数学代写|计算复杂度理论代写Computational complexity theory代考|COMP90038

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• 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|Incomplete Problems in NP

We have seen many $N P$-complete problems in Chapter 2. Many natural problems in $N P$ turn out to be $N P$-complete. There are, however, a few interesting problems in $N P$ that are not likely to be solvable in deterministic polynomial time but also are not known to be $N P$-complete. The study of these problems is thus particularly interesting, because it not only can classify the inherent complexity of the problems themselves but can also provide a glimpse of the internal structure of the class $N P$. We start with some examples.

Example 4.1 GRAPH ISOMORPHISM (GIso): Given two graphs $G_{1}=$ $\left(V_{1}, E_{1}\right)$ and $G_{2}=\left(V_{2}, E_{2}\right)$, determine whether they are isomorphic, that is, whether there is a bijection $f: V_{1} \rightarrow V_{2}$ such that ${u, v} \in E_{1}$ if and only if ${f(u), f(v)} \in E_{2}$.

The problem SuBGRAPH IsomORPHISM, which asks whether a given graph $G_{1}$ is isomorphic to a subgraph of another given graph $G_{2}$, can be proved to be $N P$-complete easily. However, the problem GIso is neither known to be $N P$-complete nor known to be in $P$, despite extensive studies in recent years. We will prove in Chapter 10, through the notion of interactive proof systems, that GIso is not $N P$-complete unless the polynomial-time hierarchy collapses to the level $\Sigma_{2}^{P}$. This result suggests that GIso is probably not $N P$-complete.

There are many number-theoretic problems in $N P$ that are neither known to be $N P$-complete nor known to be in $P$. We list three of them that have major applications in cryptography. An integer $x \in \mathbb{Z}{n}^{}$ is called a quadratic residue modulo $n$ if $x \equiv y^{2} \bmod n$ for some $y \in \mathbb{Z}{n}^{}$. We write $x \in Q R_{n}$ to denote this fact.

数学代写|计算复杂度理论代写Computational complexity theory代考|One-Way Functions and Cryptography

One-way functions are a fundamental concept in cryptography, having a number of important applications, including public-key cryptosystems,pseudorandom generators, and digital signatures. Intuitively, a one-way function is a function that is easy to compute but its inverse is hard to compute. Thus it can be applied to develop cryptosystems that need easy encoding but difficult decoding. If we identify the intuitive notion of “easiness” with the mathematical notion of “polynomial-time computability,” then one-way functions are subproblems of $N P$, because the inverse function of a polynomial-time computable function is computable in polynomial-time relative to an oracle in $N P$, assuming that the functions are polynomially honest. Indeed, all problems in $N P$ may be viewed as one-way functions.

Example 4.5 Define a function $f_{\mathrm{SAT}}$ as follows: For each Boolean function $F$ over variables $x_{1}, \ldots, x_{n}$ and each Boolean assignment $\tau$ on $x_{1}, \ldots, x_{n}$
$$f_{\mathrm{SAT}}(F, \tau)= \begin{cases}\langle F, 1\rangle & \text { if } \tau \text { satisfies } F, \ \langle F, 0\rangle & \text { otherwise. }\end{cases}$$
It is easily seen that $f_{\text {SAT }}$ is computable in polynomial time. Its inverse mapping $\langle F, 1\rangle$ to $\langle F, \tau\rangle$ is exactly the search problem of finding a truth assignment for a given Boolean formula. Using the notion of polynomialtime Turing reducibility and the techniques developed in Chapter 2, we can see that the inverse function of $f_{\mathrm{SAT}}$ is polynomial-time equivalent to the decision problem SAT. Thus, the inverse of $f_{\mathrm{SAT}}$ is not polynomial-time computable if $P \neq N P$.

Strictly speaking, function $f_{\mathrm{SAT}}$ is, however, not really a one-way function because it is not a one-to-one function and its inverse is really a multivalued function. In the following, we define one-way functions for one-to-one functions. We say that a function $f: \Sigma^{} \rightarrow \Sigma^{}$ is polynomially honest if there is a polynomial function $q$ such that for each $x \in \Sigma^{*}$, $|f(x)| \leq q(|x|)$ and $|x| \leq q(|f(x)|)$.

数学代写|计算复杂度理论代写Computational complexity theory代考|Relativization

The concept of relativization originates from recursive function theory. Consider, for example, the halting problem. We may formulate it in the following form: $K=\left{x \mid M_{x}(x)\right.$ halts $}$, where $M_{x}$ is the $x$ th TM in a standard enumeration of all TMs. Now, if we consider all oracle TMs, we may ask whether the set $K_{A}=\left{x \mid M_{x}^{A}(x)\right.$ halts $}$ is recursive relative to $A$. This is the halting problem relative to set $A$. It is easily seen from the original proof for the nonrecursiveness of $K$ that $K_{A}$ is nonrecursive relative to $A$ (i.e., no oracle TM can decide $K_{A}$ using $A$ as an oracle). Indeed, most results in recursive function theory can be extended to hold relative to any oracle set. We say that such results relativize. In this section, we investigate the problem of whether $P=N P$ in the relativized form. First, we need to define what is meant by relativizing the question of whether $P=N P$. For any set $A$, recall that $P^{A}(\circ r P(A))$ is the class of sets computable in polynomial time by oracle DTMs using $A$ as the oracle and, similarly, NPA (or $N P(A))$ is the class of sets accepted in polynomial time by oracle NTMs classes $P$ and $N P$, we show that the relativized $P=? N P$ question has both the positive and negative answers, depending on the oracle set $A$.
Theorem $4.14$ (a) There exists a recursive set $A$ such that $P^{A}=N P^{A}$.
(b) There exists a recursive set $B$ such that $P^{B} \neq N P^{B}$.
Proof. (a): Let $A$ be any set that is $\leq_{m}^{P}$-complete for PSPACE. Then, by Savitch’s theorem, we have
$$N P^{A} \subseteq N P S P A C E=P S P A C E \subseteq P^{A} .$$

数学代写|计算复杂度理论代写Computational complexity theory代考|Incomplete Problems in NP

SubGRAPH IsomORPHISM 的问题，它询问给定的图是否G1与另一个给定图的子图同构G2, 可以证明是ñ磷- 轻松完成。然而，问题 GIso 不为人所知ñ磷-完成也不知道在磷，尽管近年来进行了广泛的研究。我们将在第 10 章通过交互式证明系统的概念证明 GIso 不是ñ磷-完成，除非多项式时间层次结构崩溃到该级别Σ2磷. 该结果表明 GIso 可能不是ñ磷-完全的。

数学代写|计算复杂度理论代写Computational complexity theory代考|One-Way Functions and Cryptography

F小号一个吨(F,τ)={⟨F,1⟩ 如果 τ 满足 F, ⟨F,0⟩ 否则。

数学代写|计算复杂度理论代写Computational complexity theory代考|Relativization

(b) 存在一个递归集乙这样磷乙≠ñ磷乙.

ñ磷一个⊆ñ磷小号磷一个C和=磷小号磷一个C和⊆磷一个.

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。