Test Generation by Fault Sampling

This paper presents a novel technique of generating tests from a random sample of faults. The entire fault population of the circuit is randomly divided into two groups. Only one group, usually the smaller one, is used for test generation by the test-generator and faultsimulator programs. This group is known as the sample and its coverage is deterministic. The coverage of faults in the remaining group is similar to that of random vectors and is estimated from the distribution of fault detection probabilities in the circuit. As the sample size increases, the fraction of unslmpled faults reduces. At the same time, a larger sample yields more test vectors to increase the random coverage. The analysis in the paper determines the coverage of random and deterministic vectors from the detection probability distribution. However, a two-pass test generation process requires no prior knowledge of this distribution. Test generation in the first pnss is started with an arbitrary sample size. From the deterministic coverage in the sample, the detection probability distribution is determined using the Bayes’ theorem. Based on this distribution, the random coverage in the unsampled fault population is estimated and, if necessary, a second pass of test generation is executed with an appropriately larger sample. The sampling procedure is illustrated by several examples. Introduction A typical test generation process is summarized as follows: select an as-yet-undetected fault, generate a test for it, and simulate all other faults; update the fault list by dropping the faults detected by the test; repeat until the desired fault coverage is reached. The total cost of test generation has two easily identified components, namely, the costs of test generation and fault simulation, respectively. The second component could predominate if the circuit is very large or is sequential. The cost here refers to the use of computing resources (CPU, memory, etc.). Reducing the relative cost of fault simulation in the test generation process is the primary motivation for the present work. Suppose only a randomly chosen sample of faults is initially placed on the fault list which is used to generate tests by the above procedure. Two questions need answers: 1) For a given coverage of the generated tests in the sample, what is the fault coverage for all faults? and 2) Can we determine the smallest sample size for tests to have a given coverage of all faults? Notice, this problem is different from that of fault sampling for coverage determination [ 1 I. In sampling for coverage determination, we take a random sample of faults and determine the coverage of faults in the sample by the given tests. This coverage is a statistical estimate of the coverage over all faults. The accuracy of the estimate is dependent only on the sample size. In coverage determination, the tests and the fault sample are derived through two independent processes. In test generation, on the other hand, we use some sample faults for generating tests. Thus our tests are somewhat biased toward detecting the faults in the sample. It is for this reason that the answers to the above questions are not obvious. In this paper, we will provide a mathematical framework for probabilistic analysis and formulate procedures for test generation by fault sampling. Detection Probability The detection probability of a fault is the probability of detecting that fault by a random vector. Detection probabilities of faults in a circuit can be represented by a distribution p (X I : p (x)dx = Fraction of detectable faults with probability of detection between x and x +dx Since x represents probability, p ( x ) is non-zero (and positive) only for values of x between 0 and 1. Also,