Implementation of Perceptron Algorithm for NOR Logic Gate with 2-bit Binary Input


In the field of Machine Learning, the Perceptron is a Supervised Learning Algorithm for binary classifiers. The Perceptron Model implements the following function:

    \[ \begin{array}{c} \hat{y}=\Theta\left(w_{1} x_{1}+w_{2} x_{2}+\ldots+w_{n} x_{n}+b\right) \\ =\Theta(\mathbf{w} \cdot \mathbf{x}+b) \\ \text { where } \Theta(v)=\left\{\begin{array}{cc} 1 & \text { if } v \geqslant 0 \\ 0 & \text { otherwise } \end{array}\right. \end{array} \]

For a particular choice of the weight vector $\boldsymbol{w}$ and bias parameter $\boldsymbol{b}$, the model predicts output $\boldsymbol{\hat{y}}$ for the corresponding input vector $\boldsymbol{x}$.

NOR logical function truth table for 2-bit binary variables, i.e, the input vector $\boldsymbol{x} : (\boldsymbol{x_{1}}, \boldsymbol{x_{2}})$ and the corresponding output $\boldsymbol{y}$

$\boldsymbol{x_{1}}$ $\boldsymbol{x_{2}}$ $\boldsymbol{y}$
0 0 1
0 1 0
1 0 0
1 1 0

We can observe that, $NOR(\boldsymbol{x_{1}}, \boldsymbol{x_{2}}) = NOT(OR(\boldsymbol{x_{1}}, \boldsymbol{x_{2}}))$
Now for the corresponding weight vector $\boldsymbol{w} : (\boldsymbol{w_{1}}, \boldsymbol{w_{2}})$ of the input vector $\boldsymbol{x} : (\boldsymbol{x_{1}}, \boldsymbol{x_{2}})$ to the OR node, the associated Perceptron Function can be defined as:



    \[$\boldsymbol{\hat{y}\prime} = \Theta\left(w_{1} x_{1}+w_{2} x_{2}+b_{OR}\right)$ \]

Later on, the output of OR node $\boldsymbol{\hat{y}\prime}$ is the input to the NOT node with weight $\boldsymbol{w_{NOT}}$. Then the corresponding output $\boldsymbol{\hat{y}}$ is the final output of the NOR logic function and the associated Perceptron Function can be defined as:

    \[$\boldsymbol{\hat{y}} = \Theta\left(w_{NOT}  \boldsymbol{\hat{y}\prime}+b_{NOT}\right)$\]


For the implementation, considered weight parameters are $\boldsymbol{w_{1}} = 1, \boldsymbol{w_{2}} = 1, \boldsymbol{w_{NOT}} = -1$ and the bias parameters are $\boldsymbol{b_{OR}} = -0.5, \boldsymbol{b_{NOT}} = 0.5$.

Python Implementation:

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# 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)))

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Output:

NOR(0, 1) = 0
NOR(1, 1) = 0
NOR(0, 0) = 1
NOR(1, 0) = 0

Here, the model predicted output ($\boldsymbol{\hat{y}}$) for each of the test inputs are exactly matched with the NOR logic gate conventional output ($\boldsymbol{y}$) 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.

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Improved By : Akanksha_Rai