Monte Carlo Analysis of Uncertain Digital Circuits

1. Monte Carlo Analysis of Uncertain Digital Circuits Houssain Kettani, Ph.D. Department of Computer ScienceJackson State UniversityJackson, MS houssain.kettani@jsums.edu http://www.jsums.edu/~houssain.kettani September 2004

2. General Setup • Consider the following digital network x1 x2 Digital Network . . . f(xn, xn-1, …, x1) xn 179/MAPLD2004

3. Assumptions • The inputs xi’s and the output f are binary variables taking the values 0 and 1. • The xi’s are independent Bernoulli random variables with P(xi = 1) = E[xi] = pi. 179/MAPLD2004

4. Mission • Let P = P(f(xn, xn-1, …, x1) = 1) = E[f(xn, xn-1, …, x1)] • Questions: • Given a logic function, f(xn, xn-1, . . . , x1), with known probabilities pi’s, what can we say about the probability P? • How can we address the problem of maximizing or minimizing P? 179/MAPLD2004

5. Motivating Example • Consider the following simple digital circuit: • P = p2p1 x1 f(x2, x1) = x2.x1 x2 179/MAPLD2004

6. Theorem 1 Let f(xn, xn-1, . . . , x1) be a binary function of n independent binary random variables with P(xj = 1) = pj . Let I be the set of minterm indices for which f(xn, xn-1, . . . , x1) is 1. Then 179/MAPLD2004

7. Stochastic Optimization • Suppose that the probabilities pi can be picked from intervals Ii = [pi- , pi+]. • Consequently, the tuple (p1, p2, . . . , pn) can be picked from the hypercube I = I1 × I2 × . . . × In. • Then, what value should we set the probabilities pi to in order to maximize or minimize P? 179/MAPLD2004

8. Essential Variables • A binary variable xk is said to be essential if there does not exist admissible values of the (n−1) remaining variables xj In, j ≠ k, making the probability P independent of xkIk. • If xk is essential, then the partial derivative ∂P / ∂pk is non-zero over I. • Hence, if the variable xk is essential, the partial derivative ∂P / ∂pk has one sign over I. 179/MAPLD2004

9. Essential Variables (Cont.) • Let us denote this invariant sign by • Hence, sk is constant over I having the value sk=−1 or sk=1. 179/MAPLD2004

10. Theorem 2 Let P be a function of some pj ’s. Then, • For the case of maximizing P, if the variable xk is essential, then pick pk = p−k when sk = −1, and pick pk = p+k when sk = 1. • For the case of minimizing P, if the variable xk is essential, then pick pk = p+k when sk = −1, and pick pk = p−k when sk = 1. • If xk is not essential, then for either case pick pk = p−k or pk = p+k. 179/MAPLD2004

11. Numerical Examples (1/3) • f1(x3, x2, x1) = x3 + x1, and f2(x3, x2, x1) = x2x1 + x3x1. • p1 [0.4, 0.6], p2 [0.1, 0.5], and p3 [0.2, 0.8]. • We have, I1 = {0, 2, 4, 5, 6, 7}, and I2 = {1, 4, 5, 6}. • Hence, we have from Theorem 1: • P1 = (1 − p3)(1 − p1) + p3, and • P2 = (1 − p2)p1 + (1 − p1)p3. 179/MAPLD2004

12. Numerical Examples (2/3) Suppose we would like to maximize P1. Then • Note that both x1 and x3 are essential with s(1)1 = −1 and s(1)3 = 1. • Thus, the maximum P+1 is obtained with p1 = 0.4 and p3 = 0.8. • Consequently, P+1 = 0.92. 179/MAPLD2004

13. Numerical Examples (3/3) Suppose that we would like to minimize P2. Then • Note that both x2 and x3 are essential with s(2)2 = −1 and s(2)3 = 1. • Thus, the minimum P−2 is obtained with p2 = 0.5 and p3 = 0.2. • However, the variable x1 is not essential. • Thus, we try both values 0.4 and 0.6 for p1. • This results in P2 = 0.32 and P2 = 0.38. • Consequently, P−2 = 0.32. 179/MAPLD2004

14. Summary • Considered the case of digital circuits with uncertain input variables. • Presented a probabilistic measure of the output function in terms of the probabilities of the input. • The result is a multilinear function, which facilitates the optimization problem of the probability of the output. 179/MAPLD2004

15. Further Research • What if the input variables are dependent? • What if we consider b-ary logic instead of binary? • What if we broaden the concept of uncertain digital networks to include uncertain logic gates and extend our results to such case? 179/MAPLD2004