1. Mintrue & Maxfalse method
MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES
Mintrue & Maxfalse method
Mintrue and Maxfalse method

2. Mintrue & Maxfalse method
Comparing bit by bit and determining which is greater
Mintrue & Maxfalse method
Mintrue and Maxfalse method

3. Steps involved in finding Minimum True and Maximum False vertices
Determine the positive function
Determine Minimum True vertices
Anything greater than these are True Vertices
Remaining are all False Vertices
Determine maximal false vertices
Anything less than false are neglected or crossed
Others which are not compared are false
For false we start from bottom to top
Mintrue & Maxfalse method

4. Mintrue & Maxfalse method
Example: Finding Minimum True vertices
Mintrue & Maxfalse method
Mintrue and Maxfalse method

5. Mintrue & Maxfalse method
Example 1: Find the weights and threshold for the following fn.
Step1: Determine the Positive Function
Step2: Find all Minumum True and Maximum false
vertices
Mintrue & Maxfalse method
Mintrue and Maxfalse method

6. Mintrue & Maxfalse method
Determining the Mintrue and Maxfalse vertices
Mintrue & Maxfalse method
Mintrue and Maxfalse method

7. Mintrue & Maxfalse method
Mintrue and Maxfalse method

8. Mintrue & Maxfalse method
Step3: p = Number of Minimum True Vertices
q = Number of Maximum false Vertices
Mintrue & Maxfalse method
Mintrue and Maxfalse method

9. Mintrue & Maxfalse method
Inequalities:
Mintrue & Maxfalse method
Mintrue and Maxfalse method

10. Mintrue & Maxfalse method
Solving the inequalities:
Substituting the weights in Min True vertices:
Mintrue & Maxfalse method
Mintrue and Maxfalse method

11. Mintrue & Maxfalse method
Similarly, substituting in Maximum false vertices:
Mintrue & Maxfalse method
Mintrue and Maxfalse method

12. Mintrue & Maxfalse method
The Threshold T must be smaller than 5 but larger than 4.
(Min Limit < Threshold < Max Limit)
Hence T = 4.5
Mintrue & Maxfalse method
Mintrue and Maxfalse method

13. Mintrue & Maxfalse method
In the example, the inputs X3 and X4 appear in f
in complimented form, hence the new weighted vectors
are given by:
* Complimenting the weights of X3 and X4
* Subtracting the Threshold obtained by the
weights of these two inputs
Ans:
Mintrue & Maxfalse method
Mintrue and Maxfalse method

14. Mintrue & Maxfalse method
Example 2: Find the Threshold and Weights of the following fn.
Mintrue & Maxfalse method
Mintrue and Maxfalse method

15. Mintrue & Maxfalse method
Mintrue and Maxfalse method

16. Mintrue & Maxfalse method
We obtain a system of 12 inequalities
Mintrue & Maxfalse method
Mintrue and Maxfalse method

17. Mintrue & Maxfalse method
These impose several constraints on the weights
associatedwith the function f.
Mintrue & Maxfalse method
Mintrue and Maxfalse method

18. Mintrue & Maxfalse method
Mintrue and Maxfalse method

19. Mintrue & Maxfalse method
Mintrue and Maxfalse method

20. Mintrue & Maxfalse method
T must be smaller than 4 but larger than 3.
Weight Threshold vector is given by:
(3, 2, 2, 1; 3.5)
To find the corresponding vector for the original
function, X1 and X3 must be complimented
Mintrue & Maxfalse method
Mintrue and Maxfalse method

21. Mintrue & Maxfalse method
Example 1:
Mintrue & Maxfalse method
Mintrue and Maxfalse method

22. Mintrue & Maxfalse method
* f is not unate
* We have to synthesize it as a cascade of two
threshold elements
Mintrue & Maxfalse method
Mintrue and Maxfalse method

23. Mintrue & Maxfalse method
g(X1,X2,X3,X4) = {2,3,6,7,15}
The weight-threshold vector for the function g is
Vg = (-2,1,3,1 ; 2.5)
h(X1,X2,X3,X4) = {10,12,14,15}
The weight-threshold vector for the function h is
Vh = (2,1,1,-1 ; 2.5)
Mintrue & Maxfalse method

24. Mintrue & Maxfalse method
CASCADE REALIZATION OF THE FUNCTION
Mintrue & Maxfalse method
Mintrue and Maxfalse method

25. Mintrue & Maxfalse method
* f must have a value 1 whenever g does, the minimum
weighted sum must be larger than the threshold of
the second element.
* Negative values affect the most
Mintrue & Maxfalse method
Mintrue and Maxfalse method

26. Mintrue & Maxfalse method
Calculating Wg
Wg + 0 > 5/2 is the vital case
Wg > 5/2
Wg = 3
This is the minimum weighted sum.
So Wg is calculated from this.
As a general rule, the weight of Wg should be the sum
of the threshold of second element and the absolute value
of all negative weights of the second element
Mintrue & Maxfalse method

27. Mintrue & Maxfalse method
Example 2:
Mintrue & Maxfalse method
Mintrue and Maxfalse method

28. Mintrue & Maxfalse method
* f is not unate
* We have to synthesize it as a cascade of two
threshold elements
Mintrue & Maxfalse method
Mintrue and Maxfalse method

29. Mintrue & Maxfalse method
g(X1,X2,X3,X4) = {3,5,7,15}
The weight-threshold vector for the function g is
Vg = (-1,1,1,2 ; 2.5)
h(X1,X2,X3,X4) = {10,12,14,15}
The weight-threshold vector for the function h is
Vh = (2,1,1,-1 ; 2.5)
Mintrue & Maxfalse method

30. Mintrue & Maxfalse method
CASCADE REALIZATION OF THE FUNCTION
Mintrue & Maxfalse method
Mintrue and Maxfalse method

31. Mintrue & Maxfalse method
Example 3: f(X1,X2,X3,X4 ) = (2,3,6,7,8,9,13,15)
Realize this as a cascade of two functions
Mintrue & Maxfalse method
Mintrue and Maxfalse method

32. Mintrue & Maxfalse method
Mintrue and Maxfalse method

33. Mintrue & Maxfalse method
Mintrue and Maxfalse method

34. Mintrue & Maxfalse method
Solving the inequalities, we get the weights as:
Mintrue & Maxfalse method
Mintrue and Maxfalse method

35. Mintrue & Maxfalse method
Mintrue and Maxfalse method

36. Mintrue & Maxfalse method
Cascade Realization of the given function
Mintrue & Maxfalse method
Mintrue and Maxfalse method

37. Mintrue & Maxfalse method
questions???
Mintrue & Maxfalse method