Implementation of Artificial Neural Network for NOR Logic Gate with 2-bit Binary Input


Artificial Neural Network (ANN) is a computational model based on the biological neural networks of animal brains. ANN is modeled with three types of layers: an input layer, hidden layers (one or more), and an output layer. Each layer comprises nodes (like biological neurons) are called Artificial Neurons. All nodes are connected with weighted edges (like synapses in biological brains) between two layers. Initially, with the forward propagation function, the output is predicted. Then through backpropagation, the weight and bias to the nodes are updated to minimizing the error in prediction to attain the convergence of cost function in determining the final output.

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

Approach:
Step1: Import the required Python libraries
Step2: Define Activation Function : Sigmoid Function
Step3: Intialize neural network parameters (weights, bias)
and define model hyperparameters (number of iterations, learning rate)
Step4: Forward Propagation
Step5: Backward Propagation
Step6: Update weight and bias parameters
Step7: Train the learning model
Step8: Plot Loss value vs Epoch
Step9: Test the model performance

Python Implementation:

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# import Python Libraries
import numpy as np
from matplotlib import pyplot as plt
  
# Sigmoid Function
def sigmoid(z):
    return 1 / (1 + np.exp(-z))
  
# Initialization of the neural network parameters
# Initialized all the weights in the range of between 0 and 1
# Bias values are initialized to 0
def initializeParameters(inputFeatures, neuronsInHiddenLayers, outputFeatures):
    W1 = np.random.randn(neuronsInHiddenLayers, inputFeatures)
    W2 = np.random.randn(outputFeatures, neuronsInHiddenLayers)
    b1 = np.zeros((neuronsInHiddenLayers, 1))
    b2 = np.zeros((outputFeatures, 1))
      
    parameters = {"W1" : W1, "b1": b1,
                  "W2" : W2, "b2": b2}
    return parameters
  
# Forward Propagation
def forwardPropagation(X, Y, parameters):
    m = X.shape[1]
    W1 = parameters["W1"]
    W2 = parameters["W2"]
    b1 = parameters["b1"]
    b2 = parameters["b2"]
  
    Z1 = np.dot(W1, X) + b1
    A1 = sigmoid(Z1)
    Z2 = np.dot(W2, A1) + b2
    A2 = sigmoid(Z2)
  
    cache = (Z1, A1, W1, b1, Z2, A2, W2, b2)
    logprobs = np.multiply(np.log(A2), Y) + np.multiply(np.log(1 - A2), (1 - Y))
    cost = -np.sum(logprobs) / m
    return cost, cache, A2
  
# Backward Propagation
def backwardPropagation(X, Y, cache):
    m = X.shape[1]
    (Z1, A1, W1, b1, Z2, A2, W2, b2) = cache
      
    dZ2 = A2 - Y
    dW2 = np.dot(dZ2, A1.T) / m
    db2 = np.sum(dZ2, axis = 1, keepdims = True)
      
    dA1 = np.dot(W2.T, dZ2)
    dZ1 = np.multiply(dA1, A1 * (1- A1))
    dW1 = np.dot(dZ1, X.T) / m
    db1 = np.sum(dZ1, axis = 1, keepdims = True) / m
      
    gradients = {"dZ2": dZ2, "dW2": dW2, "db2": db2,
                 "dZ1": dZ1, "dW1": dW1, "db1": db1}
    return gradients
  
# Updating the weights based on the negative gradients
def updateParameters(parameters, gradients, learningRate):
    parameters["W1"] = parameters["W1"] - learningRate * gradients["dW1"]
    parameters["W2"] = parameters["W2"] - learningRate * gradients["dW2"]
    parameters["b1"] = parameters["b1"] - learningRate * gradients["db1"]
    parameters["b2"] = parameters["b2"] - learningRate * gradients["db2"]
    return parameters
  
# Model to learn the NOR truth table 
X = np.array([[0, 0, 1, 1], [0, 1, 0, 1]]) # NOR input
Y = np.array([[1, 0, 0, 0]]) # NOR output
  
# Define model parameters
neuronsInHiddenLayers = 2 # number of hidden layer neurons (2)
inputFeatures = X.shape[0] # number of input features (2)
outputFeatures = Y.shape[0] # number of output features (1)
parameters = initializeParameters(inputFeatures, neuronsInHiddenLayers, outputFeatures)
epoch = 100000
learningRate = 0.01
losses = np.zeros((epoch, 1))
  
for i in range(epoch):
    losses[i, 0], cache, A2 = forwardPropagation(X, Y, parameters)
    gradients = backwardPropagation(X, Y, cache)
    parameters = updateParameters(parameters, gradients, learningRate)
  
# Evaluating the performance
plt.figure()
plt.plot(losses)
plt.xlabel("EPOCHS")
plt.ylabel("Loss value")
plt.show()
  
# Testing
X = np.array([[1, 1, 0, 0], [0, 1, 0, 1]]) # NOR input
cost, _, A2 = forwardPropagation(X, Y, parameters)
prediction = (A2 > 0.5) * 1.0
# print(A2)
print(prediction)

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Output:


[[ 0.  0.  1.  0.]]

Here, the model predicted output for each of the test inputs are exactly matched with the NOR logic gate conventional output ($\boldsymbol{y}$) according to the truth table and the cost function is also continuously converging.
Hence, it signifies that the Artificial Neural Network for the NOR logic gate is correctly implemented.




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