550 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 12, NO. zyxw 4, APRIL zyxw 1993 Short Papers A Beta Model for Estimating the Testability and Coverage Distributions of a VLSI Circuit H. A. Farhat and S. G. From Abstract-In “A Statistical Theory of Digital Circuit Testability,” a relation between testability and fault coverage was established. The main contribution of this paper is in using this relation to estimate the testability and coverage distribution by modeling testability as a mix- ture of a discrete impulse function and a continuous beta distribution. Moments and maximum likelihood estimators are used to estimate the parameters of the beta component of the modeled testability profile. From the computed parameters and the testability/coverage relation, the coverage distribution is estimated. When compared with a previous Bayesian approach, experimental results on three of the large ISCAS- 89 circuit showed that the beta model provides, in general, better es- timates of population coverage. I. INTRODUCTION The problem of test generation for a given combinational circuit is to find a subset of the input space that detects (covers) a desired fraction of modeled faults. The standard method determines the fraction of detected faults by simulating the test inputs on the entire fault population. When all faults are simulated, the coverage ob- tained is accurate on a fault list that is usually reduced by fault- collapsing techniques. This method of fault simulation, however, predominates the overall costs of test generation for sequential and large combinational circuits. In [l], [12] a relationship between probabilistic testability and fault coverage was developed. This relation was used to propose a new method of test generation called “Test Generation by Fault Sampling. zyxwvutsrqpon ” The proposed method was a standard test generation process restricted to a sample of faults. Only the sample faults were simulated; the population coverage of the sample test was zyxwvutsrq esti- mated. We briefly describe the process of test generation by fault sampling below. Pass one: 1) Select a random sample of faults; 2) use a standard test generation procedure to generate tests on the sample of faults; while generating tests 3) remove identified redundant faults from the sample; zyxwvutsrqp 4) estimate the testability profile from fault coverage data; and 5) from the estimated testability profile compute the pop- ulation coverage; if the computed coverage is less than the desired population coverage, then determine the needed sample size and go to “pass two.” Pass zyxwvutsrqpo two: 1) Augment the initial random sample by the needed number of faults computed in steps 5 of “pass one”; and 2) gen- erate test sets on augmented sample. The main contribution of the proposed work is in modeling test- ability as a mixture of a discrete impulse function at zero and a Manuscript received June 25, 1991; revised May 21, 1992. This paper The authors are with the Department of Mathematics and Computer Sci- IEEE Log Number 9208604. was recommended by Associate Editor S. Seth. ence, University of Nebraska, Omaha, NE 68 182-2423. continuous beta distribution. The impulse function is used to model redundant faults in the circuit. For the detectable faults, the beta distribution is chosen because of its flexible nature (from a statis- tics points of view, this distribution provides the most tractable flexible model). The graph of the beta distribution is a function of its parameters. It can assume many different shapes. The case where the testability profile is maximized at the vertical asymptote of zero detection probability is of interest, since the distribution of the test- ability near zero can be used as a measure of the “ease” or “dif- ficulty” of test generation. In estimating the parameters of the test- ability distribution we select two samples (one for generating tests and another to estimate detection counts of faults). A second sam- ple is chosen in order to simulate the random behavior of tests. The strength of the impulse function is estimated by the fraction of sam- pled faults that are either redundant or aborted. The paper is presented as follows: Section I1 includes the needed definitions and background material. Section 111 contains the test- ability model. The experimental results for three of the large ISCAS-89 benchmark circuits are given in Section IV. An empir- ical discussion on the confidence intervals and the validity of the testability model is presented in Section V. A summary of the work is given in the conclusions in Section VI. The Appendix contains two methods of estimating the testability parameters. 11. DEFINITIONS AND BACKGROUND zyxw A. Dejnitions The detection probability of a fault a is the probability of de- tecting a by a random vector. Note that the detection probability of a fault is affected by the distribution from which the random inputs are selected. For example, when deterministic test genera- tors are used to generate the input tests, the input vectors can be considered “random” from some unknown distribution. For this type of input distribution, target faults have a detection probability equal to one (assuming no fault is redundant and no time limits are imposed on the deterministic test generator). The detection probability distribution (testability projile), p (t), of a combinational circuit is the probability density function of the detection probabilities of its faults (several algorithms that estimate p (t) exist [3], [9], [I I]). Since the testability profile is a probability density function we have p(t) dt = 1. Note that p (t) df corre- sponds to the fraction of detectable faults with detection probabil- ities between t and t + dt. The undetectability projile, zyxwv Z(n). is defined as I(n) = 1 ; (I - The fuult coveruge by k vectors, ykr is defined as the fraction of t)”p (t) dr. faults detected by the k vectors. B. Background Assume n random vectors are applied to a circuit with detection probability distribution p (t). From [12], the random coverage, yn, by n vectors is zyxwvu y,, = 1 - zyx 5’ (1 - r)”p(t) dt = 1 - I(n). (1) 0278-0070/93$03.00 zyxwvut 0 1993 IEEE