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-f98cbf4744582c2b3309f1b0ceb8a313_l3-5.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 
.
NOR 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 | 0 |
We can observe that, 
Now for the corresponding weight vector 
of the input vector to the OR node, the associated Perceptron Function can be defined as:
![\[$\boldsymbol{\hat{y}\prime} = \Theta\left(w_{1} x_{1}+w_{2} x_{2}+b_{OR}\right)$ \]](https://geeksforgeeks.org/wp-content/uploads/2023/10/quicklatex.com-c19eecbb6b1a49319b47abb2e01728a9_l3.png "Rendered by QuickLaTeX.com")
Later on, the output of OR node 
is the input to the NOT node with weight 
. Then the corresponding output is the final output of the NOR logic function and the associated Perceptron Function can be defined as:
![\[$\boldsymbol{\hat{y}} = \Theta\left(w_{NOT} \boldsymbol{\hat{y}\prime}+b_{NOT}\right)$\]](https://geeksforgeeks.org/wp-content/uploads/2023/10/quicklatex.com-176aa68936dc6b2a6618fc23c6708820_l3-1.png "Rendered by QuickLaTeX.com")
For the implementation, considered weight parameters are 
and the bias parameters are 
.
Python Implementation:
# importing Python library import numpy as np # define Unit Step Function def unitStep(v): if v >= 0: return 1 else: return 0 # design Perceptron Model def perceptronModel(x, w, b): v = np.dot(w, x) + b y = unitStep(v) return y # NOT Logic Function # wNOT = -1, bNOT = 0.5 def NOT_logicFunction(x): wNOT = -1 bNOT = 0.5 return perceptronModel(x, wNOT, bNOT) # OR Logic Function # w1 = 1, w2 = 1, bOR = -0.5 def OR_logicFunction(x): w = np.array([1, 1]) bOR = -0.5 return perceptronModel(x, w, bOR) # NOR Logic Function # with OR and NOT # function calls in sequence def NOR_logicFunction(x): output_OR = OR_logicFunction(x) output_NOT = NOT_logicFunction(output_OR) return output_NOT # testing the Perceptron Model test1 = np.array([0, 1]) test2 = np.array([1, 1]) test3 = np.array([0, 0]) test4 = np.array([1, 0]) print("NOR({}, {}) = {}".format(0, 1, NOR_logicFunction(test1))) print("NOR({}, {}) = {}".format(1, 1, NOR_logicFunction(test2))) print("NOR({}, {}) = {}".format(0, 0, NOR_logicFunction(test3))) print("NOR({}, {}) = {}".format(1, 0, NOR_logicFunction(test4))) |
NOR(0, 1) = 0 NOR(1, 1) = 0 NOR(0, 0) = 1 NOR(1, 0) = 0
Here, the model predicted output () for each of the test inputs are exactly matched with the NOR logic gate conventional output () according to the truth table for 2-bit binary input.
Hence, it is verified that the perceptron algorithm for NOR logic gate is correctly implemented.
