Reliable cellular automata with self-organization

Abstract

In a probabilistic cellular automaton in which all local transitions have positive probability, the problem of keeping a bit of information inde nitely is nontrivial, even in an in nite automaton. Still, there is a solution in 2 dimensions, and this solution can be used to construct a simple 3-dimensional discrete-time universal fault-tolerant cellular automaton. This technique does not help much to solve the following problems: remembering a bit of information in 1 dimension; computing in dimensions lower than 3; computing in any dimension with non-synchronized transitions. Our more complex technique organizes the cells in blocks that perform a reliable simulation of a second (generalized) cellular automaton. The cells of the latter automaton are also organized in blocks, simulating even more reliably a third automaton, etc. Since all this (a possibly in nite hierarchy) is organized in \software, it must be under repair all the time from damage caused by errors. A large part of the problem is essentially self-stabilization recovering from a mess of arbitrary size and content. The present paper constructs an asynchronous one-dimensional faulttolerant cellular automaton, with the further feature of \self-organization. The latter means that unless a large amount of input information must be given, the initial con guration can be chosen homogeneous. Date: May 7, 1998.