Consider the function $f$ expressed in the CNF form:

$f={}_1{}_2$cdots${}_8$

where

${}_1=a_1+a_2,$

${}_2=a_1+a_2+a_3+{}_4,$

${}_3={}_2+a_4,$

${}_4=a_1+{}_2+{}_4,$

${}_5={}_1+a_2+a_4,$

${}_6={}_1+a_2+a_3,$

${}_7={}_1+{}_2+{}_4+{}_5,$

${}_8={}_1+a_2+{}_3+a_4+{}_5.$

Assume that we have decided that $a_4=0.$ Make this assignment and then run Boolean

constraint propagation $($BCP$)$ by applying the unit clause rule. You should continue BCP

until you cannot make any additional progress on the formula.

Suppose that we are optimizing an SOP Boolean expression $F(a,b,c,d)$ using the reduce$-$

expand$-$irredundant loop. Assume that we have just done a REDUCE step and we have an

intermediate, non$-$prime $4-$cube cover of $F$ as

$F=\frac{b}{b}ar\frac{c}{b}ar(d)+\frac{b}{b}ar(c)d+cd+b\frac{c}{b}ar(d)$

We want to perform an EXPAND operation on the cube $\frac{b}{b}ar\frac{c}{b}ar(d).$ As we learned in lecture,

ESPRESSO does this by building a cover of the OFF$-$SET for this current cover, then

building a blocking matrix for the cube we seek to expand, and then computing a cover of

this matrix, which tells us how to grow this cube. Apply the recipe and show the result of

EXPAND. $($Hint: you can draw Karnaugh map to verify if you answer is correct.$)$