Memory in Media with Manufacturing Faults

As computational systems scale to arbitrarily large sizes, and as they are expected to function reliably for arbitrary lengths of time, there are certain realities of the physical world that can not be ignored in the design and modeling of computational systems. For example, considerations for the delivery of power and removal of heat seem to limit the system to a two-dimensional surface. Additionally, it can be argued that all of the following restrictions are desirable and reasonable: there is a finite variety of components, no components (or external agents) are immune to failure, and components can communicate only with a bounded number of other components over a bounded distance. A very general model, with a reasonable ability to capture many of these features, is that of a cellular automaton. We extend the widely studied model of transient faults (which occur independently at different places and different times) in cellular automata to consider manufacturing faults (which occur independently at different places, but affect cells for all time). Although a well known monotone binary transition rule (known as Toom’s Rule) in two dimensions can remember a bit (that is, the system can be used to preserve a single Boolean value for all time with probability one) in the presence of transient faults, we show that no monotone binary transition rule in two dimensions can remember a bit when both manufacturing faults and transient faults are present. On the other hand, we give a monotone binary transition rule in three dimensions that can remember a bit in the presence of both manufacturing faults and transient faults. (By adding one or two further dimensions, one can reduce the problem of performing an arbitrary computation reliably to the problem of remembering a single bit.) We also study cellular automata that are based on hyperbolic (rather than Euclidean) tessellations (including infinite regular trees), and we completely classify the cases in which majority voting among all nearest neighbors can tolerate manufacturing faults and/or transient faults.