Emulating Logical Gates with a Neural Network by James Fulton

Let’s go with a single hidden layer with two nodes in it. We’ll be using the sigmoid function in each of our hidden layer nodes and of course, our output node. There are no fixed rules on the number of hidden layers or the number of nodes in each layer of a network. The best performing models are obtained through trial and error. To train our perceptron, we must ensure that we correctly classify all of our train data.

Training algorithm

The only difference is that we have engineered the third feature x3_torch which is equal to element-wise product of the first feature x1_torch and the second feature x2_torch. The next step is to create the LogisticRegression() class. To be able to use it as a PyTorch model, we will pass torch. Then, we will define the init() function by passing the parameter self. This exercise brings to light the importance of representing a problem correctly.

  1. The output unit also parses the sum of its input values through an activation function — again, the sigmoid function is appropriate here — to return an output value falling between 0 and 1.
  2. First, we will create our decision table were x1 and x2 are two NumPy arrays consisting of four numbers.
  3. Just by looking at this graph, we can say that the data was almost perfectly classified.
  4. In the image above we see the evolution of the elements of \(W\).
  5. The most important thing to remember from this example is the points didn’t move the same way (some of them did not move at all).
  6. It means that from the four possible combinations only two will have 1 as output.

Deep learning neural networks are trained using the stochastic gradient descent optimization algorithm. As part of the…

The Output of XOR logic is yielded by the equation as shown below. Neural network researchers find the XOR problem particularly intriguing because it is a complicated binary function that cannot be resolved by a neural network. In the case of XOR problem, unsupervised learning techniques like clustering or dimensionality reduction can be used to find patterns in data without any labeled examples.


Empirically, it is better to use the ReLU instead of the softplus. Furthermore, the dead ReLU is a more important problem than the non-differentiability at the origin. Then, at the end, the pros (simple evaluation and simple slope) outweight the cons (dead neuron and non-differentiability at the origin). If you want to read another explanation on why a stack of linear layers is still linear, please access this Google’s Machine Learning Crash Course page.

Weight initialization is an important aspect of a neural network architecture. But, Similar to the case of input parameters, for many practical problems the output data available with us may have missing values to some given inputs. And it could be dealt with the same approaches described above. We are running 1000 iterations to fit the model to given data.

However, with the 1969 book named ‘Perceptrons’, written by Minsky and Paper, the limitations of using linear classification became more apparent. So these new weights gave us a small adjustment, and our new output is $0.538$. We’ll give our inputs, which is either 0 or 1, and they both will be multiplied by the synaptic weight.

This is important because it allows the network to start learning from scratch. Neural networks have been proven to solve complex problems, and one of the most challenging ones is the XOR problem. In this article, we will explore how neural networks can solve this problem and provide a better xor neural network understanding of their capabilities. For each of the element of the input vector \(x\), i.e., \(x_1\) and \(x_2\), we will multiply a weight vector \(w\) and a bias \(w_0\). We, then, convert the vector \(x\) into a scalar value represented by \(z\) and passed onto the sigmoid function.

Traditional neural networks also use a single layer of neurons which makes it difficult for them to learn complex patterns in data. To solve complex problems like the XOR problem with traditional neural networks, we would need to add more layers and neurons which can lead to overfitting and slow learning. Traditional neural networks use linear activation functions that can only model linear relationships between input variables. In other words, they can only learn patterns that are directly proportional or inversely proportional to each other. The information of a neural network is stored in the interconnections between the neurons i.e. the weights.

In this tutorial I want to show you how you can train a neural networks to perform the function of a network of logical gates. I’m am going to dive into the purpose of each individual neuron in the network and show that none are wasted. We’ll ground this by using a problem example from Tensorflow Playground so that you can implement and experiment with this idea yourself. Then you can model this problem as a neural network, a model that will learn and will calibrate itself to provide accurate solutions. Some machine learning algorithms like neural networks are already a black box, we enter input in them and expect magic to happen.

It happened because their negative coordinates were the y ones. It happened due to the fact their x coordinates were negative. Note every moved coordinate became zero (ReLU effect, right?) and the orange’s non negative coordinate was zero (just like the black’s one). The black and orange points ended up in the same place (the origin), and the image just shows the black dot.

Then we will store loss inside this all_loss list that we have created. Now we can call the function create_dataset() and plot our data. We will create an https://forexhero.info/ index_shuffle variable and apply np.arrange() function on x1.shape[0]. Next, we will use the function  np.random.shuffle() on the variable index_shuffle.

Millions of these neural connections exist throughout our bodies, collectively referred to as neural networks. The loss function we used in our MLP model is the Mean Squared loss function. Though this is a very popular loss function, it makes some assumptions on the data (like it being gaussian) and isn’t always convex when it comes to a classification problem. It was used here to make it easier to understand how a perceptron works, but for classification tasks, there are better alternatives, like binary cross-entropy loss.

If we represent the problem at hand in a more suitable way, many difficult scenarios become easy to solve as we saw in the case of the XOR problem. In the 1950s and the 1960s, linear classification was widely used and in fact, showed significantly good results on simple image classification problems such as the Perceptron. To speed things up with the beauty of computer science – when we run this iteration 10,000 times, it gives us an output of about $.9999$.

It is the weights that determine where the classification line, the line that separates data points into classification groups, is drawn. If all data points on one side of a classification line are assigned the class of 0, all others are classified as 1. Hence, it signifies that the Artificial Neural Network for the XOR logic gate is correctly implemented. In this tutorial I am going to use a neural network to emulate the behaviour of a network of logical gates.