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# 计算机代写|机器学习代写Machine Learning代考|CNF Functions

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## 计算机代写|机器学习代写Machine Learning代考|CNF Functions

Disjunctive normal form has a dual: conjunctive normal form (CNF). A Boolean function is said to be in CNF if it can be written as a conjunction of clauses. An example in CNF is: $f=\left(x_1+x_2\right)\left(x_2+x_3+x_4\right)$. A CNF expression is called a $k$-clause CNF expression if it is a conjunction of $k$ clauses; it is in the class $k$-CNF if the size of its largest clause is $k$. The example is a 2-clause expression in 3-CNF. If $f$ is written in DNF, an
Rivest has proposed a class of Boolean functions called decision lists [Rivest, 1987]. A decision list is written as an ordered list of pairs:
\begin{aligned} & \left(t_q, v_q\right) \ & \left(t_{q-1}, v_{q-1}\right) \ & \ldots \ & \left(t_i, v_i\right) \ & \cdots \ & \left(t_2, v_2\right) \ & \left(T, v_1\right) \end{aligned}
where the $v_i$ are either 0 or 1 , the $t_i$ are terms in $\left(x_1, \ldots x_n\right)$, and $T$ is a term whose value is 1 (regardless of the values of the $x_i$ ). The value of a decision list is the value of $v_i$ for the first $t_i$ in the list that has value 1. (At least one $t_i$ will have value 1 , because the last one does; $v_1$ can be regarded as a default value of the decision list.) The decision list is of size $k$, if the size of the largest term in it is $k$. The class of decision lists of size $k$ or less is called $k$-D L.
An example decision list is:
\begin{aligned} & f= \ & \left(\overline{x_1} x_2, 1\right) \ & \left(\overline{x_1} \overline{x_2} x_3, 0\right) \ & \left.\overline{x_2} x_3, 1\right) \ & (1,0) \end{aligned}
$f$ has value 0 for $x_1=0, x_2=0$, and $x_3=1$. It has value 1 for $x_1=1$, $x_2=0$, and $x_3=1$. This function is in 3 -DL.
It has been show $n$ that the class $k$-DL is a strict superset of the union of $k$-D NF and $k$-C NF. There are $2^{O\left[n^k k \log (n)\right]}$ functions in $k$-D L [Rivest, 1987].
Interesting generalizations of decision lists use other Boolean functions in place of the terms, $t_i$. For example we might use linearly separable functions in place of the $t_i$

## 计算机代写|机器学习代写Machine Learning代考|Symmetric and Voting Functions

A Boolean function is called symmetric if it is invariant under permutations of the input variables. For example, any function that is dependent only on the number of input variables whose values are 1 is a symmetric function. The parity functions, which have value 1 depending on whether or not the number of input variables with value 1 is even or odd is a symmetric function. (The exclusive or function, illustrated in Fig. 2.1, is an odd-parity function of two dimensions. The or and and functions of two dimensions are also symmetric.)
An important subclass of the symmetric functions is the class of voting functions (also called $m$-of- $n$ functions). A $k$-voting function has value 1 if and only if $k$ or more of its $n$ inputs has value 1 . If $k=1$, a voting function is the same as an $n$-sized clause; if $k=n$, a voting function is the same as an $n$-sized term; if $k=(n+1) / 2$ for $n$ odd or $k=1+n / 2$ for $n$ even, we have the majority function.
The linearly separable functions are those that can be expressed as follows:
$$f=\operatorname{thresh}\left(\sum_{i=1}^n w_i x_i, \theta\right)$$
where $w_i, i=1, \ldots, n$, are real-valued numbers called weights, $\theta$ is a realvalued number called the threshold, and thresh $(\sigma, \theta)$ is 1 if $\sigma \geq \theta$ and 0 otherwise. (Note that the concept of linearly separable functions can be extended to non-Boolean inputs.) The $k$-voting functions are all members of the class of linearly separable functions in which the weights all have unit value and the threshold depends on $k$. Thus, terms and clauses are special cases of linearly separable functions.
A convenient way to write linearly separable functions uses vector notation:
$$f=\operatorname{thresh}(\mathbf{X} \cdot \mathbf{W}, \theta)$$
where $\mathbf{X}=\left(x_1, \ldots, x_n\right)$ is an $n$-dimensional vector of input variables, $\mathbf{W}=$ $\left(w_1, \ldots, w_n\right)$ is an $n$-dimensional vector of weight values, and $\mathbf{X} . \mathbf{W}$ is the dot (or inner) product of the two vectors. Input vectors for which $f$ has value 1 lie in a half-space on one side of (and on) a hyperplane whose orientation is normal to $\mathbf{W}$ and whose position with respect to the origin is determined by $\theta$. We saw an example of such a separating plane in Fig. 1.6. With this idea in mind, it is easy to see that two of the functions in Fig. 2.1 are linearly separable, while two are not. Also note that the terms in Figs. 2.3 and 2.4 are linearly separable functions as evidenced by the separating planes shown.

## 计算机代写|机器学习代写Machine Learning代考|CNF Functions

\begin{aligned} & \left(t_q, v_q\right) \ & \left(t_{q-1}, v_{q-1}\right) \ & \ldots \ & \left(t_i, v_i\right) \ & \cdots \ & \left(t_2, v_2\right) \ & \left(T, v_1\right) \end{aligned}

\begin{aligned} & f= \ & \left(\overline{x_1} x_2, 1\right) \ & \left(\overline{x_1} \overline{x_2} x_3, 0\right) \ & \left.\overline{x_2} x_3, 1\right) \ & (1,0) \end{aligned}
$f$对于$x_1=0, x_2=0$和$x_3=1$的值为0。对于$x_1=1$、$x_2=0$和$x_3=1$，其值为1。该函数为3 -DL格式。

## 计算机代写|机器学习代写Machine Learning代考|Symmetric and Voting Functions

where $w_i, i=1, \ldots, n$，是实数，称为权值， $\theta$ 重新估值的数字是否称为阈值和脱粒 $(\sigma, \theta)$ 等于1 $\sigma \geq \theta$ 否则为0。(注意，线性可分函数的概念可以扩展到非布尔输入。)该 $k$-投票函数是一类线性可分函数的所有成员，其中权重都具有单位值，阈值取决于 $k$． 因此，项和分句是线性可分函数的特殊情况。线性可分函数的一个方便的写法是使用向量表示法$$f=\operatorname{thresh}(\mathbf{X} \cdot \mathbf{W}, \theta)$$
where $\mathbf{X}=\left(x_1, \ldots, x_n\right)$ 是吗? $n$输入变量的-维向量， $\mathbf{W}=$ $\left(w_1, \ldots, w_n\right)$ 是吗? $n$权重值的-维向量，和 $\mathbf{X} . \mathbf{W}$ 是两个向量的点积(或内积)输入向量 $f$ 值1是否位于方向与之垂直的超平面的一侧(和上)的半空间中 $\mathbf{W}$ 它相对于原点的位置由 $\theta$． 我们在图1.6中看到了这样一个分离平面的例子。有了这个想法，很容易看出图2.1中的两个函数是直线的

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。