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# 数学代考|计算复杂性理论代写computational complexity theory代考|The Polynomial-Time Hierarchy and Polynomial Space

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## 数学代考|计算复杂性理论代写computatiknal complexity theory代考|Nondeterministic Oracle Turing Machines

We have defined in Chapter 2 the notions of polynomial-time Turing reducibility and oracle TMs, and have seen that many optimization problems, when formulated in the search problem form, are solvable in polynomial time relative to a set in $N P$. We now extend this notion to nondeterministic oracle TMs and study problems that are solvable in nondeterministic polynomial time relative to sets in N P.

A nondeterministic (function-)oracle Turing machine (oracle NTM) is an NTM equipped with an additional query tape and two additional states: the query state and the answer state. The computation of an oracle NTM is similar to that of an oracle DTM, except that at each nonquery state an oracle NTM can make a nondeterministic move. We require that the query step of the computation be a deterministic move determined by the oracle. Let $M$ be an oracle NTM and $f$ an oracle function. We write $M^{f}(x)$ to denote the computation of $M$ on input $x$, using $f$ as the oracle function (note that this is a computation tree). If the oracle function is a characteristic function of a set $A$, we say $M$ is a set-oracle NTM and write $M^{A}$ to denote $M^{f}$, and write $L(M, A)$ to denote the set of strings accepted by $M^{A}$.

## 数学代考|计算复杂性理论代写computatiknal complexity theory代考|Polynomial-Time Hierarchy

The polynomial-time hierarchy is the polynomial analog of the arithmetic hierarchy in recursion theory (Rogers, 1967). It can be defined inductively by oracle NTMs.

Definition 3.3 For integers $n \in \mathbb{N}$, complexity classes $\Delta_{n}^{P}$, $\Sigma_{n}^{P}$, and $\Pi_{n}^{P}$ are defined as follows:
\begin{aligned} \Sigma_{0}^{P} &=\Pi_{0}^{P}=\Delta_{0}^{P}=P, \ \Sigma_{n+1}^{P} &=N P\left(\Sigma_{n}^{P}\right), \ \Pi_{n+1}^{P} &=c o-\Sigma_{n+1}^{P}, \ \Delta_{n+1}^{P} &=P\left(\Sigma_{n}^{P}\right), \quad n \geq 0 . \end{aligned}
The class $P H$ is defined to be the union of $\Sigma_{n}^{P}$ over all $n \geq 0$.
Thus, $\Sigma_{1}^{P}=N P, \Sigma_{2}^{P}=N P^{N P}, \Sigma_{3}^{P}=N P\left(N P^{N P}\right)$, and so on. It is easy to verify that these classes form a hierarchy.

## 数学代考|计算复杂性理论代写computatiknal complexity theory代考|Complete Problems in PH

We have proved in Example $2.20$ that the problem EXACT-CLIQUE is in $P(N P)=\Delta_{2}^{P}$, and it is $\leq_{T}^{P}$-hard for $N P$. Therefore, it is $\leq_{T}^{P}$-complete for $\Delta_{2}^{P}$, because all problems in $\Delta_{2}^{P}$ are $\leq_{T}^{P}$-reducible to a problem in $N P$. For most $N P$-complete optimization problems, it can be easily shown that the corresponding version of the decision problem of determining whether a given integer $K$ is the size of the optimum solutions is $\leq_{T}^{P}$-complete for $\Delta_{2}^{P}$.

In addition to complete problems in $\Delta_{2}^{P}$, there are also natural problems complete for the classes in the higher levels of the polynomial-time hierarchy. First, we show that the generic $N P$-complete problem BOUNDED HALTING PROBLEM, or BHP, has relativized versions that are complete for classes $\Sigma_{k}^{P}$ for $k>1$. Let $A$ be an arbitrary set.
BHP relative to set $A\left(\mathrm{BHP}^{A}\right)$ : Given an oracle NTM $M$, an input $w$, and a time bound t, written in the unary form $0^{t}$, determine whether $M^{A}$ accepts w in t moves.

## 数学代考|计算复杂性理论代写COMPUTATIKNAL COMPLEXITY THEORY代考|POLYNOMIAL-TIME HIERARCHY

\begin{aligned} \Sigma_{0}^{P} &=\Pi_{0}^{P}=\Delta_{0}^{P}=P, \ \Sigma_{n+1}^{P} &=N P\left(\Sigma_{n}^{P}\right), \ \Pi_{n+1}^{P} &=c o-\Sigma_{n+1}^{P}, \ \Delta_{n+1}^{P} &=P\left(\Sigma_{n}^{P}\right), \quad n \geq 0 . \end{aligned}

## 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。