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

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

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

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

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

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

Mintrue & Maxfalse method Mintrue and Maxfalse method

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

Mintrue & Maxfalse method Inequalities: Mintrue & Maxfalse method Mintrue and Maxfalse method

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

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

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

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

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

Mintrue & Maxfalse method Mintrue and Maxfalse method

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

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

Mintrue & Maxfalse method Mintrue and Maxfalse method

Mintrue & Maxfalse method Mintrue and Maxfalse method

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

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

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

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

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

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

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

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

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

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

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

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

Mintrue & Maxfalse method Mintrue and Maxfalse method

Mintrue & Maxfalse method Mintrue and Maxfalse method

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

Mintrue & Maxfalse method Mintrue and Maxfalse method

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

Mintrue & Maxfalse method questions??? Mintrue & Maxfalse method