2位二进制输入的XNOR逻辑门感知器算法的实现

在机器学习领域，感知器是一种用于二元分类器的监督学习算法。感知器模型实现以下函数：

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

对于权重向量的特定选择$$\boldsymbol{w}$$和偏置参数$$\boldsymbol{b}$$ ，模型预测输出$$\boldsymbol{\hat{y}}$$对于相应的输入向量$$\boldsymbol{x}$$ .

2位二进制变量的XNOR逻辑函数真值表，即输入向量$$\boldsymbol{x} : (\boldsymbol{x_{1}}, \boldsymbol{x_{2}})$$和相应的输出$$\boldsymbol{y}$$ –

$$\boldsymbol{x_{1}}$$	$$\boldsymbol{x_{2}}$$	$$\boldsymbol{y}$$
0	0	1
0	1	0
1	0	0
1	1	1

我们可以观察到， $$XNOR(\boldsymbol{x_{1}}, \boldsymbol{x_{2}}) = OR(NOT(OR(\boldsymbol{x_{1}}, \boldsymbol{x_{2}})), AND(\boldsymbol{x_{1}}, \boldsymbol{x_{2}}))$$
设计感知器网络：

1. Step1：现在为对应的权重向量$$\boldsymbol{w} : (\boldsymbol{w_{1}}, \boldsymbol{w_{2}})$$输入向量的$$\boldsymbol{x} : (\boldsymbol{x_{1}}, \boldsymbol{x_{2}})$$到 OR 和 AND 节点，相关的感知函数可以定义为：

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

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

2. 第二步：输出$($\boldsymbol{\hat{y}}_{1}$)$来自 OR 节点的将被输入到具有权重的 NOT 节点$$\boldsymbol{w_{NOT}}$$并且相关的感知函数可以定义为：

       $\[$\boldsymbol{\hat{y}_{3}} = \Theta\left(w_{NOT} \boldsymbol{\hat{y}_{1}}+b_{NOT}\right)$\]$

3. 第三步：输出$($\boldsymbol{\hat{y}}_{2}$)$从 AND 节点和输出$($\boldsymbol{\hat{y}}_{3}$)$ Step2中提到的来自NOT节点将被输入到具有权重的OR节点$$(\boldsymbol{w_{OR1}}, \boldsymbol{w_{OR2}})$$ .然后对应的输出$$\boldsymbol{\hat{y}}$$是 XNOR 逻辑函数的最终输出。相关的感知函数可以定义为：

       $\[$\boldsymbol{\hat{y}} = \Theta\left(w_{OR1} \boldsymbol{\hat{y}_{3}}+w_{OR2} \boldsymbol{\hat{y}_{2}}+b_{OR}\right)$\]$

为了实现，权重参数被认为是$$\boldsymbol{w_{1}} = 1, \boldsymbol{w_{2}} = 1, \boldsymbol{w_{NOT}} = -1, \boldsymbol{w_{OR1}} = 1, \boldsymbol{w_{OR2}} = 1$$和偏置参数是$$\boldsymbol{b_{AND}} = -1.5, \boldsymbol{b_{OR}} = -0.5, \boldsymbol{b_{NOT}} = 0.5$$ .

Python实现：

# importing Python library
import numpy as np
  
# define Unit Step Function
def unitStep(v):
    if v >= 0:
        return 1
    else:
        return 0
  
# design Perceptron Model
def perceptronModel(x, w, b):
    v = np.dot(w, x) + b
    y = unitStep(v)
    return y
  
# NOT Logic Function
# wNOT = -1, bNOT = 0.5
def NOT_logicFunction(x):
    wNOT = -1
    bNOT = 0.5
    return perceptronModel(x, wNOT, bNOT)
  
# AND Logic Function
# w1 = 1, w2 = 1, bAND = -1.5
def AND_logicFunction(x):
    w = np.array([1, 1])
    bAND = -1.5
    return perceptronModel(x, w, bAND)
  
# OR Logic Function
# here w1 = wOR1 = 1, 
# w2 = wOR2 = 1, bOR = -0.5
def OR_logicFunction(x):
    w = np.array([1, 1])
    bOR = -0.5
    return perceptronModel(x, w, bOR)
  
# XNOR Logic Function
# with AND, OR and NOT  
# function calls in sequence
def XNOR_logicFunction(x):
    y1 = OR_logicFunction(x)
    y2 = AND_logicFunction(x)
    y3 = NOT_logicFunction(y1)
    final_x = np.array([y2, y3])
    finalOutput = OR_logicFunction(final_x)
    return finalOutput
  
# testing the Perceptron Model
test1 = np.array([0, 1])
test2 = np.array([1, 1])
test3 = np.array([0, 0])
test4 = np.array([1, 0])
  
print("XNOR({}, {}) = {}".format(0, 1, XNOR_logicFunction(test1)))
print("XNOR({}, {}) = {}".format(1, 1, XNOR_logicFunction(test2)))
print("XNOR({}, {}) = {}".format(0, 0, XNOR_logicFunction(test3)))
print("XNOR({}, {}) = {}".format(1, 0, XNOR_logicFunction(test4)))

输出：

XNOR(0, 1) = 0
XNOR(1, 1) = 1
XNOR(0, 0) = 1
XNOR(1, 0) = 0

这里，模型预测输出（ $$\boldsymbol{\hat{y}}$$ ) 每个测试输入都与 XNOR 逻辑门常规输出 ( $$\boldsymbol{y}$$ ) 根据真值表。
因此，验证了 XNOR 逻辑门的感知器算法是正确实现的。