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Lecture Six Chapter 5: Quine-McCluskey Method

Lecture Six Chapter 5: Quine-McCluskey Method. COMP 370. Dr. S.V. Providence. Computer Minimization Techniques. Boolean Algebra Karnaugh Maps Quine-McCluskey Method. COMP 370. Dr. S.V. Providence. Boolean Algebra. Review of Boolean Postulates Review of Boolean Identities

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Lecture Six Chapter 5: Quine-McCluskey Method

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

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

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

  4. Review of Boolean Postulates A & B = B & A A # B = B # A Commutative Laws A & (B # C) = (A & B) # (A & C) A # (B & C) = (A # B) & (A # C) Distributive Laws (not like ordinary algebra) 1 & A = A 0 # A = A Identity Elements A & !A = 0 A # !A = 1 Inverse Elements A # A & B = A A & ( A # B ) = A Absorption COMP 370 Dr. S.V. Providence

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

  6. 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

  7. Example2 (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 COMP 370 Dr. S.V. Providence

  8. 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 COMP 370 Dr. S.V. Providence

  9. 2-Variable K -map Y 0 1 X m1 0 m0 m2 m3 1 X Y F(X,Y) = (0,1,2,3) COMP 370 Dr. S.V. Providence

  10. 3-Variable K -map Y YZ 00 01 11 10 X m1 m2 0 m0 m3 m5 m6 m4 m7 1 X Z (0,1,2,3,4,5,6,7) COMP 370 Dr. S.V. Providence

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

  12. Example1 YZ 00 01 11 10 X 1 1 0 1 1 1 1 1 F(X,Y,Z) = (1,3,4,5,6,7) COMP 370 Dr. S.V. Providence

  13. Example1 YZ 00 01 11 10 X 1 1 0 1 1 1 1 1 F(X,Y,Z) = (1,3,4,5,6,7) = m1 # m3 # m4 # m5 # m6 # m7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z COMP 370 Dr. S.V. Providence

  14. Example1 YZ 00 01 11 10 X 1 1 0 1 1 1 1 1 F(X,Y,Z) = (1,3,4,5,6,7) = m1 # m3 # m4 # m5 # m6 # m7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z COMP 370 Dr. S.V. Providence

  15. Example1 YZ 00 01 11 10 X 1 1 0 1 1 1 1 1 F(X,Y,Z) = X # Z COMP 370 Dr. S.V. Providence

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

  17. Example2 YZ 00 01 11 10 X 0 1 1 1 1 1 F(X,Y,Z) = (0,2,4,6) COMP 370 Dr. S.V. Providence

  18. Example2 YZ 00 01 11 10 X 0 1 1 1 1 1 F(X,Y,Z) = COMP 370 Dr. S.V. Providence

  19. Example2 YZ 00 01 11 10 X 0 1 1 1 1 1 F(X,Y,Z) = !Z COMP 370 Dr. S.V. Providence

  20. 4-Variable K -map Y YZ 00 01 11 10 WX 00 m0 m1 m3 m2 01 m4 m5 m7 m6 X m12 m13 m15 m14 11 W 10 m8 m9 m11 m10 Z F(W,X,Y,Z) = (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15) COMP 370 Dr. S.V. Providence

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

  22. Example1 Y YZ 00 01 11 10 F(W,X,Y,Z) = (5,7,9,11,13,15) WX 00 01 1 1 X 1 1 11 W 10 1 1 Z COMP 370 Dr. S.V. Providence

  23. Example1 Y YZ 00 01 11 10 F(W,X,Y,Z) = X & Z # W & Z = (X # W) & Z WX 00 01 1 1 X 1 1 11 W 10 1 1 Z COMP 370 Dr. S.V. Providence

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

  25. Example2 Y YZ F(W,X,Y,Z) = (2,3,6,7,8,10,11,12,14,15) 00 01 11 10 WX 1 1 00 01 1 1 X 1 1 1 11 W 10 1 1 1 Z COMP 370 Dr. S.V. Providence

  26. Example2 Y YZ F(W,X,Y,Z) = W & !Z # Y 00 01 11 10 WX 1 1 00 01 1 1 X 1 1 1 11 W 10 1 1 1 Z COMP 370 Dr. S.V. Providence

  27. 5-Variable K -map V=0 V=1 Y Y YZ YZ 00 01 11 10 00 01 11 10 WX WX m17 m18 m1 m2 m16 m19 m0 m3 00 00 m21 m22 m20 m23 m4 m5 m7 m6 01 01 X X m28 m29 m31 m30 m12 m13 m15 m14 11 11 W W m25 m26 m9 m10 10 m24 m27 10 m8 m11 Z Z COMP 370 Dr. S.V. Providence

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

  29. Finding Prime Implicants (PIs) F(W,X,Y,Z) = (5,7,9,11,13,15) 2 3 4 List minterms by the number of 1s it contains. COMP 370 Dr. S.V. Providence

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

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

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

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

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

  35. Finding Essential Prime Implicants (EPIs) Enter the Prime Implicants and their minterms. COMP 370 Dr. S.V. Providence

  36. Finding Essential Prime Implicants (EPIs) Enter Xs for the minterms covered. COMP 370 Dr. S.V. Providence

  37. Finding Essential Prime Implicants (EPIs) Circle Xs that are in a column singularly. COMP 370 Dr. S.V. Providence

  38. Finding Essential Prime Implicants (EPIs) The circled Xs are the Essential Prime Implicants, so we check them off. COMP 370 Dr. S.V. Providence

  39. Finding Essential Prime Implicants (EPIs) We check off the minterms covered by each of the EPIs. COMP 370 Dr. S.V. Providence

  40. Finding Essential Prime Implicants (EPIs) EPIs: F = X & Z # W & Z = (X # W) & Z COMP 370 Dr. S.V. Providence

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

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

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

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

  45. Finding Essential Prime Implicants (EPIs) COMP 370 Dr. S.V. Providence

  46. Finding Essential Prime Implicants (EPIs) COMP 370 Dr. S.V. Providence

  47. Finding Essential Prime Implicants (EPIs) COMP 370 Dr. S.V. Providence

  48. Finding Essential Prime Implicants (EPIs) COMP 370 Dr. S.V. Providence

  49. Finding Essential Prime Implicants (EPIs) COMP 370 Dr. S.V. Providence

  50. Finding Essential Prime Implicants (EPIs) EPIs: F = (W & !Z) # Y COMP 370 Dr. S.V. Providence

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