Exploring Completeness: Proofs and Minimization in Circuit Design Diversity

1. Proving Incompleteness • {NAND} is a complete system • Is {XOR,0} a complete system?

2. Intuition X’Y + Y’X

3. Intuition xor(X,Y) = xor(Y,X)

4. Intuition • xor(X,Y) = xor(Y,X) • xor(x,x) = 0 • xor(x,0) = x A single layer circuit that includes {XOR,0} cannot produce the gate not(X)

5. Proof for n-layered circuit A circuit with minimal number of gates X 0 X Something is going on Here

6. Proof for n-layered circuitCase A: the other input is 0 A circuit with minimal number of gates X 0 Something is going on Here

7. Proof for n-layered circuitCase A: the other input is 0 A circuit with minimal number of gates X X 0 Something is going on Here Contradiction to minimality!!! X

8. Proof for n-layered circuitCase B: the other input is X A circuit with minimal number of gates 0 X X Something is going on Here Contradiction to minimality!!! 0

9. Proof for n-layered circuit (II)Proof in induction • For circuit with 1 layer we already prooved. • Induction assumption: • There is not circuit with n layers that can produce not with xor and 0. • Proof that there is no circuit with n+1 layers that implements not with xor.

10. Proof in induction for n-layered circuit A circuit with n+1 layer X 0 X Something is going on Here

11. Proof in induction for n-layered circuit A circuit with n+1 layer 0 X Something is going on Here Change to circuit with n layers using similar consderations A proof using the induction assumption.

12. Minimizing to sum of products and product of sums How to write in minimal form?

13. When do we minimize? ABC + ABC’ = AB(C+C’) = AB When there are two terms that differ in only one literal!!

14. Minimizing to sum of products = X’Y’Z + X’YZ’ + XY’Z’ + XYZ Nothing to minimize!

15. Minimizing to product of sums F’ = X’Y’Z’ + X’YZ+XY’Z + XYZ’ Nothing to minimize!

16. The table Method:Example Minimize : F = w’x’y’z’ + w’x’y’z + w’x’yz’ + wx’y’z’ + wx’yz’ + wx’yz + wxyz’ + wxyz Very difficult!!

17. The table method for minimizing ABC + ABC’ ABC + AB’C’

18. The table method for minimizing ABC + ABC’ ABC + AB’C’ 111110 11 11 0 0

19. The table method for minimizing ABC + ABC’ AB C + AB’C’ 111110 11 11 0 0 = 1 = 3 7 - 6 7 - 4

20. The table method for minimizing ABC + ABC’ AB C + AB’C’ 111110 11 11 0 0 = 1 = 3 7 - 6 7 - 4 20 We can minimize only if the difference is a power of 2

21. The table method for minimizing ABC + ABC’ AB C + AB’C’ 111110 11 11 0 0 = 1 = 3 7 - 6 7 - 4 20 We can minimize only if the difference is a power of 2 IS IT SUFFICIENT? No

22. The table method for minimizing AB’C + A’BC 1010 11 = 2 5 - 3 21

23. The table method for minimizing AB’C + A’BC We can minimize only if the difference is a power of 2 and the number of 1 is different! 1010 11 = 2 5 - 3 22

24. The table Method:Example Minimize : F = w’x’y’z’ + w’x’y’z + w’x’yz’ + wx’y’z’ + wx’yz’ + wx’yz + wxyz’ + wxyz = (0,1,2,8,10,11,14,15)

25. The table method

26. The table method

27. The table method

28. The table method • The minimal term: F = w’x’y’ + x’z’ + wy

29. The table method - faster

30. Choosing Minimal term F= (1,4,6,7,8,9,10,11,15)

31. The minimal terms

32. The minimal function F = x’y’z + w’xz’ + w’xy + xyz + wyz + wx’ Is it really the minimum ? No

33. The minimal function All the three account for Minterms 7,15 – maybe we can dispose one of them? F = x’y’z + w’xz’ + w’xy + xyz + wyz + wx’ Is it really the minimum ? No

34. Essential Primary Element

35. Essential Primary Element

36. Essential Primary Element

37. Choosing the other Essential Primary Element

38. The minimal function is F = x’y’z + w’xz’ + wx’ + xyz