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Monte Carlo Analysis
of Uncertain Digital Circuits
Houssain Kettani, Ph.D.
Department of Computer Science
Jackson State University
Jackson, MS
[email protected]
http://www.jsums.edu/~houssain.kettani
September 2004
General Setup
• Consider the following digital network
x1
x2
.
.
.
Digital
f(xn, xn-1, …, x1)
Network
xn
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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.
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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?
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Motivating Example
• Consider the following simple digital
circuit:
x1
x2
f(x2, x1) = x2.x1
• P = p2p1
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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
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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 hypercubeI = I1 × I2 × . . . × In.
• Then, what value should we set the probabilities
pi to in order to maximize or minimize P?
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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.
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Essential Variables (Cont.)
• Let us denote this invariant sign by
• Hence, sk is constant over I having the value
sk=−1 or sk=1.
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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.
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Numerical Examples (1/3)
• f1(x3, x2, x1) = x3 + x1, and f2(x3, x2, x1) = x2x1 +
x 3x 1.
• 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.
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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.
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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.
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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.
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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?
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