On the Theory of Stochastic Processors

Traditional architecture design approaches hide hardware uncertainties from the software stack through over-design, which is often expensive in terms of power consumption. The recently proposed quantitative alternative of stochastic computing requires circuits and processors to be correct only probabilistically and use less power. In this paper, we present the first step towards a theory of stochastic computing. Specifically, a formal model of a device which computes a deterministic function with stochastic delays is presented; the semantics of a stochastic circuit is obtained by composing such devices; finally, a quantitative notion of stochastic correctness, called correctness factor (CF), is introduced. For random data sources, a closed form expression is derived for CF of devices, which shows that there are two probabilities that contribute positively, namely, the probability of being timely with current inputs and the probability of being lucky with past inputs. We show the characteristic graphs obtained from the analytical expressions for the variation of correctness factor with clock period, for several simple circuits and sources.