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U. Glaeser

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Example<br />

Given y = x 1 + x 2. Find P(y(t − T)y(t)) and a(y).<br />

P(y(t − T)y(t)) is given by the product of two terms, namely<br />

Expanding the product of these two terms, we obtain the following four terms:<br />

After rearranging the above equation, we obtain<br />

Partitioning Algorithm<br />

Accurate calculation of the symbolic probability is important to subsequent computation of the activity at<br />

internal node of a circuit; however, not only the exact calculation of the symbolic probability is NP-hard,<br />

but also the size of symbolic probability expression grows exponentially with the number of the inputs.<br />

Thus, a technique to partition the circuit network by utilizing the circuit topology information is used [6].<br />

Using this partitioning scheme, the size of the probability expression is limited as each node in the circuit<br />

network is now only dependent upon its minimum set of topologically independent inputs (MSTII). MSTII<br />

is a set of independent inputs (or internal nodes) that logically determines the logic function of a node.<br />

This partition scheme can trade off accuracy with computation speed and data storage size.<br />

Figure 20.6 shows the MSTII of a logic gate Z. The MSTII of y 1, x 4, w, and x 7 are used instead of x 1,<br />

x 2,…, x 7. Hence, we only deal with four inputs instead of the original seven.<br />

FIGURE 20.6 MSTII of a logic gate.<br />

© 2002 by CRC Press LLC<br />

P( y)<br />

= P( x1) + P( x1)P( x2) and P( x1) = P( x1) = 1 – P( x1) P( x1( t– T)<br />

) + P( x1( t– T)<br />

)P( x2( t– T)<br />

) and P( x1() t ) + P( x1() t )P( x2() t )<br />

P( x1( t– T)<br />

)P( x1() t ) + P( x1( t– T)<br />

)P( x1() t )P( x2() t ) + P( x1( t– T)<br />

)P( x1() t )P( x2( t– T)<br />

)<br />

+ P( x1( t– T)<br />

)P( x1() t )P( x2( t– T)<br />

)P( x2() t )<br />

= P( x1( t– T)x1()<br />

t ) + P( x1( t– T)x1()<br />

t )P( x2() t ) + P( x1( t– T)x1()<br />

t )P( x2( t– T)<br />

)<br />

=<br />

+ P( x1( t– T)x1()<br />

t )P( x2( t– T)x2()<br />

t )<br />

1<br />

P( x1) – -- a( x1) 2<br />

1<br />

1<br />

-- a( x1)P( x2) -- a( x1)P( x2) ⎛ 1<br />

1 – P( x1) – -- a( x1) ⎞ 1<br />

+ + +<br />

⎛P( x2)<br />

– -- a( x<br />

2<br />

2<br />

⎝ 2 ⎠<br />

2)<br />

⎞<br />

⎝ 2 ⎠<br />

1<br />

a( y)<br />

= ( 1 – P( x1) )a( x2) + ( 1 – P( x2) )a( x1) – -- a( x1)a( x2). 2<br />

X1<br />

X2<br />

X3<br />

X4<br />

X5<br />

X6<br />

X7<br />

W<br />

Y1<br />

Y2<br />

Y3<br />

Z

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