On The Capacity Of Noisy Computations

This paper presents an analysis of the concept of capacity for noisy com- putations, i.e. functions implemented by unreliable or random devices. An information theoretic model of noisy computation of a perfect function f (measurable function between sequence spaces) thanks to an unreliable device (random channel) F is given: a noisy computation is a product fxF of channels. A model of reliable computation based on input encoding and output decoding is also proposed. These models extend those of noisy communication channel and of reliable communication through a noisy channel. The capacity of a noisy computation is defined and justified by a coding theorem and a converse. Under some constraints on the encoding process, capacity is the upper bound of input rates allowing reliable computation, i.e. decodability of noisy outputs into expected outputs. These results hold when the one-sided random processes under concern are asymptotic mean stationary (AMS) and ergodic. In addition, some characterizations of AMS and ergodic noisy computations are given based on stability properties of the perfect function f and of the random channel F. These results are derived from the more general framework of channel products. Finally, a way to apply the noisy and reliable computation models to cases where the perfect function f is defined according to a formal computational model is proposed

Computing Linear Transformations with Unreliable Components

We consider the problem of computing a binary linear transformation using unreliable components when all circuit components are unreliable. Two noise models of unreliable components are considered: probabilistic errors and permanent errors. We introduce the "ENCODED" technique that ensures that the error probability of the computation of the linear transformation is kept bounded below a small constant independent of the size of the linear transformation even when all logic gates in the computation are noisy. Further, we show that the scheme requires fewer operations (in order sense) than its "uncoded" counterpart. By deriving a lower bound, we show that in some cases, the scheme is order-optimal. Using these results, we examine the gain in energy-efficiency from use of "voltage-scaling" scheme where gate-energy is reduced by lowering the supply voltage. We use a gate energy-reliability model to show that tuning gate-energy appropriately at different stages of the computation ("dynamic" voltage scaling), in conjunction with ENCODED, can lead to order-sense energy-savings over the classical "uncoded" approach. Finally, we also examine the problem of computing a linear transformation when noiseless decoders can be used, providing upper and lower bounds to the problem.Comment: Accepted by Transactions on Information Theory for future publicatio