19th Ave New York, NY 95822, USA

计算机代写|机器学习代写Machine Learning代考|Boolean Algebra

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

计算机代写|机器学习代写Machine Learning代考|Boolean Algebra

Many important ideas about learning of functions a re most easily presented using the special case of Boolean functions. There are several important subclasses of Boolean functions that are used as hypothesis classes for function learning. Therefore, we digress in this chapter to present a review of Boolean functions and their properties. (For a more thorough treatment see, for example, [Unger, 1989].)
A Boolean function, $f\left(x_1, x_2, \ldots, x_n\right)$ maps an $n$-tuple of $(0,1)$ values to ${0,1}$. Boolean algebra is a convenient notation for representing Boolean functions. Boolean algebra uses the connectives :, + , and – For example, the and function of two variables is written $x_1 \cdot x_2$. By convention, the connective, “.” is usually suppressed, and the and function is written $x_1 x_2$. $x_1 x_2$ has value 1 if and only if both $x_1$ and $x_2$ have value 1 ; if either $x_1$ or $x_2$ has value $0, x_1 x_2$ has value 0 . The (inclusive) or function of two variables is written $x_1+x_2 . x_1+x_2$ has value 1 if and only if either or both of $x_1$ or $x_2$ has value 1 ; if both $x_1$ and $x_2$ have value $0, x_1+x_2$ has value 0 . The complement or negation of a variable, $x$, is written $\bar{x}, \bar{x}$ has value 1 if and only if $x$ has value 0 ; if $x$ has value 1 has value $1, \bar{x}$ has value 0 .
These definitions are compactly given by the following rules for Boolean a lgebra:
\begin{aligned} & 1+1=1,1+0=1,0+0=0, \ & 1 \cdot 1=1,1 \cdot 0=0,0 \cdot 0=0, \text { and } \end{aligned}
$\overline{1}=0, \overline{0}=1$.
Sometimes the arguments and values of Boolean functions are expressed in terms of the constants $T$ (True) and $F$ (False) instead of 1 and 0 , respectively.
The connectives . and + are each commutative and associative. Thus, for example, $x_1\left(x_2 x_3\right)=\left(x_1 x_2\right) x_3$, and both can be written simply as $x_1 x_2 x_3$. Similarly for + .
A Boolean formula consisting of a single variable, such as $x_1$ is called an atom. One consisting of either a single variable or its complement, such as $\overline{x_1}$, is called a literal.
The operators a and + do not commute between themselves. Instead, we have DeMorgan’s laws (which can be verified by using the above definitions):
$\overline{x_1 x_2}=\overline{x_1}+\overline{x_2}$, and $\overline{x_1+x_2}=\overline{x_1} \overline{x_2}$.

计算机代写|机器学习代写Machine Learning代考|Diagrammatic Representations

We saw in the last chapter that a Boolean function could be represented by labeling the vertices of a cube. For a function of $n$ variables, we would need an n-dimensional hypercube. In Fig. 2.1 we show some 2- and 3dimensional examples. Vertices having value 1 are labeled with a small square, and vertices having value 0 are labeled with a small circle.
Using the hypercube representations, it is easy to see how many Boolean functions of $n$ dimensions there are. A 3-dimensional cube has $2^3=8$ vertices, and each may be labeled in two different ways; thus there are $2^{\left(2^3\right)}=256$ different Boolean functions of 3 variables. In general, there are $2^{2^n}$ Boolean functions of $n$ variables.
We will be using 2- and 3-dimensional cubes later to provide some intuition about the properties of certain Boolean functions. Of course, we cannot visualize hypercubes (for $n>3$ ), and there are many surprising properties of higher dimensional spaces, so we must be careful in using intuitions gained in low dimensions. One diagrammatic technique for dimensions slightly higher than 3 is the Karnaugh map. A Karnaugh map is an array of values of a Boolean function in which the horizontal rows are indexed by the values of some of the variables and the vertical columns are indexed by the rest. The rows and columns are arranged in such a way that entries that are adjacent in the map correspond to vertices that are adjacent in the hypercube representation. We show an example of the 4-dimensional even parity function in Fig. 2.2. An even parity function is a Boolean function that has value 1 if there are an even number of its arguments that have value 1 ; otherwise it has value 0 . Note that all adjacent cells in the table correspond to inputs differing in only one component.

计算机代写|机器学习代写Machine Learning代考|Boolean Algebra

\begin{aligned} & 1+1=1,1+0=1,0+0=0, \ & 1 \cdot 1=1,1 \cdot 0=0,0 \cdot 0=0, \text { and } \end{aligned}
$\overline{1}=0, \overline{0}=1$ .

$\overline{x_1 x_2}=\overline{x_1}+\overline{x_2}$和$\overline{x_1+x_2}=\overline{x_1} \overline{x_2}$ .

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。