mintrue & maxfalse method1 minimum true and maximum false vertices method for realization of...
Post on 19-Dec-2015
240 views
TRANSCRIPT
![Page 1: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/1.jpg)
Mintrue & Maxfalse method 1
MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION
OF THRESHOLD GATES
![Page 2: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/2.jpg)
Mintrue & Maxfalse method 2
Comparing bit by bit and determining which is greaterComparing bit by bit and determining which is greater
![Page 3: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/3.jpg)
Mintrue & 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
![Page 4: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/4.jpg)
Mintrue & Maxfalse method 4
Example: Finding Minimum True vertices
![Page 5: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/5.jpg)
Mintrue & Maxfalse method 5
Example 1: Find the weights and threshold for the following fn.
Step1: Determine the Positive Function
Step2: Find all Minumum True and Maximum falsevertices
![Page 6: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/6.jpg)
Mintrue & Maxfalse method 6
Determining the Mintrue and Maxfalse vertices
![Page 7: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/7.jpg)
Mintrue & Maxfalse method 7
![Page 8: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/8.jpg)
Mintrue & Maxfalse method 8
Step3: p = Number of Minimum True Verticesq = Number of Maximum false Vertices
![Page 9: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/9.jpg)
Mintrue & Maxfalse method 9
Inequalities:
![Page 10: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/10.jpg)
Mintrue & Maxfalse method 10
Substituting the weights in Min True vertices:
Solving the inequalities:
![Page 11: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/11.jpg)
Mintrue & Maxfalse method 11
Similarly, substituting in Maximum false vertices:
![Page 12: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/12.jpg)
Mintrue & Maxfalse method 12
The Threshold T must be smaller than 5 but larger than 4. (Min Limit < Threshold < Max Limit)
Hence T = 4.5
![Page 13: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/13.jpg)
Mintrue & Maxfalse method 13
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:
![Page 14: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/14.jpg)
Mintrue & Maxfalse method 14
Example 2: Find the Threshold and Weights of the following fn.
![Page 15: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/15.jpg)
Mintrue & Maxfalse method 15
![Page 16: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/16.jpg)
Mintrue & Maxfalse method 16
We obtain a system of 12 inequalities
![Page 17: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/17.jpg)
Mintrue & Maxfalse method 17
These impose several constraints on the weights associatedwith the function f.
![Page 18: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/18.jpg)
Mintrue & Maxfalse method 18
![Page 19: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/19.jpg)
Mintrue & Maxfalse method 19
![Page 20: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/20.jpg)
Mintrue & Maxfalse method 20
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
![Page 21: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/21.jpg)
Mintrue & Maxfalse method 21
Example 1:
![Page 22: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/22.jpg)
Mintrue & Maxfalse method 22
* f is not unate* We have to synthesize it as a cascade of two
threshold elements
![Page 23: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/23.jpg)
Mintrue & Maxfalse method 23
g(X1,X2,X3,X4) = {2,3,6,7,15}
The weight-threshold vector for the function g isVg = (-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)
![Page 24: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/24.jpg)
Mintrue & Maxfalse method 24
CASCADE REALIZATION OF THE FUNCTION
![Page 25: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/25.jpg)
Mintrue & Maxfalse method 25
* f must have a value 1 whenever g does, the minimumweighted sum must be larger than the threshold ofthe second element.
* Negative values affect the most
![Page 26: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/26.jpg)
Mintrue & Maxfalse method 26
Calculating Wg
Wg + 0 > 5/2 is the vital case
Wg > 5/2Wg = 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 valueof all negative weights of the second element
![Page 27: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/27.jpg)
Mintrue & Maxfalse method 27
Example 2:
![Page 28: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/28.jpg)
Mintrue & Maxfalse method 28
* f is not unate* We have to synthesize it as a cascade of two
threshold elements
![Page 29: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/29.jpg)
Mintrue & Maxfalse method 29
g(X1,X2,X3,X4) = {3,5,7,15}
The weight-threshold vector for the function g isVg = (-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)
![Page 30: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/30.jpg)
Mintrue & Maxfalse method 30
CASCADE REALIZATION OF THE FUNCTION
![Page 31: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/31.jpg)
Mintrue & Maxfalse method 31
Example 3: f(X1,X2,X3,X4 ) = (2,3,6,7,8,9,13,15)
Realize this as a cascade of two functions
![Page 32: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/32.jpg)
Mintrue & Maxfalse method 32
![Page 33: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/33.jpg)
Mintrue & Maxfalse method 33
![Page 34: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/34.jpg)
Mintrue & Maxfalse method 34
Solving the inequalities, we get the weights as:
![Page 35: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/35.jpg)
Mintrue & Maxfalse method 35
![Page 36: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/36.jpg)
Mintrue & Maxfalse method 36
Cascade Realization of the given function
![Page 37: Mintrue & Maxfalse method1 MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION OF THRESHOLD GATES](https://reader035.vdocuments.us/reader035/viewer/2022081421/56649d3e5503460f94a172e8/html5/thumbnails/37.jpg)
Mintrue & Maxfalse method 37
questions???