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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Python XOR Neural Network"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**(C) 2018 by Oren Baldinger**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**License:** [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/) ([CA BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Perceptrons"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "A perceptron, or single layer neural network with a linear activation function, cannot model functions that are not linearly separable. Minsky and Papert (1969) demonstrated that the XOR function in particular cannot be modeled this way."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Neural Networks"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Goodfellow et al. (2016) showed that by using two layers with the non-linear ReLU activation function, XOR can be modeled by a neural network."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The basic idea is that the first layer maps the XOR function onto a new, linearly separable space that the output layer can successfully classify (as 0 or 1)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This network can be implemented as follows:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "def relu(z):\n",
+    "    return max(0, z)\n",
+    "\n",
+    "def xor(x1, x2):\n",
+    "    activaiton = np.vectorize(relu)\n",
+    "\n",
+    "    # hidden layer\n",
+    "    W = np.array([[1, 1], [1, 1]])\n",
+    "    b = np.array([0, -1])\n",
+    "    x = np.array([x1, x2])\n",
+    "    h = activaiton(W.dot(x) + b)\n",
+    "\n",
+    "    # output layer\n",
+    "    return relu(np.array([1, -2]).dot(h))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0 XOR 0: 0\n",
+      "0 XOR 1: 1\n",
+      "1 XOR 0: 1\n",
+      "1 XOR 1: 0\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(\"0 XOR 0:\", xor(0, 0))\n",
+    "print(\"0 XOR 1:\", xor(0, 1))\n",
+    "print(\"1 XOR 0:\", xor(1, 0))\n",
+    "print(\"1 XOR 1:\", xor(1, 1))"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.5.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}