1. 1 Quine-McCluskey Method

2. 2 Motivation Karnaugh maps are very effective for the minimization of expressions with up to 5 or 6 inputs. However they are difficult to use and error prone for circuits with many inputs. Karnaugh maps depend on our ability to visually identify prime implicants and select a set of prime implicants that cover all minterms. They do not provide a direct algorithm to be implemented in a computer. For larger systems, we need a programmable method!!

3. 3 Quine-McCluskey Quine, Willard, “A way to simplify truth functions.” American Mathematical Monthly, vol. 62, 1955. Quine, Willard, “The problem of simplifying truth functions.” American Mathematical Monthly, vol. 59, 1952. Willard van Orman Quine 1908-2000, Edgar Pierce Chair of Philosophy at Harvard University. http://members.aol.com/drquine/wv-quine.html McCluskey Jr., Edward J. “Minimization of Boolean Functions.” Bell Systems Technical Journal, vol. 35, pp. 1417-1444, 1956 Edward J. McCluskey, Professor of Electrical Engineering and Computer Science at Stanford http://www-crc.stanford.edu/users/ejm/McCluskey_Edward.html

4. 4 Outline of the Quine-McCluskey Method 1. Produce a minterm expansion (standard sum-of-products form) for a function F 2. Eliminate as many literals as possible by systematically applying XY + XY’ = X. 3. Use a prime implicant chart to select a minimum set of prime implicants that when ORed together produce F, and that contains a minimum number of literals.

5. 5 Determination of Prime Implicants AB’CD’ + AB’CD = AB’C 1 0 1 0 + 1 0 1 1 = 1 0 1 - (The dash indicates a missing variable) A’BC’D + A’BCD’ 0 1 0 1 + 0 1 1 0 We can combine the minterms above because they differ by a single bit. The minterms below won’t combine

6. 6 Quine-McCluskey Method An Example 1. Find all the prime implicants group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Group the minterms according to the number of 1s in the minterm. This way we only have to compare minterms from adjacent groups.

7. 7 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II Combining group 0 and group 1:

8. 8 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000-  Combining group 0 and group 1:

9. 9 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0   Combining group 0 and group 1:

10. 10 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000    Does it make sense to no combine group 0 with group 2 or 3? No, there are at least two bits that are different.

11. 11 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000    Does it make sense to no combine group 0 with group 2 or 3? No, there are at least two bits that are different. Thus, next we combine group 1 and group 2.

12. 12 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01     Combine group 1 and group 2.

13. 13 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01     Combine group 1 and group 2.

14. 14 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001      Combine group 1 and group 2.

15. 15 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001      Combine group 1 and group 2.

16. 16 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001      Combine group 1 and group 2.

17. 17 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001      Combine group 1 and group 2.

18. 18 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10       Combine group 1 and group 2.

19. 19 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10       Combine group 1 and group 2.

20. 20 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010        Combine group 1 and group 2.

21. 21 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010        Combine group 1 and group 2.

22. 22 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010        Combine group 1 and group 2.

23. 23 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100-        Combine group 1 and group 2.

24. 24 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0        Again, there is no need to try to combine group 1 with group 2.

25. 25 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0        Again, there is no need to try to combine group 1 with group 3. Lets try to combine group 2 with group 23.

26. 26 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1         Combine group 2 and group 3.

27. 27 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1         Combine group 2 and group 3.

28. 28 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011-         Combine group 2 and group 3.

29. 29 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110          Combine group 2 and group 3.

30. 30 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110          Combine group 2 and group 3.

31. 31 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110          Combine group 2 and group 3.

32. 32 Quine-McCluskey Method An Example group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110 Column I Column II  0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110          Combine group 2 and group 3.

33. 33 Quine-McCluskey Method An Example Column I Column II 0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           We have now completed the first step. All minterms in column I were included. We can divide column II into groups.

34. 34 Quine-McCluskey Method An Example Column I Column II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10

35. 35 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III

36. 36 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III

37. 37 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III

38. 38 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III

39. 39 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III

40. 40 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00-  

41. 41 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00-  

42. 42 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00-  

43. 43 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00-  

44. 44 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00-  

45. 45 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00-  

46. 46 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00-  

47. 47 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0    

48. 48 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0    

49. 49 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00-      

50. 50 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00-      

51. 51 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0       

52. 52 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0       

53. 53 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0       

54. 54 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0       

55. 55 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0       

56. 56 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10         

57. 57 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10          

58. 58 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10          

59. 59 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10          

60. 60 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10           No more combinations are possible, thus we stop here.

61. 61 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 0,8,1,9 -00- 0,8,2,10 -0-0 2,6,10,14 --10 2,10,6,14 --10           We can eliminate repeated combinations

62. 62 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10           f = a’c’d Now we form f with the terms not checked

63. 63 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10           f = a’c’d + a’bd

64. 64 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10           f = a’c’d + a’bd + a’bc

65. 65 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10           f = a’c’d + a’bd + a’bc + b’c’

66. 66 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10           f = a’c’d + a’bd + a’bc + b’c’ + b’d’

67. 67 Quine-McCluskey Method An Example Column IColumn II group 0 group 1 group 2 group 3 0 0000 1 0001 2 0010 8 1000 5 0101 6 0110 9 1001 10 1010 7 0111 14 1110           0,1 000- 0,2 00-0 0,8 -000 1,5 0-01 1,9 -001 2,6 0-10 2,10 -010 8,9 100- 8,10 10-0 5,7 01-1 6,7 011- 6,14 -110 10,14 1-10 Column III 0,1,8,9 -00- 0,2,8,10 -0-0 2,6,10,14 --10           f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’

68. 68 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ But, the form below is not minimized, using a Karnaugh map we can obtain: a 1 b c d 1

69. 69 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ But, the form below is not minimized, using a Karnaugh map we can obtain: a 1 1 b c d 1

70. 70 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ But, the form below is not minimized, using a Karnaugh map we can obtain: a 1 1 1 b c d 1

71. 71 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ But, the form below is not minimized, using a Karnaugh map we can obtain: a 1 1 1 1 1 1 b c d 1

72. 72 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ But, the form below is not minimized, using a Karnaugh map we can obtain: a 1 1 1 1 1 1 11 b c d 1

73. 73 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ But, the form below is not minimized. Using a Karnaugh map we can obtain: a 1 1 1 1 1 1 111 b c d 1

74. 74 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ But, the form below is not minimized, using a Karnaugh map we can obtain: a 1 1 1 1 1 1 111 b c d 1 F = a’bd

75. 75 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ But, the form below is not minimized, using a Karnaugh map we can obtain: a 1 1 1 1 1 1 111 b c d 1 F = a’bd + cd’

76. 76 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ But, the form below is not minimized, using a Karnaugh map we can obtain: a 1 1 1 1 1 1 111 b c d 1 F = a’bd + cd’ + b’c’

77. 77 Quine-McCluskey Method An Example f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’ What are the extra terms in the solution obtained with the Quine-McCluskey method? a 1 1 1 1 1 1 111 b c d 1 F = a’bd + cd’ + b’c’ Thus, we need a method to eliminate this redundant terms from the Quine-McCluskey solution.

78. 78 The Prime Implicant Chart The prime implicant chart is the second part of the Quine-McCluskey procedure. It is used to select a minimum set of prime implicants. Similar to the Karnaugh map, we first select the essential prime implicants, and then we select enough prime implicants to cover all the minterms of the function.

79. 79 Prime Implicant Chart (Example) Question: Given the prime implicant chart above, how can we identify the essential prime implicants of the function? minterms Prime Implicants

80. 80 Prime Implicant Chart (Example) Similar to the Karnaugh map, all we have to do is to look for minterms that are covered by a single term. minterms Prime Implicants

81. 81 Prime Implicant Chart (Example) Once a term is included in the solution, all the minterms covered by that term are covered. Therefore we may now mark the covered minterms and find terms that are no longer useful. minterms Prime Implicants

82. 82 Prime Implicant Chart (Example) minterms Prime Implicants

83. 83 Prime Implicant Chart (Example) As we have not covered all the minterms with essential prime implicants, we must choose enough non-essential prime implicants to cover the remaining minterms. minterms Prime Implicants

84. 84 Prime Implicant Chart (Example) What strategy should we use to find a minimum cover for the remaining minterms? minterms Prime Implicants

85. 85 Prime Implicant Chart (Example) We choose first prime implicants that cover the most minterms. Should this strategy always work?? minterms Prime Implicants

86. 86 Prime Implicant Chart (Example) Therefore our minimum solution is: f(a,b,c,d) = b’c’ + cd’ + a’bd minterms Prime Implicants

87. 87 Cyclic Prime Implicant Chart F(a,b,c) =  m(0, 1, 2, 5, 6, 7) 0 000  1 001  2 010  5 101  6 110  7 111  0,1 00- 0,2 0-0 1,5 -01 2,6 -10 5,7 1-1 6,7 11- Which ones are the essential prime implicants in this chart? There is no essential prime implicants, how we proceed? minterms Prime Implicants

88. 88 Cyclic Prime Implicant Chart F(a,b,c) =  m(0, 1, 2, 5, 6, 7) 0 000  1 001  2 010  5 101  6 110  7 111  0,1 00- 0,2 0-0 1,5 -01 2,6 -10 5,7 1-1 6,7 11- Also, all implicants cover the same number of minterms. We will have to proceed by trial and error. minterms Prime Implicants F(a,b,c) = a’b’

89. 89 Cyclic Prime Implicant Chart F(a,b,c) =  m(0, 1, 2, 5, 6, 7) 0 000  1 001  2 010  5 101  6 110  7 111  0,1 00- 0,2 0-0 1,5 -01 2,6 -10 5,7 1-1 6,7 11- Also, all implicants cover the same number of minterms. We will have to proceed by trial and error. minterms Prime Implicants F(a,b,c) = a’b’ + bc’

90. 90 Cyclic Prime Implicant Chart F(a,b,c) =  m(0, 1, 2, 5, 6, 7) 0 000  1 001  2 010  5 101  6 110  7 111  0,1 00- 0,2 0-0 1,5 -01 2,6 -10 5,7 1-1 6,7 11- Thus, we get the minimization: F(a,b,c) = a’b’ + bc’ + ac minterms Prime Implicants

91. 91 Cyclic Prime Implicant Chart F(a,b,c) =  m(0, 1, 2, 5, 6, 7) 0 000  1 001  2 010  5 101  6 110  7 111  0,1 00- 0,2 0-0 1,5 -01 2,6 -10 5,7 1-1 6,7 11- Lets try another set of prime implicants. minterms Prime Implicants

92. 92 Cyclic Prime Implicant Chart F(a,b,c) =  m(0, 1, 2, 5, 6, 7) 0 000  1 001  2 010  5 101  6 110  7 111  0,1 00- 0,2 0-0 1,5 -01 2,6 -10 5,7 1-1 6,7 11- Lets try another set of prime implicants. minterms Prime Implicants F(a,b,c) = a’c

93. 93 Cyclic Prime Implicant Chart F(a,b,c) =  m(0, 1, 2, 5, 6, 7) 0 000  1 001  2 010  5 101  6 110  7 111  0,1 00- 0,2 0-0 1,5 -01 2,6 -10 5,7 1-1 6,7 11- Lets try another set of prime implicants. minterms Prime Implicants F(a,b,c) = a’c + b’c’

94. 94 Cyclic Prime Implicant Chart F(a,b,c) =  m(0, 1, 2, 5, 6, 7) 0 000  1 001  2 010  5 101  6 110  7 111  0,1 00- 0,2 0-0 1,5 -01 2,6 -10 5,7 1-1 6,7 11- Lets try another set of prime implicants. minterms Prime Implicants F(a,b,c) = a’c + b’c’+ ab

95. 95 Cyclic Prime Implicant Chart F(a,b,c) =  m(0, 1, 2, 5, 6, 7) 0 000  1 001  2 010  5 101  6 110  7 111  0,1 00- 0,2 0-0 1,5 -01 2,6 -10 5,7 1-1 6,7 11- This time we obtain: F(a,b,c) = a’c + b’c’+ ab minterms Prime Implicants

96. 96 Cyclic Prime Implicant Chart Which minimal form is better? F(a,b,c) = a’b’ + bc’ + ac F(a,b,c) = a’c + b’c’+ ab Depends on what terms we must form for other functions that we must also implement. Often we are interested in examining all minimal forms for a given function. Thus we need an algorithm to do so.

97. 97 Petrick’s Method S. R. Petrick. A direct determination of the irredundant forms of a boolean function from the set of prime implicants. Technical Report AFCRC-TR-56-110, Air Force Cambridge Research Center, Cambridge, MA, April, 1956. Goal: Given a prime implicant chart, determine all minimum sum-of-products solutions.

98. 98 Petrick’s Method An Example Step 1: Label all the rows in the chart. Step 2: Form a logic function P with the logic variables P 1, P 2, P 3 that is true when all the minterms in the chart are covered. minterms Prime Implicants

99. 99 Petrick’s Method An Example The first column has an X in rows P 1 and P 2. Therefore we must include one of these rows in order to cover minterm 0. Thus the following term must be in P: (P 1 + P 2 ) minterms Prime Implicants

100. 100 Petrick’s Method An Example Following this technique, we obtain: P = (P 1 + P 2 ) (P 1 + P 3 ) (P 2 + P 4 ) (P 3 + P 5 ) (P 4 + P 6 ) (P 5 + P 6 ) P = (P 1 + P 2 ) (P 1 + P 3 ) (P 4 + P 2 ) (P 5 + P 3 ) (P 4 + P 6 ) (P 5 + P 6 ) P = (P 1 + P 2 ) (P 1 + P 3 ) (P 4 + P 2 ) (P 4 + P 6 ) (P 5 + P 3 ) (P 5 + P 6 ) P = (P 1 + P 2 P 3 ) (P 4 + P 2 P 6 ) (P 5 + P 3 P 6 ) minterms Prime Implicants

101. 101 Petrick’s Method An Example P = (P 1 + P 2 ) (P 1 + P 3 ) (P 2 + P 4 ) (P 3 + P 5 ) (P 4 + P 6 ) (P 5 + P 6 ) P = (P 1 + P 2 ) (P 1 + P 3 ) (P 4 + P 2 ) (P 5 + P 3 ) (P 4 + P 6 ) (P 5 + P 6 ) P = (P 1 + P 2 ) (P 1 + P 3 ) (P 4 + P 2 ) (P 4 + P 6 ) (P 5 + P 3 ) (P 5 + P 6 ) P = (P 1 + P 2 P 3 ) (P 4 + P 2 P 6 ) (P 5 + P 3 P 6 ) P = (P 1 P 4 + P 1 P 2 P 6 + P 2 P 3 P 4 + P 2 P 3 P 6 ) (P 5 + P 3 P 6 ) P = P 1 P 4 P 5 + P 1 P 2 P 5 P 6 + P 2 P 3 P 4 P 5 + P 2 P 3 P 5 P 6 + P 1 P 3 P 4 P 6 + P 1 P 2 P 3 P 6 + P 2 P 3 P 4 P 6 + P 2 P 3 P 6 P = P 1 P 4 P 5 + P 1 P 2 P 5 P 6 + P 2 P 3 P 4 P 5 + P 1 P 4 P 3 P 6 + P 2 P 3 P 6

102. 102 Petrick’s Method An Example P = P 1 P 4 P 5 + P 1 P 2 P 5 P 6 + P 2 P 3 P 4 P 5 + P 1 P 4 P 3 P 6 + P 2 P 3 P 6 This expression says that to cover all the minterms we must include the terms in line P 1 and line P 4 and line P 5, or we must include line P 1, and line P 2, and line P 5, and line P 6, or … Considering that all the terms P 1, P 2, … have the same cost, how many minimal forms the function has? The two minimal forms are P 1 P 4 P 5 and P 2 P 3 P 6: F = a’b’ + bc’ + ac F = a’c’ + b’c + ab