Reliable Computations Based on Locally Decodable Codes

We investigate the coded model of fault-tolerant computations introduced by D. Spielman. In this model the input and the output of a computation circuit is treated as words in some error-correcting code. A circuit is said to compute some function correctly if for an input which is a encoded argument of the function, the output, been decoded, is the value of the function on the given argument. We consider two models of faults. In the first one we suppose that an elementary processor at each step can be corrupted with some small probability, and faults of different processors are independent. For this model, we prove that a parallel computation running on n elementary non-faulty processors in time t = poly(n) can be simulated on O(n log n/ log log n) faulty processors in time O(t log log n). Note that we get sub-logarithmic blow-up of the memory, which cannot be achieved in the classic model of faulty boolean circuit, where the input is not encoded. In the second model, we assume that at each step some fixed fraction of elementary processors can be corrupted by an adversary, who is free to chose these processors arbitrarily. We show that in this model any computation can be made reliable with exponential blow-up of the memory. Our method employs a sort of mixing mappings, which enjoy some properties of expanders. Based on mixing mappings, we implement an effective selfcorrecting procedure for an array of faulty processors.