Generation of complete test sets

Abstract

We use testing to check if a combinational circuit N always evaluates to 0. The usual point of view is that to prove that N always evaluates to 0 one has to check the value of N for all 2^|X| input assignments where X is the set of input variables of N. We use the notion of a Stable Set of Assignments (SSA) to show that one can build a complete test set (i.e. a test set proving that N always evaluates to 0) that consists of less than 2^|X| tests. Given an unsatisfiable CNF formula H(W), an SSA of H is a set of assignments to W proving unsatisfiability of H. A trivial SSA is the set of all 2^|W| assignments to W. Importantly, real-life formulas can have SSAs that are much smaller than 2^|W|. Generating a complete test set for N using only the machinery of SSAs is inefficient. We describe a much faster algorithm that combines computation of SSAs with resolution derivation and produces a complete test set for a "projection" of N on a subset of variables of N. We give experimental results and describe potential applications of this algorithm.