Fault Tolerance in Cellular Automata at High Fault Rates

TL;DR

This paper investigates the fault tolerance of cellular automata at high fault rates, demonstrating that certain automata can tolerate nearly 50% faults with relatively low degree, and establishing bounds on fault tolerance.

Contribution

It introduces new fault tolerance results for cellular automata at high fault rates, including models with adversarial and probabilistic faults, and provides bounds on the necessary degree for fault tolerance.

Findings

Automata can tolerate fault rates close to 1/2 with degree O((1/ξ^2) log(1/ξ)).

Results apply to structures like infinite regular trees and hyperbolic tessellations.

Lower bound of Ω(1/ξ^2) for fault tolerance under probabilistic faults.

Abstract

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate $1/2 - \xi$ (with $\xi>0$) with degree $O((1/\xi^2)\log(1/\xi))$, even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of $\Omega(1/\xi^2)$, even with purely probabilistic transient faults only.