A commonly used approach to compute the error rate at the primary outputs (POs) of a circuit is to compare the fault-free and faulty copies of the circuit using XOR gates. This model results in poor accuracies with nonsampling-based methods for reliability estimation. An alternative is to use a single copy of the circuit with a four-valued representation for each net corresponding to the correct and incorrect signals. One problem in this formulation is the accurate propagation of associated probabilities. We use the framework of Bayesian inference (BI) to address this issue. We derive the conditional probability distribution (CPD) corresponding to the four-valued signals and find the output error rate using various approximate BI techniques. With our formulation, we demonstrate that the output error rate scales with the gate error probabilities. It is guaranteed to be zero when the gate error probability is zero, provided approximate BI algorithms based on sum-product belief propagation (BP) are used. Although inaccuracies increase at very low gate error probabilities, it is able to capture the relative reliability of outputs with respect to each other. We also propose a new method for finding the overall circuit error rate as the partition function for a fixed state of POs. This method provides a significant improvement in accuracy when compared with the existing method using OR gates.
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A Framework for Reliability Analysis of Combinational Circuits Using Approximate Bayesian Inference