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

3. Kettani179/MAPLD20042 General Setup Consider the following digital network x1x1 f(x n, x n-1, …, x 1 ) x2x2 xnxn...... Digital Network

4. Kettani179/MAPLD20043 Assumptions The inputs x i ’s and the output f are binary variables taking the values 0 and 1. The x i ’s are independent Bernoulli random variables with P(x i = 1) = E[x i ] = p i.

5. Kettani179/MAPLD20044 Mission Let P = P(f(x n, x n-1, …, x 1 ) = 1) = E[f(x n, x n-1, …, x 1 )] Questions: –Given a logic function, f(x n, x n-1,..., x 1 ), with known probabilities p i ’s, what can we say about the probability P? –How can we address the problem of maximizing or minimizing P?

6. Kettani179/MAPLD20045 Motivating Example Consider the following simple digital circuit: P = p 2 p 1 x1x1 x2x2 f(x 2, x 1 ) = x 2.x 1

7. Kettani179/MAPLD20046 Theorem 1 Let f(x n, x n-1,..., x 1 ) be a binary function of n independent binary random variables with P(x j = 1) = p j. Let I be the set of minterm indices for which f(x n, x n-1,..., x 1 ) is 1. Then

8. Kettani179/MAPLD20047 Stochastic Optimization Suppose that the probabilities p i can be picked from intervals I i = [p i -, p i + ]. Consequently, the tuple (p 1, p 2,..., p n ) can be picked from the hypercubeI = I 1 × I 2 ×... × I n. Then, what value should we set the probabilities p i to in order to maximize or minimize P?

9. Kettani179/MAPLD20048 Essential Variables A binary variable x k is said to be essential if there does not exist admissible values of the (n−1) remaining variables xj  I n, j ≠ k, making the probability P independent of x k  I k. If x k is essential, then the partial derivative ∂P / ∂p k is non-zero over I. Hence, if the variable x k is essential, the partial derivative ∂P / ∂p k has one sign over I.

10. Kettani179/MAPLD20049 Essential Variables (Cont.) Let us denote this invariant sign by Hence, s k is constant over I having the value s k =−1 or s k =1.

11. Kettani179/MAPLD200410 Theorem 2 Let P be a function of some p j ’s. Then, –For the case of maximizing P, if the variable x k is essential, then pick p k = p − k when s k = −1, and pick p k = p + k when s k = 1. –For the case of minimizing P, if the variable x k is essential, then pick p k = p + k when s k = −1, and pick p k = p − k when s k = 1. –If x k is not essential, then for either case pick p k = p − k or p k = p + k.

12. Kettani179/MAPLD200411 Numerical Examples (1/3) f 1 (x 3, x 2, x 1 ) = x 3 + x 1, and f 2 (x 3, x 2, x 1 ) = x 2 x 1 + x 3 x 1. p 1  [0.4, 0.6], p 2  [0.1, 0.5], and p 3  [0.2, 0.8]. We have, I 1 = {0, 2, 4, 5, 6, 7}, and I 2 = {1, 4, 5, 6}. Hence, we have from Theorem 1: – P 1 = (1 − p 3 )(1 − p 1 ) + p 3, and – P 2 = (1 − p 2 )p 1 + (1 − p 1 )p 3.

13. Kettani179/MAPLD200412 Numerical Examples (2/3) Suppose we would like to maximize P 1. Then Note that both x 1 and x 3 are essential with s (1) 1 = −1 and s (1) 3 = 1. Thus, the maximum P + 1 is obtained with p 1 = 0.4 and p 3 = 0.8. Consequently, P + 1 = 0.92.

14. Kettani179/MAPLD200413 Numerical Examples (3/3) Suppose that we would like to minimize P 2. Then Note that both x 2 and x 3 are essential with s (2) 2 = −1 and s (2) 3 = 1. Thus, the minimum P − 2 is obtained with p 2 = 0.5 and p 3 = 0.2. However, the variable x 1 is not essential. Thus, we try both values 0.4 and 0.6 for p 1. This results in P 2 = 0.32 and P 2 = 0.38. Consequently, P − 2 = 0.32.

15. Kettani179/MAPLD200414 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.

16. Kettani179/MAPLD200415 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?