Complete Test Sets And Their Approximations

TL;DR

This paper introduces a method to construct smaller, approximate complete test sets for combinational circuits using Stable Sets of Assignments, improving efficiency over traditional exhaustive testing.

Contribution

It proposes a novel approach to generate non-trivial, approximate test sets from SSAs of circuit projections, reducing testing complexity.

Findings

Non-trivial SSAs can be significantly smaller than trivial ones.

Tests derived from SSAs effectively approximate complete test sets.

Experimental results demonstrate improved efficiency in circuit testing.

Abstract

We use testing to check if a combinational circuit $N$ always evaluates to 0 (written as $N \equiv 0$). We call a set of tests proving $N \equiv 0$ a complete test set (CTS). The conventional point of view is that to prove $N \equiv 0$ 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.