In the field of Machine Learning, the Perceptron is a Supervised Learning Algorithm for binary classifiers. The Perceptron Model implements the following function:
![\[ \begin{array}{c} \hat{y}=\Theta\left(w_{1} x_{1}+w_{2} x_{2}+\ldots+w_{n} x_{n}+b\right) \\ =\Theta(\mathbf{w} \cdot \mathbf{x}+b) \\ \text { where } \Theta(v)=\left\{\begin{array}{cc} 1 & \text { if } v \geqslant 0 \\ 0 & \text { otherwise } \end{array}\right. \end{array} \]](https://geeksforgeeks.org/wp-content/uploads/2023/10/quicklatex.com-f7626bb7672b45e3187df50ef64cd4bc_l3-1.png "Rendered by QuickLaTeX.com")
For a particular choice of the weight vector 
and bias parameter 
, the model predicts output 
for the corresponding input vector 
. XNOR logical function truth table for 2-bit binary variables, i.e, the input vector 
and the corresponding output 
–
 |
 |
 |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
We can observe that, 
Designing the Perceptron Network:
- Step1: Now for the corresponding weight vector
of the input vector 
to the OR and AND node, the associated Perceptron Function can be defined as:
![\[$\boldsymbol{\hat{y}_{1}} = \Theta\left(w_{1} x_{1}+w_{2} x_{2}+b_{OR}\right)$ \]](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20332%2027'%3E%3C/svg%3E "Rendered by QuickLaTeX.com")
[Tex]\[$\boldsymbol{\hat{y}_{2}} = \Theta\left(w_{1} x_{1}+w_{2} x_{2}+b_{AND}\right)$ \] [/Tex]
- Step2: The output
from the OR node will be inputted to the NOT node with weight 
and the associated Perceptron Function can be defined as:
![\[$\boldsymbol{\hat{y}_{3}} = \Theta\left(w_{NOT} \boldsymbol{\hat{y}_{1}}+b_{NOT}\right)$\]](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20301%2027'%3E%3C/svg%3E "Rendered by QuickLaTeX.com")
- Step3: The output from the AND node and the output from NOT node as mentioned in Step2 will be inputted to the OR node with weight
. Then the corresponding output 
is the final output of the XNOR logic function. The associated Perceptron Function can be defined as:
![\[$\boldsymbol{\hat{y}} = \Theta\left(w_{OR1} \boldsymbol{\hat{y}_{3}}+w_{OR2} \boldsymbol{\hat{y}_{2}}+b_{OR}\right)$\]](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20387%2027'%3E%3C/svg%3E "Rendered by QuickLaTeX.com")
of the input vector ![\[$\boldsymbol{\hat{y}_{1}} = \Theta\left(w_{1} x_{1}+w_{2} x_{2}+b_{OR}\right)$ \]](https://geeksforgeeks.org/wp-content/uploads/2023/10/quicklatex.com-977101357b34fb859ced599bcefae030_l3.png "Rendered by QuickLaTeX.com")
from the OR node will be inputted to the NOT node with weight 
and the associated Perceptron Function can be defined as:
![\[$\boldsymbol{\hat{y}_{3}} = \Theta\left(w_{NOT} \boldsymbol{\hat{y}_{1}}+b_{NOT}\right)$\]](https://geeksforgeeks.org/wp-content/uploads/2023/10/quicklatex.com-ba58779f2294aab77613511a51fec2fe_l3-1.png "Rendered by QuickLaTeX.com")
from the AND node and the output 
from NOT node as mentioned in Step2 will be inputted to the OR node with weight 
. Then the corresponding output ![\[$\boldsymbol{\hat{y}} = \Theta\left(w_{OR1} \boldsymbol{\hat{y}_{3}}+w_{OR2} \boldsymbol{\hat{y}_{2}}+b_{OR}\right)$\]](https://geeksforgeeks.org/wp-content/uploads/2023/10/quicklatex.com-b4ec4a2e01d87a282e4912f727867ea5_l3.png "Rendered by QuickLaTeX.com")
For the implementation, the weight parameters are considered to be 
and the bias parameters are 
. Python Implementation:
Python3
# importing Python libraryimport numpy as np# define Unit Step Functiondef unitStep(v): if v >= 0: return 1 else: return 0# design Perceptron Modeldef perceptronModel(x, w, b): v = np.dot(w, x) + b y = unitStep(v) return y# NOT Logic Function# wNOT = -1, bNOT = 0.5def NOT_logicFunction(x): wNOT = -1 bNOT = 0.5 return perceptronModel(x, wNOT, bNOT)# AND Logic Function# w1 = 1, w2 = 1, bAND = -1.5def AND_logicFunction(x): w = np.array([1, 1]) bAND = -1.5 return perceptronModel(x, w, bAND)# OR Logic Function# here w1 = wOR1 = 1, # w2 = wOR2 = 1, bOR = -0.5def OR_logicFunction(x): w = np.array([1, 1]) bOR = -0.5 return perceptronModel(x, w, bOR)# XNOR Logic Function# with AND, OR and NOT # function calls in sequencedef XNOR_logicFunction(x): y1 = OR_logicFunction(x) y2 = AND_logicFunction(x) y3 = NOT_logicFunction(y1) final_x = np.array([y2, y3]) finalOutput = OR_logicFunction(final_x) return finalOutput# testing the Perceptron Modeltest1 = np.array([0, 1])test2 = np.array([1, 1])test3 = np.array([0, 0])test4 = np.array([1, 0])print("XNOR({}, {}) = {}".format(0, 1, XNOR_logicFunction(test1)))print("XNOR({}, {}) = {}".format(1, 1, XNOR_logicFunction(test2)))print("XNOR({}, {}) = {}".format(0, 0, XNOR_logicFunction(test3)))print("XNOR({}, {}) = {}".format(1, 0, XNOR_logicFunction(test4))) |
XNOR(0, 1) = 0 XNOR(1, 1) = 1 XNOR(0, 0) = 1 XNOR(1, 0) = 0
Here, the model predicted output () for each of the test inputs are exactly matched with the XNOR logic gate conventional output () according to the truth table. Hence, it is verified that the perceptron algorithm for XNOR logic gate is correctly implemented.
