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MINIMUM TRUE AND MAXIMUM FALSE VERTICES METHOD FOR REALIZATION

OF THRESHOLD GATES

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Comparing bit by bit and determining which is greaterComparing bit by bit and determining which is greater

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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

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Example: Finding Minimum True vertices

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Example 1: Find the weights and threshold for the following fn.

Step1: Determine the Positive Function

Step2: Find all Minumum True and Maximum falsevertices

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Determining the Mintrue and Maxfalse vertices

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Step3: p = Number of Minimum True Verticesq = Number of Maximum false Vertices

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Inequalities:

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Substituting the weights in Min True vertices:

Solving the inequalities:

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Similarly, substituting in Maximum false vertices:

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The Threshold T must be smaller than 5 but larger than 4. (Min Limit < Threshold < Max Limit)

Hence T = 4.5

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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:

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Example 2: Find the Threshold and Weights of the following fn.

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We obtain a system of 12 inequalities

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These impose several constraints on the weights associatedwith the function f.

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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

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Example 1:

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* f is not unate* We have to synthesize it as a cascade of two

threshold elements

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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)

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CASCADE REALIZATION OF THE FUNCTION

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* 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

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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

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Example 2:

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* f is not unate* We have to synthesize it as a cascade of two

threshold elements

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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)

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CASCADE REALIZATION OF THE FUNCTION

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Example 3: f(X1,X2,X3,X4 ) = (2,3,6,7,8,9,13,15)

Realize this as a cascade of two functions

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Solving the inequalities, we get the weights as:

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Cascade Realization of the given function

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