### 数学代写|计算复杂度理论代写Computational complexity theory代考|Models of Computation and Complexity Classes

statistics-lab™ 为您的留学生涯保驾护航 在代写计算复杂度理论Computational complexity theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计算复杂度理论Computational complexity theory代写方面经验极为丰富，各种代写计算复杂度理论Computational complexity theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Strings, Coding, and Boolean Functions

Our basic data structure is a string. All other data structures are to be encoded and represented by strings. A string is a finite sequence of symbols. For instance, the word string is a string over the symbols of English letters; the arithmetic expression ” $3+4-5$ ” is a string over symbols $3,4,5,+$, and -. Thus, to describe a string, we must specify the set of symbols to occur in that string. We call a finite set of symbols to be used to define strings an alphabet. Note that not every finite set can be an alphabet. A finite set $S$ can be an alphabet if and only if the following condition holds.

Property 1.1 Two finite sequences of elements in $S$ are identical if and only if the elements in the two sequences are identical respectively in ordering.
For example, ${0,1}$ and ${00,01}$ are alphabets, but ${1,11}$ is not an alphabet because 11 can be formed by either 11 or ( 1 and 1$)$.

Assume that $\Sigma$ is an alphabet. A set of strings over the alphabet $\Sigma$ is called a language. A collection of languages is called a language class, or simply a class.

The length of a string $x$ is the number of symbols in the string $x$, denoted by $|x|$. For example, $\mid$ string $\mid=6$ and $|3+4-5|=5$. For convenience, we allow a string to contain no symbol. Such a string is called the empty string, which is denoted by $\lambda$. So, $|\lambda|=0$. (The notation $|\cdot|$ is also used on sets. If $S$ is a finite set, we write $|S|$ to denote its cardinality.)

There is a fundamental operation on strings. The concatenation of two strings $x$ and $y$ is the string $x y$. The concatenation follows associative law, that is, $x(y z)=(x y) z$. Moreover, $\lambda x=x \lambda=x$. Thus, all strings over an alphabet form a monoid under concatenation. ${ }^{1}$ We denote $x^{0}=\lambda$ and $x^{n}=x x^{n-1}$ for $n \geq 1$.

The concatenation operation on strings can be extended to languages. The concatenation of two languages $A$ and $B$ is the language $A B={a b$ : $a \in A, b \in B}$. We also denote $A^{0}={\lambda}$ and $A^{n}=A A^{n-1}$ for $n \geq 1$. In addition, we define $A^{}=\bigcup_{i=0}^{\infty} A^{i}$. The language $A^{}$ is called the Kleene closure of $A$. The Kleene closure of an alphabet is the set of all strings over the alphabet.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Deterministic Turing Machines

Turing machines (TMs) are simple and yet powerful enough computational models. Almost all reasonable general-purpose computational models have been known to be equivalent to TMs, in the sense that they define the same class of computable functions. There are many variations of TMs studied in literature. We are going to introduce, in this section,

the simplest model of TMs, namely, the deterministic Turing machine (DTM). Another model, the nondeterministic Turing machine (NTM), is to be defined in the next section. Other generalized TM models, such as deterministic and nondeterministic oracle TMs, will be defined in later chapters. In addition, we will introduce in Part II other nonuniform computational models which are not equivalent to TMs.

A deterministic (one-tape) TM (DTM) consists of two basic units: the control unit and the memory unit. The control unit contains a finite number of states. The memory unit is a tape that extends infinitely to both ends. The tape is divided into an infinite number of tape squares (or, tape cells). Each tape square stores one of a finite number of tape symbols. The communication between the control unit and the tape is through a readlwrite tape head that scans a tape square at a time (See Figure 1.1).
A normal move of a TM consists of the following actions:
(1) Reading the tape symbol from the tape square currently scanned by the tape head;
(2) Writing a new tape symbol on the tape square currently scanned by the tape head;
(3) Moving the tape head to the right or left of the current square; and
(4) Changing to a new control state.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Nondeterministic Turing Machines

The TMs we defined in the last section are deterministic, because from each configuration of a machine there is at most one move to make, and hence, there is at most one next configuration. If we allow more than one moves for some configurations, and hence those configurations have more than one next configurations, then the machine is called a nondeterministic Turing machine (NTM).

Formally, an NTM $M$ is defined by the following information: states $Q$; initial state $q_{0}$; accepting states $F$; input symbols $\Sigma$; tape symbols $\Gamma$, including the blank symbol $\mathrm{B}$; and the transition relation $\Delta$. All information except the transition relation $\Delta$ is defined in the same form as a DTM. The transition relation $\Delta$ is a subset of $(Q-F) \times \Gamma \times Q \times \Gamma \times$ ${\mathrm{L}, \mathrm{R}}$. Each quintuple $\left(q_{1}, s_{1}, q_{2}, s_{2}, D\right)$ in $\Delta$ indicates that one of the possible moves of $M$, when it is in state $q_{1}$ and scanning symbol $s_{1}$, is to change the current state to $q_{2}$, to overwrite symbol $s_{1}$ by $s_{2}$, and to move the tape head to the direction $D$.

The computation of an NTM can be defined similar to that of a DTM. First, we consider a way of restricting an NTM to a DTM. Let $M$ be an NTM defined by $\left(Q, q_{0}, F, \Sigma, \Gamma, \Delta\right)$ as above. We say $M_{1}$ is a restricted DTM of $M$ if $M_{1}$ has the same components $Q, q_{0}, F, \Sigma, \Gamma$ as $M$ and it has a transition function $\delta_{1}$ that is a subrelation of $\Delta$ satisfying the property that for each $q_{1} \in Q$ and $s_{1} \in \Gamma$, there is at most one triple $\left(q_{2}, s_{2}, D\right)$, $D \in{\mathrm{L}, \mathrm{R}}$, such that $\left(q_{1}, s_{1}, q_{2}, s_{2}, D\right) \in \delta_{1}$. Now we can define the notion of the next configurations of an NTM easily: For each configuration $\alpha=$ $\left(q_{1}, x_{1}, y_{1}\right)$ of $M$, we let $\vdash_{M}(\alpha)$ be the set of all configurations $\beta$ such that $\alpha \vdash_{M_{1}} \beta$ for some restricted DTM $M_{1}$ of $M$. We write $\alpha \vdash_{M} \beta$ if $\beta \in \vdash_{M}(\alpha)$. As each configuration of $M$ may have more than one next configurations, the computation of an NTM on an input $w$ is, in general, a computation tree rather than a single computation path (as it is in the case of DTMs). In the computation tree, each node is a configuration $\alpha$ and all its next configurations are its children. The root of the tree is the initial configuration.

We say an NTM $M$ halts on an input string $w \in \Sigma^{*}$ if there exists a finite sequence of configurations $\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n}$ such that
(1) $\alpha_{0}=\left(q_{0}, \lambda, w\right)$;
(2) $\alpha_{i} \vdash_{M} \alpha_{i+1}$ for all $i=0,1, \ldots, n-1$; and
(3) $\vdash_{M}\left(\alpha_{n}\right)$ is undefined (i.e., it is an empty set).

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Deterministic Turing Machines

(1) 从磁带头当前扫描的磁带方格中读取磁带符号；
(2)在磁带头当前扫描的磁带方格上写入一个新的磁带符号；
(3) 将磁带头移动到当前方格的右侧或左侧；(
4) 转变为新的控制状态。

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Nondeterministic Turing Machines

NTM 的计算可以定义为类似于 DTM 的计算。首先，我们考虑一种将 NTM 限制为 DTM 的方法。让米是由以下定义的 NTM(问,q0,F,Σ,Γ,Δ)如上。我们说米1是一个受限制的 DTM米如果米1具有相同的组件问,q0,F,Σ,Γ作为米并且有过渡功能d1这是一个子关系Δ满足对于每个q1∈问和s1∈Γ, 最多有一个三元组(q2,s2,D), D∈大号,R, 这样(q1,s1,q2,s2,D)∈d1. 现在我们可以轻松定义 NTM 的下一个配置的概念： 对于每个配置一个= (q1,X1,是1)的米，我们让⊢米(一个)是所有配置的集合b这样一个⊢米1b对于一些受限的 DTM米1的米. 我们写一个⊢米b如果b∈⊢米(一个). 作为每个配置米可能有多个下一个配置，在输入上计算 NTM在通常，它是一个计算树，而不是单个计算路径（就像在 DTM 的情况下一样）。在计算树中，每个节点都是一个配置一个它的所有下一个配置都是它的孩子。树的根是初始配置。

(1)一个0=(q0,λ,在);
(2) 一个一世⊢米一个一世+1对所有人一世=0,1,…,n−1; (
3)⊢米(一个n)是未定义的（即，它是一个空集）。

## 有限元方法代写

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## 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。