逻辑运算 神经网络

Hello everyone!! Before starting with part 2 of implementing logic gates using Neural networks, you would want to go through part1 first.

大家好！！ 与实施基于神经网络的逻辑门的第二部分开始之前，你可能想通过第一部分第一。

From part 1, we had figured out that we have two input neurons or x vector having values as x1 and x2 and 1 being the bias value. The input values, i.e., x1, x2, and 1 is multiplied with their respective weight matrix that is W1, W2, and W0. The corresponding value is then fed to the summation neuron where we have the summed value which is

从第1部分中，我们发现我们有两个输入神经元或x向量，其值分别为x1和x2，其中1为偏差值。 输入值x1，x2和1乘以它们各自的权重矩阵W1，W2和W0。 然后将相应的值馈入求和神经元，其中我们得到的总和为

Now, this value is fed to a neuron which has a non-linear function(sigmoid in our case) for scaling the output to a desirable range. The scaled output of sigmoid is 0 if the output is less than 0.5 and 1 if the output is greater than 0.5. Our main aim is to find the value of weights or the weight vector which will enable the system to act as a particular gate.

现在，该值被馈送到具有非线性函数(在本例中为Sigmoid)的神经元，用于将输出缩放到所需范围。 如果输出小于0.5，则Sigmoid的缩放输出为0，如果输出大于0.5，则为1。 我们的主要目的是找到权重值或权重向量，以使系统充当特定的门。

实施与门 (Implementing AND gate)

AND gate operation is a simple multiplication operation between the inputs. If any of the input is 0, the output is 0. In order to achieve 1 as the output, both the inputs should be 1. The truth table below conveys the same information.

与门运算是输入之间的简单乘法运算。 如果任何输入为0，则输出为0。为了使输出为1，两个输入都应为1。下面的真值表传达了相同的信息。

Truth Table of AND gate and the values of weights that make the system act as AND and NAND gate, Image by Author AND门的真值表和使系统充当AND和NAND门的权重值，作者提供

As we have 4 choices of input, the weights must be such that the condition of AND gate is satisfied for all the input points.

由于我们有4种输入选择，因此权重必须满足所有输入点的AND门条件。

(0,0)案例 ((0,0) case)

Consider a situation in which the input or the x vector is (0,0). The value of Z, in that case, will be nothing but W0. Now, W0 will have to be less than 0 so that Z is less than 0.5 and the output or ŷ is 0 and the definition of the AND gate is satisfied. If it is above 0, then the value after Z has passed through the sigmoid function will be 1 which violates the AND gate condition. Hence, we can say with a resolution that W0 has to be a negative value. But what value of W0? Keep reading…

考虑输入或x向量为(0,0)的情况。 在这种情况下，Z的值不过是W0。 现在，W0必须小于0，以使Z小于0.5且输出或ŷ为0，并且满足AND门的定义。 如果它大于0，则Z通过S型函数后的值将为1，这违反了AND门条件。 因此，我们可以说，分辨率W0必须为负值。 但是W0的值是多少？ 继续阅读…

(0,1)案件 ((0,1) case)

Now, consider a situation in which the input or the x vector is (0,1). Here the value of Z will be W0+0+W2*1. This being the input to the sigmoid function should have a value less than 0 so that the output is less than 0.5 and is classified as 0. Henceforth, W0+W2<0. If we take the value of W0 as -3(remember the value of W0 has to be negative) and the value of W2 as +2, the result comes out to be -3+2 and that is -1 which seems to satisfy the above inequality and is at par with the condition of AND gate.

现在，考虑输入或x向量为(0,1)的情况。 Z的值将为W0 + 0 + W2 * 1。 作为S形函数的输入，其值应小于0，以使输出小于0.5，并分类为0。此后，W0 + W2 <0。 如果我们将W0的值设为-3(请记住W0的值必须为负)，而W2的值为+2，则结果为-3 + 2，即-1，这似乎满足不等式以上，并且与AND门的条件相同。

(1,0)案件 ((1,0) case)

Similarly, for the (1,0) case, the value of W0 will be -3 and that of W1 can be +2. Remember you can take any values of the weights W0, W1, and W2 as long as the inequality is preserved.

同样，对于(1,0)情况，W0的值为-3，W1的值为+2。 请记住，只要保留不等式，就可以采用权重W0，W1和W2的任何值。

(1,1)案例 ((1,1) case)

In this case, the input or the x vector is (1,1). The value of Z, in that case, will be nothing but W0+W1+W2. Now, the overall output has to be greater than 0 so that the output is 1 and the definition of the AND gate is satisfied. From previous scenarios, we had found the values of W0, W1, W2 to be -3,2,2 respectively. Placing these values in the Z equation yields an output -3+2+2 which is 1 and greater than 0. This will, therefore, be classified as 1 after passing through the sigmoid function.

在这种情况下，输入或x向量为(1,1)。 在这种情况下，Z的值不过是W0 + W1 + W2。 现在，总输出必须大于0，以便输出为1并满足“与”门的定义。 从以前的场景中，我们发现W0，W1，W2的值分别为-3、2、2。 将这些值放在Z方程中将产生输出-3 + 2 + 2，该输出为1且大于0。因此，在通过S型函数后，将其分类为1。

关于AND和NAND实现的最后说明 (A final note on AND and NAND implementation)

The line separating the above four points, therefore, be an equation W0+W1*x1+W2*x2=0 where W0 is -3, and both W1 and W2 are +2. The equation of the line of separation of four points is therefore x1+x2=3/2. The implementation of the NOR gate will, therefore, be similar to the just the weights being changed to W0 equal to 3, and that of W1 and W2 equal to -2

因此，将上述四个点分开的线是等式W0 + W1 * x1 + W2 * x2 = 0，其中W0为-3，W1和W2均为+2。 因此，四点分离线的等式为x1 + x2 = 3/2。 因此，或非门的实现将类似于权重变为W0等于3，而W1和W2等于-2的情况。

移至XOR门 (Moving on to XOR gate)

For the XOR gate, the truth table on the left side of the image below depicts that if there are two complement inputs, only then the output will be 1. If the input is the same(0,0 or 1,1), then the output will be 0. The points when plotted in the x-y plane on the right gives us the information that they are not linearly separable like in the case of OR and AND gates(at least in two dimensions).

对于XOR门，下图左侧的真值表显示，如果有两个补码输入，则只有输出为1。如果输入相同(0,0或1,1)，则输出将为0。这些点在右侧的xy平面上绘制时，会向我们提供以下信息：它们不能像OR和AND门(至少在二维上)那样线性分离。

XOR gate truth table and plotting of values on the x-y plane, Image by Author XOR门真值表并在xy平面上绘制值，作者提供

两种可能的解决方案 (Two possible solution)

To solve the above problem of separability, two techniques can be employed i.e Adding non-linear features also known as the Kernel trick or adding extra layers also known as Deep network

为了解决上述可分离性问题，可以采用两种技术，即添加非线性特征(也称为内核技巧)或添加额外的层(也称为深度网络)

XOR(x1,x2) can be thought of as NOR(NOR(x1,x2),AND(x1,x2))

XOR(x1，x2)可以被认为是NOR(NOR(x1，x2)，AND(x1，x2))

The solution to implementing XOR gate, Image by Author 实现XOR门的解决方案，作者提供

XOR网络的权重 (Weights of the XOR network)

Here we can see that the layer has increased from 2 to 3 as we have added a layer where AND and NOR operation is being computed. The inputs remain the same with an additional bias input of 1. The table on the right below displays the output of the 4 inputs taken as the input. An interesting thing to notice here is that the total number of weights has increased to 9. A total of 6 weights from the input layer to the 2nd layer and a total of 3 weights from the 2nd layer to the output layer. The 2nd layer is also termed as a hidden layer.

在这里，我们可以看到该层已从2增加到3，因为我们添加了正在计算AND和NOR运算的层。 输入保持不变，附加偏置输入为1。右下方的表显示了作为输入的4个输入的输出。 这里要注意的一个有趣的事情是，权重的总数已增加到9。从输入层到第二层的权重总计为6，从第二层到输出层的权重总计为3。 第二层也称为隐藏层。

Weights of the network for it to act as an XOR gate, Image by Author 网络充当异或门的权重，作者提供

Talking about the weights of the overall network, from the above and part 1 content we have deduced the weights for the system to act as an AND gate and as a NOR gate. We will be using those weights for the implementation of the XOR gate. For layer 1, 3 of the total 6 weights would be the same as that of the NOR gate and the remaining 3 would be the same as that of the AND gate. Therefore, the weights for the input to the NOR gate would be [1,-2,-2], and the input to the AND gate would be [-3,2,2]. Now, the weights from layer 2 to the final layer would be the same as that of the NOR gate which would be [1,-2,-2].

在讨论整个网络的权重时，从以上内容和第1部分的内容中，我们推导出了系统充当“与”门和“或非”门的权重。 我们将使用这些权重来实现XOR门。 对于第1层，总共6个权重中的3个将与NOR门的权重相同，其余3个权重将与AND门的权重相同。 因此，“或非”门输入的权重将为[1，-2，-2]，而“与”门的输入权重将为[-3,2,2]。 现在，从第2层到最后一层的权重将与NOR门的权重相同，即[1、2，-2]。

通用逼近定理 (Universal approximation theorem)

It states that any function can be expressed as a neural network with one hidden layer to achieve the desired accuracy

它指出，任何函数都可以表示为具有一个隐藏层的神经网络，以实现所需的精度

几何解释 (Geometrical interpretation)

Linear separability of the two classes in 3D, Image by Author 两类3D的线性可分离性，作者提供

With this, we can think of adding extra layers as adding extra dimensions. After visualizing in 3D, the X’s and the O’s now look separable. The red plane can now separate the two points or classes. Such a plane is called a hyperplane. In conclusion, the above points are linearly separable in higher dimensions.

这样，我们可以考虑将额外的图层添加为额外的尺寸。 在3D模式下可视化之后，X和O现在看起来是可分离的。 现在，红色飞机可以将两个点或两个类分开。 这样的平面称为超平面。 总之，以上各点在较高维度上是线性可分离的。

翻译自: https://towardsdatascience.com/implementing-logic-gates-using-neural-networks-part-2-b284cc159fce

逻辑运算 神经网络