Lecture Six Chapter 5: Quine-McCluskey Method Dr. S.V. Providence COMP 370.

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Presentation transcript:

Lecture Six Chapter 5: Quine-McCluskey Method Dr. S.V. Providence COMP 370

Computer Minimization Techniques Boolean Algebra Karnaugh Maps Quine-McCluskey Method Dr. S.V. Providence COMP 370

Boolean Algebra  Review of Boolean Postulates  Review of Boolean Identities  Example1  Example2 Dr. S.V. Providence COMP 370

Review of Boolean Postulates A & B = B & AA # B = B # ACommutative Laws A & (B # C) = (A & B) # (A & C)A # (B & C) = (A # B) & (A # C)Distributive Laws (not like ordinary algebra) 1 & A = A0 # A = AIdentity Elements A &!A = 0A # !A = 1Inverse Elements A #A & B = AA & ( A # B ) = AAbsorption Dr. S.V. Providence COMP 370

Review Boolean Identities 0 & A = 0, A & 0 = 0 Contradiction (always false) A # 1 = 1, 1 # A = 1 Tautology (always true) A & A = AA # A = AIdempotence A & (B & C) = (A & B) & C0 # A = AAssociative Laws !(A & B) = !A # !B or A NAND B = !A OR !B !(A # B) = !A & !B or A NOR B = !A AND !B DeMorgan’s Theorem !!A = AInvolution Dr. S.V. Providence COMP 370

Example1 A #A & B = A Proof: 1. A # A & B = A & 1 # A & B Identity 2. = A & ( 1 # B ) Distribution 3. = A & 1 Identity 4. = A

(X # Y) & (!X # Y) = (X & !X) # (!X & Y) # (X & Y) # (Y & Y) = 0 # (!X & Y) # (X & Y) # Y = (!X # X) & Y # Y = 1 & Y = Y Proof: 1. (X # Y) & (!X # Y) = !![(X # Y) & (!X # Y)] 2. = ![(!X & !Y) # (X & !Y)] DeMorgan’s 3. = ![(!X # X) & !Y] Distribution 4. = ![1 & !Y] Identity 5. = ![!Y] = Y Involution Example2 Dr. S.V. Providence COMP 370

Karnaugh Maps  A 2 Variable K - map  Review 3 Variable K - maps  Example1  Example2  Review 4 Variable K - maps  Example1  Example2  A 5 Variable K - map Dr. S.V. Providence COMP 370

2-Variable K -map Dr. S.V. Providence COMP 370 m0m0 m1m1 m2m2 m3m X Y F(X,Y) =  (0,1,2,3) X Y

3-Variable K -map Dr. S.V. Providence COMP 370 m0m0 m1m1 m4m4 m5m5 m3m3 m2m2 m7m7 m6m X YZ  (0,1,2,3,4,5,6,7) X Y Z

Example1 Dr. S.V. Providence COMP 370 F(X,Y,Z) =  (1,3,4,5,6,7)

Example1 Dr. S.V. Providence COMP X YZ F(X,Y,Z) =  (1,3,4,5,6,7)

Example1 Dr. S.V. Providence COMP X YZ F(X,Y,Z) =  (1,3,4,5,6,7) = m 1 # m 3 # m 4 # m 5 # m 6 # m 7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z

Example1 Dr. S.V. Providence COMP X YZ F(X,Y,Z) =  (1,3,4,5,6,7) = m 1 # m 3 # m 4 # m 5 # m 6 # m 7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z

Example1 Dr. S.V. Providence COMP X YZ F(X,Y,Z) = X # Z

Example2 Dr. S.V. Providence COMP 370 F(X,Y,Z) =  (0,2,4,6)

Example2 Dr. S.V. Providence COMP X YZ 1 11 F(X,Y,Z) =  (0,2,4,6)

Example2 Dr. S.V. Providence COMP X YZ 1 11 F(X,Y,Z) =

Example2 Dr. S.V. Providence COMP X YZ 1 11 F(X,Y,Z) = !Z

4-Variable K -map Dr. S.V. Providence COMP 370 m0m0 m1m1 m4m4 m5m5 m3m3 m2m2 m7m7 m6m WX YZ m8m8 m9m9 m 11 m 10 m 12 m 13 m 15 m 14 W Y X Z F(W,X,Y,Z) =  (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)

Dr. S.V. Providence COMP 370 F(W,X,Y,Z) =  (5,7,9,11,13,15) Example1

Dr. S.V. Providence COMP WX YZ W Y X Z F(W,X,Y,Z) =  (5,7,9,11,13,15)

Example1 Dr. S.V. Providence COMP WX YZ W Y X Z F(W,X,Y,Z) = X & Z # W & Z = (X # W) & Z

Example2 Dr. S.V. Providence COMP 370 F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15)

Example2 Dr. S.V. Providence COMP WX YZ W Y X Z F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15)

Example2 Dr. S.V. Providence COMP WX YZ W Y X Z F(W,X,Y,Z) = W & !Z # Y

5-Variable K -map Dr. S.V. Providence COMP 370 m0m0 m1m1 m4m4 m5m5 m3m3 m2m2 m7m7 m6m6 00 WX YZ m8m8 m9m9 m 11 m 10 m 12 m 13 m 15 m 14 W Y X Z m 16 m 17 m 20 m 21 m 19 m 18 m 23 m WX YZ m 24 m 25 m 27 m 26 m 28 m 29 m 31 m 30 W Y X Z V=0V=1

Quine-McCluskey Method  Prime Implicants Table 3 or 4 steps  Essential Prime Implicants Table Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step List minterms by the number of 1s it contains Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step , ,13 9, , ,15 11, ,15 Enter combinations of minterms by the number of 1s it contains. 2 3 Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step 3  ,701-1  , ,  ,   , ,  , Check off elements used from Step 1. Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step 3  ,701-15,7,13,  , ,13,7, , ,11,13,  , ,13,11,   , ,  , Enter combinations of minterms by the number of 1s it contains. Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (5,7,9,11,13,15) Step 1Step 2Step 3   5,701-15,7,13,   5, ,13,7,  9, ,11,13,   9, ,13,11,    7,  11,   13, The entries left unchecked are Prime Implicants. Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms ,7,13, ,13,11,15 Enter the Prime Implicants and their minterms. Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms ,7,13,15XXXX ,13,11,15XXXX Enter Xs for the minterms covered. Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms ,7,13,15XXXX ,13,11,15XXXX Circle Xs that are in a column singularly. Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms  - 1 5,7,13,15XXXX  ,13,11,15XXXX The circled Xs are the Essential Prime Implicants, so we check them off. Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms  - 1 5,7,13,15XXXX  ,13,11,15XXXX  We check off the minterms covered by each of the EPIs. Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms  - 1 5,7,13,15XXXX  ,13,11,15XXXX  WXYZ EPIs: F = X & Z # W & Z = (X # W) & Z Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15) Step 1Step 2Step 3Step Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15) Step 1Step 2Step 3Step 4  ,3001-  , ,  ,  ,   , ,  ,7011-  ,  , ,  , , , , Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15) Step 1Step 2Step 3Step 4   2,3001-2,3,6,70-1-   2,60-102,6,3,70-1-  2, ,3,10,   8, ,6,10,   8, ,10,3,  ,10,6,   3,70-118,10,12,  3, ,12,10,   6,7011-   6, ,7,11,   10, ,11,7,  10, ,7,14,   12, ,14,7, ,14,11,151 -  7, ,11,14,151 -  11,  14, Dr. S.V. Providence COMP 370

Finding Prime Implicants (PIs) F(W,X,Y,Z) =  (2,3,6,7,8,10,11,12,14,15) Step 1Step 2Step 3Step 4   2,3001-  2,3,6,70-1-2,3,6,7,10,14,11,   2,60-10  2,6,3,70-1-2,3,10,11,6,14,7,  2,  2,3,10, ,6,3,7,10,11,14,   8,  2,6,10, ,6,10,14,3,7,11,   8,  2,10,3, ,10,3,11,6,7,14,   2,10,6, ,10,6,14,3,11,7,   3,70-118,10,12,  3, ,12,10,   6,7011-   6,  3,7,11,   10,  3,11,7,  10,  6,7,14,   12,  6,14,7,  10,14,11,151 -  7,  10,11,14,151 -  11,  14, Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms ,12,10, ,3,6,7,10,11,14,15 Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms ,12,10,14XXXX ,3,6,7,10,11,14,15XXXXXXXX Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms ,12,10,14XXXX ,3,6,7,10,11,14,15XXXXXXXX Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms  ,12,10,14XXXX  ,3,6,7,10,11,14,15XXXXXXXX Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms  ,12,10,14XXXX  ,3,6,7,10,11,14,15XXXXXXXX  Dr. S.V. Providence COMP 370

Finding Essential Prime Implicants (EPIs) Prime ImplicantsCovered MintermsMinterms  ,12,10,14XXXX  ,3,6,7,10,11,14,15XXXXXXXX  WXYZ EPIs: F = (W & !Z) # Y Dr. S.V. Providence COMP 370