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# 数学代写|信息论代写Information Theory代考|ELEN90030

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## 数学代写|信息论代写Information Theory代考|ALGORITHMICALLY RANDOM AND INCOMPRESSIBLE SEQUENCES

From the examples in Section 14.2, it is clear that there are some long sequences that are simple to describe, like the first million bits of $\pi$. By the same token, there are also large integers that are simple to describe, such as
$$2^{2^{2^{2^{2^2}}}}$$
or $(100 !) !$.
We now show that although there are some simple sequences, most sequences do not have simple descriptions. Similarly, most integers are not simple. Hence, if we draw a sequence at random, we are likely to draw a complex sequence. The next theorem shows that the probability that a sequence can be compressed by more than $k$ bits is no greater than $2^{-k}$.

Theorem 14.5.1 Let $X_1, X_2, \ldots, X_n$ be drawn according to a Bernoulli $\left(\frac{1}{2}\right)$ process. Then
$$P\left(K\left(X_1 X_2 \ldots X_n \mid n\right)<n-k\right)<2^{-k} .$$

Proof:
\begin{aligned} P(K & \left.\left(X_1 X_2 \ldots X_n \mid n\right)<n-k\right) \ & =\sum_{x_1 x_2 \ldots x_n: K\left(x_1 x_2 \ldots x_n \mid n\right)<n-k} p\left(x_1, x_2, \ldots, x_n\right) \ & =\sum_{x_1 x_2 \ldots x_n: K\left(x_1 x_2 \ldots x_n \mid n\right)<n-k} 2^{-n} \ & =\left|\left{x_1 x_2 \ldots x_n: K\left(x_1 x_2 \ldots x_n \mid n\right)<n-k\right}\right| 2^{-n} \ & <2^{n-k} 2^{-n} \quad(\text { by Theorem 14.2.4) } \ & =2^{-k} . \end{aligned}

## 数学代写|信息论代写Information Theory代考|UNIVERSAL PROBABILITY

We now consider the tree-structured version of Lempel-Ziv, where the input sequence is parsed into phrases, each phrase being the shortest string that has not been seen so far. The proof of the optimality of this algorithm has a very different flavor from the proof for LZ77; the essence of the proof is a counting argument that shows that the number of phrases cannot be too large if they are all distinct, and the probability of any sequence of symbols can be bounded by a function of the number of distinct phrases in the parsing of the sequence.

The algorithm described in Section 13.4.2 requires two passes over the string – in the first pass, we parse the string and calculate $c(n)$, the number of phrases in the parsed string. We then use that to decide how many bits $[\log c(n)]$ to allot to the pointers in the algorithm. In the second pass, we calculate the pointers and produce the coded string as indicated above. The algorithm can be modified so that it requires only one pass over the string and also uses fewer bits for the initial pointers. These modifications do not affect the asymptotic efficiency of the algorithm. Some of the implementation details are discussed by Welch [554] and Bell et al. [41].
We will show that like the sliding window version of Lempel-Ziv, this algorithm asymptotically achieves the entropy rate for the unknown ergodic source. We first define a parsing of the string to be a decomposition into phrases.

Suppose that a computer is fed a random program. Imagine a monkey sitting at a keyboard and typing the keys at random. Equivalently, feed a series of fair coin flips into a universal Turing machine. In either case, most strings will not make sense to the computer. If a person sits at a terminal and types keys at random, he will probably get an error message (i.e., the computer will print the null string and halts). But with a certain probability she will hit on something that makes sense. The computer will then print out something meaningful. Will this output sequence look random?
From our earlier discussions, it is clear that most sequences of length $n$ have complexity close to $n$. Since the probability of an input program $p$ is $2^{-l(p)}$, shorter programs are much more probable than longer ones; and when they produce long strings, shorter programs do not produce random strings; they produce strings with simply described structure.

The probability distribution on the output strings is far from uniform. Under the computer-induced distribution, simple strings are more likely

## 数学代写|信息论代写Information Theory代考|ALGORITHMICALLY RANDOM AND INCOMPRESSIBLE SEQUENCES

$$2^{2^{2^{2^{2^2}}}}$$

$$P\left(K\left(X_1 X_2 \ldots X_n \mid n\right)<n-k\right)<2^{-k} .$$

\begin{aligned} P(K & \left.\left(X_1 X_2 \ldots X_n \mid n\right)<n-k\right) \ & =\sum_{x_1 x_2 \ldots x_n: K\left(x_1 x_2 \ldots x_n \mid n\right)<n-k} p\left(x_1, x_2, \ldots, x_n\right) \ & =\sum_{x_1 x_2 \ldots x_n: K\left(x_1 x_2 \ldots x_n \mid n\right)<n-k} 2^{-n} \ & =\left|\left{x_1 x_2 \ldots x_n: K\left(x_1 x_2 \ldots x_n \mid n\right)<n-k\right}\right| 2^{-n} \ & <2^{n-k} 2^{-n} \quad(\text { by Theorem 14.2.4) } \ & =2^{-k} . \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。