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_at_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
3
Assumptions

• The inputs xis and the output f are binary
  variables taking the values 0 and 1.
• The xis are independent Bernoulli random
  variables with P(xi 1) Exi pi.

4
Mission

• Let
• P P(f(xn, xn-1, , x1) 1)
• Ef(xn, xn-1, , x1)
• Questions
• Given a logic function, f(xn, xn-1, . . . , x1),
  with known probabilities pis, what can we say
  about the probability P?
• How can we address the problem of maximizing or
  minimizing P?

5
Motivating Example

• Consider the following simple digital circuit
• P p2p1

x1
f(x2, x1) x2.x1
x2
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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 hypercube I 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 sk1.

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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 pk when sk 1.
• For the case of minimizing P, if the variable xk
  is essential, then pick pk pk 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 pk.

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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.

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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 P1 is obtained with p1 0.4
  and p3 0.8.
• Consequently, P1 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?