Using the examples of an excitable chemical system (Belousov-Zhabotinsky medium) and plasmodium of Physarum polycephalum we show that universal computation in a geometrically unconstrained medium is only possible when resources (excitability or concentration of nutrients) are limited. In situations of limited resources the systems studied develop travelling localizations. The localizations are elementary units of dynamical logical circuits in collision-based computing architectures. Keywords: unconventional computing, collision-based computing, Belousov-Zhabotinsky system, Physarum polycephalum
1.Introduction: architecture-based vs architecture-less computing The first thoughts on the implementation of computational operations with patterns propagated in spatially extended non-linear systems date back to 1800's where Plateau experimented with the problem involving the calculation of the surface of smallest area bounded by a given closed contour in space [Courant and Robins, 1941] (the classical problem of calculating a minimal spanning tree of planar points using a soap film). These ideas were rediscovered many times but mostly in a framework of theoretical design of novel algorithms, of these the grass fire transformation [Blum, 1968;Calabi & Hartnett, 1968] is the most famous example. In 1990's the new field of reaction-diffusion computingcomputation with excitation and diffusive waves in two-dimensional chemical media --was conceived in experiments using the Belousov-Zhabotinsky medium [Kuhnert & Agladze, 1989;Rambidi, 1998] and precipitating chemical systems [Tolmachiev & Adamatzky, 1996]. A reaction-diffusion computer is a spatially extended chemical system, which process information using interacting growing patterns, excitable and diffusive waves [Adamatzky, 2001; Adamatzky, De Lacy Costello & Asai, 2005]. In reaction-diffusion processors, both the data and the results of the computation are encoded as concentration profiles of the reagents. The computation is performed via the spreading and interaction of wave fronts. Reaction-diffusion chemical processors are now `classical' examples of non-linear medium computers. A non-linear medium processor can be either specialised or general-purpose (universal). A specialized processor is built to solve only one particular problem, possibly with different data sets and with variations in the interpretation of the results. Specialised computing devices are quite handy when we deal with image processing, problems of mathematical morphology, or computation on graphs [Adamatzky, 2001; Adamatzky, De Lacy Costello, Asai, 2005]. A device is called computationally universal if it computes a functionally complete set of logical operations. To prove a medium's universality one must represent quanta of information, routes of information transmission and logical gates, where information quanta are processes, in states of the given system. This can be done in two ways: architecture-based and architecture-less. An architecture-based, or stationary, computation implies that a l...