具有3位二进制输入的逻辑门的感知器算法

在机器学习领域，感知器是一种用于二进制分类器的监督学习算法。 Perceptron模型实现以下函数：

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

对于权重向量的特定选择$$\boldsymbol{w}$$和偏差参数$$\boldsymbol{b}$$ ，模型会预测输出$$\boldsymbol{\hat{y}}$$对应的输入向量$$\boldsymbol{x}$$ 。
3位二进制变量(即输入向量)的AND，OR，NAND，NOR门的逻辑函数真值表$$\boldsymbol{x} : (\boldsymbol{x_{1}}, \boldsymbol{x_{2}}, \boldsymbol{x_{3}})$$和相应的输出$$\boldsymbol{y_{AND}}, \boldsymbol{y_{OR}}, \boldsymbol{y_{NAND}}, \boldsymbol{y_{NOR}}$$ –

$$\boldsymbol{x_{1}}$$	$$\boldsymbol{x_{2}}$$	$$\boldsymbol{x_{3}}$$	$$\boldsymbol{y_{AND}}$$	$$\boldsymbol{y_{OR}}$$	$$\boldsymbol{y_{NAND}}$$	$$\boldsymbol{y_{NOR}}$$
0	0	0	0	0	1	1
0	0	1	0	1	1	0
0	1	0	0	1	1	0
0	1	1	0	1	1	0
1	0	0	0	1	1	0
1	0	1	0	1	1	0
1	1	0	0	1	1	0
1	1	1	1	1	0	0

现在为相应的权重向量$$\boldsymbol{w} : (\boldsymbol{w_{1}}, \boldsymbol{w_{2}}, \boldsymbol{w_{3}})$$输入向量的$$\boldsymbol{x} : (\boldsymbol{x_{1}}, \boldsymbol{x_{2}}, \boldsymbol{x_{3}})$$ ，相关的感知器函数可以定义为：

    $\[$\boldsymbol{\hat{y}} = \Theta\left(w_{1} x_{1}+w_{2} x_{2}+w_{3} x_{3}+b\right)$\]$

为了实现，考虑的权重参数是$$\boldsymbol{w_{1}}, \boldsymbol{w_{2}}, \boldsymbol{w_{3}}$$偏差参数为$$\boldsymbol{b}$$对于每个逻辑门

$$\boldsymbol{Parameters}$$	$$\boldsymbol{AND}$$	$$\boldsymbol{OR}$$	$$\boldsymbol{NAND}$$	$$\boldsymbol{NOR}$$
$$\boldsymbol{w_{1}}$$	1	1	-1	-1
$$\boldsymbol{w_{2}}$$	1	1	-1	-1
$$\boldsymbol{w_{2}}$$	1	1	-1	-1
$$\boldsymbol{b}$$	-2	-0.9	3	1

Python实现：

# importing python library
import numpy as np
  
# sigmoid activation function
def activationFunction(model, type ="sigmoid"):
   return {
       "sigmoid": 1 / (1 + np.exp(-model))
   }[type]
  
# designing perceptron model
def perceptronModel(weights, inputs, bias):
   model = np.add(np.dot(inputs, weights), bias)
   logic = activationFunction(model, type ="sigmoid")
   return np.round(logic)
  
# computation model
def compute(data, logicGate, weights, bias):
   weights = np.array(weights)
   output = np.array([ perceptronModel(weights,  
            datum, bias) for datum in data ])
   return output
  
# Print Output
def printOutput(dataset, name, data):
   print("Logic Function: {}".format(name.upper()))
   print("X1\tX2\tX3\tY")
   toPrint = ["{1}\t{2}\t{3}\t{0}".format(output, *datas)  
              for datas, output in zip(dataset, data)]
   for i in toPrint:
       print(i)
  
# main function
def main():
   # 3 bit binary data
   dataset = np.array([
     [0, 0, 0],
     [0, 0, 1],
     [0, 1, 0],
     [0, 1, 1],
     [1, 0, 0],
     [1, 0, 1],
     [1, 1, 0],
     [1, 1, 1]
   ])
  
   # Parameters of every Logic Gates
   # weight parameters: w1, w2, w3
   # bias parameter: b
   logicGate = {
       "and": compute(dataset, "and", [1, 1, 1], -2),
       "or": compute(dataset, "or", [1, 1, 1], -0.9),
       "nand": compute(dataset, "nand", [-1, -1, -1], 3),
       "nor": compute(dataset, "nor", [-1, -1, -1], 1)
   }
   for gate in logicGate:
       printOutput(dataset, gate, logicGate[gate])
  
if __name__ == '__main__':
   main()

输出：

Logic Function: AND
X1    X2    X3    Y
0    0    0    0.0
0    0    1    0.0
0    1    0    0.0
0    1    1    0.0
1    0    0    0.0
1    0    1    0.0
1    1    0    0.0
1    1    1    1.0
Logic Function: OR
X1    X2    X3    Y
0    0    0    0.0
0    0    1    1.0
0    1    0    1.0
0    1    1    1.0
1    0    0    1.0
1    0    1    1.0
1    1    0    1.0
1    1    1    1.0
Logic Function: NAND
X1    X2    X3    Y
0    0    0    1.0
0    0    1    1.0
0    1    0    1.0
0    1    1    1.0
1    0    0    1.0
1    0    1    1.0
1    1    0    1.0
1    1    1    0.0
Logic Function: NOR
X1    X2    X3    Y
0    0    0    1.0
0    0    1    0.0
0    1    0    0.0
0    1    1    0.0
1    0    0    0.0
1    0    1    0.0
1    1    0    0.0
1    1    1    0.0

在此，模型的预测输出( $$\boldsymbol{\hat{y}}$$ )，因为每个测试输入均与AND，OR，NAND，NOR逻辑门的常规输出( $$\boldsymbol{y}$$ )根据3位二进制输入的真值表。
因此，可以证明所有这些逻辑门的感知器算法均已正确实现。