajectory of the traveling pattern is a momentarily wire––almost any part of the medium's space can be used as a wire. Truth values of logical variables are given by either absence or presence of a localization or by various types of localizations. State of the art of CB computing is presented in [6].

Solitons, defects in tubulin microtubules, excitons in Scheibe aggregates and breather in polymer chains are most frequently considered candidates for a role of information carrier in nature-inspired CB computers, see overview in [1]. It is experimentally difficult to reproduce all these artifacts in natural systems, therefore existence of mobile localizations in an experiment-friendly RD media would open new horizons for fabrication of CB computers. Until recently we have a little if any information about interaction of mobile localizations in 2D or 3D RD media. However the works [10], [29] demonstrated existence and rich interaction of quasi-particles (dissipative solitons) in a three-component RD system. The basis for CB universality of RD chemical media was finally laid when Sendin̋a-Nadal et al [30] experimentally proved existence of localized excitations––traveling wave fragments which behave like quasi-particles––in photosensitive sub-excitable BZ medium.

In present paper we aim to computationally demonstrate how logical circuits can be fabricated in a sub-excitable BZ medium via collisions between traveling wave fragments. While implementation CB logical operations themselves is relatively straightforward, more attention should be paid to control of signal propagation in the homogeneous medium. For example, to realize a (non-conservative) analog of Fredkin–Toffoli–Margolus billiard-ball model of interaction logic [14], [20] we must somehow `fabricate' a reflector to control information quanta trajectories. It has been demonstrated widely that applying light of varying intensity we can control excitation dynamic in BZ medium [8], [15], [16], [22], wave velocity [28], patter formation [36]. Of particular interest are experimental evidences of light-induced back propagating waves, wave-front splitting and phase shifting [37]; we can also manipulate medium's excitability by varying intensity of the medium's illumination [11]. Basing on these facts we show how to control signal-wave fragments by varying geometric configuration of excitatory and inhibitory segments of impurity-reflectors.