# Complete Test Sets And Their Approximations

We use testing to check if a combinational circuit N always evaluates to 0 (written as N ≡ 0). We call a set of tests proving N ≡ 0 a complete test set (CTS). The conventional point of view is that to prove N ≡ 0 one has to generate a trivial CTS. It consists of 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 non-trivial CTS consisting of less than 2^|X| tests. Given an unsatisfiable CNF formula H(W), an SSA of H is a set of assignments to W that proves unsatisfiability of H. A trivial SSA is the set of all 2^|W| assignments to W. Importantly, real-life formulas can have non-trivial SSAs that are much smaller than 2^|W|. In general, construction of even non-trivial CTSs is inefficient. We describe a much more efficient approach where tests are extracted from an SSA built for a `projection' of N on a subset of variables of N. These tests can be viewed as an approximation of a CTS for N. We give experimental results and describe potential applications of this approach.

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