8/18/2019 Final Exam - 1436-1437 - Jan 2016

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Cairo

University

Faculty of  

Engineering 

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ا

 

ع

 

 ت ه

 

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Electronics and

Electrical

Communications

Engineering Department

ELC 448 Advanced Topics in Logic Design - Elective Course 2Fourth Year 1436/1437 H (2015/2016) – - Term 1

Final Exam –  Rabi’a Al-Akhar 1437 H (January 2016) - 2 Hours

Question 1 Boolean Decomposition (20 minutes)

a.  [4 marks] Write the logic expression of the Boolean Difference ⁄  of the n-variable Booleanfunction  (, , ⋯ , )  with respect to an arbitrary variable   in terms of cofactors of    withrespect to that variable. Find the simplest Boolean expression of the Boolean Difference ⁄ , theUniversal Quantification ∀   ∙ , and the Existential Quantification ∃   ∙  of the n-variable XORfunction (, , ⋯ , )   =  ⊕  ⊕ ⋯ ⊕ .

 b.  [8 marks] Prove for the Boolean function ℎ(, , ⋯ , ) = (, , ⋯ , ) ⨁ (, , ⋯ , ) that for any variable  

ℎ =

  ⨁

  

Verify the correctness of this rule when = 4, = 2,  = ̅, and = ̅ + .

Question 2 Boolean Satisfiability (20 minutes)

a.  [4 marks] Explain briefly what is meant in Boolean Satisfiability by an empty clause, a unit clause, a

unate variable, and a contradiction.

 b.  [4 marks] Solve the following Boolean SAT problem:

( +

)(̅ + ̅

)( + ̅ +

)(̅ +  +

)(̅ + ̅

) 

using the Davis-Putnam Resolution-Based Algorithm.

c.  [4 marks] Solve the SAT problem given in part (b) using the Davis-Logemann-Loveland Depth-First

Search Algorithm.

Question 3 Boolean Matching (20 minutes)

a. 

[4 marks] Find the Unateness and the Onset-Size signatures of the pair of Boolean functions

 (, , ) = ̅ + ̅ + ̅ and (, , ) =  + . b.  [8 marks] Find all possible matching cases for the pair of Boolean functions given in part (a).

8/18/2019 Final Exam - 1436-1437 - Jan 2016

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Question 4 Reduced Ordered Binary Decision Diagrams (20 minutes)

a.  [4 marks] Write the logic expression of the if-then-else (ITE) operator, and show how the ITE

operator can be used to represent a 2-input NAND gate.

 b.  [4 marks] Explain briefly how the Onset of an -variable Boolean function can be computed from anROBDD representation of the function.

c.  [4 marks] Sketch the ROBDD representation of the Boolean function 

 (, , , ) = ( ∙ ) ⊕ ( + ) 

with the variable ordering  <  <  < . 

Question 5 Logic Optimization (20 minutes)

a.  [4 marks] Explain briefly what is meant in logic optimization by an irredundant cover, and an

essential prime implicant.

 b.  [8 marks] Use the Quine-McCluskey Tabular Minimization method to find the minimum cover of the

following incompletely specified Boolean function

F( , , , ) = ∑ (0,1,5,7,8,10,13) + ∑ (2,9,14,15) 

Question 6 And-Inverter Graphs (20 minutes)

a.  [4 marks] Use the AIG representation of the basic logic operations to construct a 2-level AIG

representation of the 4-input OR Boolean function ( , , ,) = + + + . 

 b.  [4 marks] Find all 3-feasible structural cuts of the AIG constructed in part (a).

c.  [4 marks] Explain briefly how structural hashing in used AIG optimization to reduce the number of

AND nodes.