338 lines
10 KiB
TeX
338 lines
10 KiB
TeX
\documentclass[UTF8,punct=kaiming]{ctexart}
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\usepackage{amsmath}
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\usepackage[a4paper,left=2cm,right=2cm,top=2.5cm,bottom=2.5cm]{geometry}
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\usepackage{multicol}
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\usepackage{makecell}
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\usepackage{fancyhdr}
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\usepackage{hyperref}
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\usepackage[type={CC},modifier={by-sa},version={4.0},lang={chinese-utf8}]{doclicense}
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\usepackage{amssymb}
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\usepackage{amsthm}
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\usepackage[many]{tcolorbox}
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\usepackage{multirow}
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\usepackage{listings}
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\input{insbox}
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\setlength{\parindent}{0pt}
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\hypersetup{hidelinks,
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colorlinks=false,
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allcolors=black,
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pdfstartview=Fit,
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breaklinks=true}
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\pagestyle{fancy}
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\fancypagestyle{myfancypagestyle}{
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\fancyhf{}
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\fancyhead[L]{计算机系统基础\uppercase\expandafter{\romannumeral1\relax} 考试用资料}
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\fancyhead[R]{Made with $\heartsuit$ by \href{https://kagurach.uk/}{kagura} and \href{https://nkid00.name/}{nkid00} \& licensed under \doclicenseNameRef}
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\fancyfoot[C]{\thepage}
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\renewcommand{\headrulewidth}{0.4pt}
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\renewcommand{\footrulewidth}{0pt}
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}
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\pagestyle{myfancypagestyle}
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\tcbuselibrary{listings}
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\tcbset{colback=white,boxrule=0.3mm,enhanced,size=small}
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\lstset{basicstyle=\ttfamily,columns=fullflexible}
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\setlength{\columnsep}{1cm}
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\setlength{\columnseprule}{0.4pt}
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\begin{document}
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\section{数据表示与存储}
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\subsection{数据的类型及大小}
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\begin{table}[h]
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\begin{tabular}{|c|c|c|c|c|}
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\hline
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类型 & 字节数 & 最小值 & 最大值(signed) & 最大值(unsigned) \\ \hline
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\texttt{char} & 1 & -128 & 127 & 255\\ \hline
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\texttt{short} & 2 & -32768 & 32767 & 65535\\ \hline
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\texttt{int} & 4 & -2147483648 & 2147483647 & 4294967295 \\ \hline
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\texttt{long} & \multirow{2}{*}{8} & \multirow{2}{*}{-9223372036854775808} & \multirow{2}{*}{9223372036854775807} & \multirow{2}{*}{18446744073709551615} \\
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\cline{1-1} \texttt{void*} & & & & \\ \hline
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\texttt{float} & 4 & 1.17549$\times10^{-38}$ & 3.40282$\times10^{38}$& \\ \hline
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\texttt{double} & 8 & 2.22507$\times10^{-308}$ & 1.79769$\times10^{308}$ & \\ \hline
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\end{tabular}
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\end{table}
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\subsection{计算值域}
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T: 用 $n$ 位表示数字
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\begin{align*}
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\text{(signed) T} \quad & \text{可表示} \quad -2^{n-1} \sim ~ 2^{n-1}-1 \\
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\text{(unsigned) T} \quad & \text{可表示} \quad 0 \sim 2^{n}-1
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\end{align*}
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\subsection{补码}
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\begin{multicols}{2}
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\qquad 对应正数补码的“各位取反、末位加1”
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\begin{align*}
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+23 & = \texttt{00010111} \\
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\text{按位取反} & = \texttt{11101000} \\
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& + \hspace{0.3em}\phantom{0000000}\texttt{1} \\
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-23_{\text{补码}} & = \texttt{11101001}
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\end{align*}
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\columnbreak
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\qquad 模($2^n$)减去该负数的绝对值
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\begin{align*}
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& \texttt{100000000} \\
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- \hspace{0.5em} & \phantom{\texttt{0}}\texttt{00010111} \\[-1em]
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& \hspace{-1em}\rule{2.5cm}{0.02em} \\[-0.5em]
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& \phantom{\texttt{0}}\texttt{11101001}
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\end{align*}
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\end{multicols}
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\subsection{GDB查看数据}
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\texttt{>(gdb) x/4xb} \\[1em]
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\begin{minipage}{12cm}
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\begin{multicols}{2}
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\texttt{b} - byte (8-bit value) \\
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\texttt{h} - halfword (16-bit value) \\
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\texttt{w} - word (32-bit value) \\
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\texttt{g} - giant word (64-bit value) \\
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\texttt{o} - octal \\
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\texttt{x} - hexadecimal \\
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\texttt{d} - decimal \\
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\texttt{u} - unsigned decimal \\
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\texttt{t} - binary \\
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\texttt{f} - floating point \\
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\texttt{a} - address \\
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\texttt{c} - char \\
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\texttt{s} - string \\
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\texttt{i} - instruction
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\end{multicols}
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\end{minipage}
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\newpage
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\subsection{浮点数}
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\vspace{-1cm}
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\InsertBoxR{0}{\tcbox[blank]{\begin{tabular}{|c|c|c|c|c|}
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\hline 二进制位数 & s 符号位 & exp 指数 & frac 尾数 & 总计 \\
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\hline \texttt{float} & 1 & 8 & 23 & 32 \\
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\hline \texttt{double} & 1 & 11 & 52 & 64 \\
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\hline
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\end{tabular}}}
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\vspace{1cm}
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浮点数表示为 $(-1)^s \cdot M \cdot 2^E$
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\InsertBoxR{0}{\tcboxmath{\begin{matrix}
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\multicolumn{2}{c}{\text{偏置值 Bias}} \\
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\texttt{float} & 127 \\
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\texttt{double} & 1023 \\
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\end{matrix}}\hspace{3cm}}
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\vspace{-1em}
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\subsubsection{规格化数 $ \text{exp} \ne 0$ 且 $\text{exp} \ne 11 \dots 1$}
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$\text{(unsigned)} \ \text{exp} \ = \ E \ + \text{Bias}$
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$\text{Bias (偏置值)} = 2^{k-1}-1$ , \(k\) 为 exp 的二进制位数
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\begin{multicols}{2}
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例1:十进制整数$\rightarrow$二进制浮点数
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\begin{align*}
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\text{float} \ F &= 15213.0 \\
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\text{化为二进制数:} \\
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15213_{10} &= 11101101101101_{2} \\
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&= 1.1101101101101_{2} \ \times \ 2^{13} \\
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\text{计算 frac:} \\
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M &= 1.\underbar{1101101101101}_{2} \\
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\text{frac} &= \underbar{1101101101101}0000000000_{2}\\
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\text{计算 exp:} \\
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E &= 13 \qquad \text{来自化为二进制时的指数} \\
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\text{Bias} &= 127 \\
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\text{exp} &= 140 = 10001100_{2}
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\end{align*}
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结果：\\
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\(\begin{matrix}
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0 & 10001100 & 11011011011010000000000 \\
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\text{s} & \text{exp} & \text{frac}
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\end{matrix}\)
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\columnbreak
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例2: 二进制浮点数$\rightarrow$十进制数
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无符号数，4位阶码(Bias\(=7\))，3个小数位 \\
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\quad \(\begin{matrix}
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1001 & 111 \\
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\text{exp} & \text{frac}
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\end{matrix}\) \\[0.5em]
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\(\begin{aligned}
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\text{计算} \space E &= \text{exp} - \text{Bias} & \\
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&= 1001_2 - 7_{10} \\
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&= 2_{10}
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\end{aligned}\) \\[0.5em]
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计算 \(M = 1.\underbar{frac} = 1.\underbar{111}\) \\[0.5em]
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化为十进制:
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\(\begin{aligned}
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1.111 \times 2^2 & = 111.1_2 \quad \text{小数点右移2位} \\
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& = \frac{15}{2} = 7.5
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\end{aligned}\)
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\end{multicols}
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\InsertBoxR{0}{\tcboxmath{\begin{matrix}
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\text{非规格化数} \ E = 1 - \text{Bias} \\
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\begin{matrix}
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\texttt{float} & -126 \\
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\texttt{double} & -1022 \\
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\end{matrix}
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\end{matrix}}
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\hspace{2cm}}
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\vspace{-5mm}
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\vspace{-0.3em}
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\subsubsection{非规格化数 $\text{exp} = 0$}
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frac \(= 00\dots0\) 表示 0
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frac \(\ne 00\dots0\) 表示接近 0 的小数 $(-1)^s \cdot M \cdot 2^{E}$
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\vspace{-0.3em}
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\subsubsection{特殊值 $\text{exp} = 11 \dots 1$}
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\begin{multicols}{2}
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frac \(= 00\dots0\) 表示 \(\infty\)
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frac \(\ne 00\dots0\) 表示 NaN
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\columnbreak
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\(1.0 / 0.0 = −1.0 / −0.0 = + \infty\)
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\(1.0 / −0.0 = - \infty\)
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\(\sqrt{–1.0} = \infty - \infty = \infty \times 0 = \text{NaN}\)
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\end{multicols}
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\subsubsection{舍入(到偶数)}
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\begin{table}[h]
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\begin{tabular}{|ccc|}
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\hline
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末两位 & 动作 & 例子（保留一位小数） \\ \hline
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01 & 舍 & $11.0\underbar{01}_2 \to 11.0_2$ \\ \hline
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11 & 入 & $10.0\underbar{11}_2 \to 10.1_2$ \\ \hline
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10 & \makecell[c]{强迫结果为偶数(末尾为0)\\010舍 , 110入} &\makecell[c]{$10.0\underbar{10}_2 \to 10.0_2$ \\ $10.1\underbar{10}_2 \to 11.0_2$} \\ \hline
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\end{tabular}
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\end{table}
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\pagebreak
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\InsertBoxR{0}{\tcboxmath{\begin{matrix}
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\text{寻址模式} & \text{p. 121} \\
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\text{栈帧结构} & \text{p. 164} \\
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\text{gdb 操作} & \text{p. 194} \\
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\end{matrix}}\hspace{2cm}}
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\vspace{-1.5em}
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\section{程序的机器级表示}
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\subsection{计算数组元素的地址}
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\begin{multicols}{2}
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计算 \texttt{T* D[R][C]} 元素 \texttt{D[i][j]}的地址： \\
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\(\texttt{\&D[i][j]} = \texttt{\&D[0][0]} + \texttt{sizeof(T)} \times \left(C \cdot i + j\right)\) \\
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假设 \texttt{sizeof(T) = k}, 将 \texttt{D[i][j]} 复制到 \%eax 中 \\
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\texttt{asm: D in \%rdi , i in \%rsi , j in \%rdx }
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\columnbreak
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\texttt{1 \ leaq (\%rsi,\%rsi,\$C-1), \%rax \\
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2 \ leaq (\%rdi,\%rax,\$k), \%rax \\
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3 \ movl (\%rax,\%rdx,\$k), \%rax \\
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}
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结果为 \texttt{D + k $\cdot$ C $\cdot$ i + k $\cdot$ j} \\
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即 \texttt{D + sizeof(T) $\times$ (C $\cdot$ i + j)}
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\end{multicols}
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\vspace{-2.5em}
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\section{链接}
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\vspace{-1.5em}
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\subsection{符号表 (.symtab)}
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\vspace{-1em}
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\begin{table}[h]
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\begin{tabular}{l|c|c|c|l}
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\hline
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C语言表示 & 类型 & 符号强度 & 节 & 说明\\ \hline
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\texttt{void swap();} & 全局 & 强 & \texttt{.text} & 非静态函数 \\ \hline
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\texttt{int *bufp0 = \&buf[0]} & 全局 & 强 & \texttt{.data} & 初始化为其他值的全局变量\\ \hline
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\texttt{int a = 0;} & 全局 & 强 & \texttt{.bss} & 初始化为 0 的全局变量 \\ \hline
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\texttt{int *bufp1;} & 全局 & 弱 & \texttt{COMMON} & 未初始化的全局变量 \\ \hline
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\texttt{extern int buf[];} & 外部 & --- & \texttt{UNDEF} & \makecell[l]{
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未解析的引用符号 \\
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位于实际定义所在位置
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} \\ \hline
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\begin{lstlisting}[language=C,gobble=8]
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void p() {
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static int i = 1; }
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\end{lstlisting}
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& 局部 & --- & \texttt{.data} & 初始化为其他值的静态局部变量 \\ \hline
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\begin{lstlisting}[language=C,gobble=8]
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void p() { static int i;
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static int j = 0; }
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\end{lstlisting}
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& 局部 & --- & \texttt{.bss} & \makecell[l]{未初始化的静态局部变量 \\ 初始化为 0 的静态局部变量} \\ \hline
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\begin{lstlisting}[language=C,gobble=8]
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static void q() {
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int j = 2; }
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\end{lstlisting}
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& --- & --- & --- & \makecell[l]{链接不涉及静态函数\\链接不涉及非静态局部变量} \\ \hline
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\end{tabular}
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\end{table}
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\vspace{-2em}
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\subsection{链接顺序}
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\vspace{-0.5em}
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\texttt{\$ gcc -static -o prog2c main2.o ./libvector.a} \\
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E 将被合并以组成可执行文件的所有目标文件集合\\
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U 当前所有未解析的引用符号的集合\\
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D 当前所有定义符号的集合\\
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开始 E、U、D为空，首先扫描 \texttt{main2.o}，将其加入 E，将未找到的符号加入 U, 定义的符号加入 D。 \\
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再扫描 \texttt{./libvector.a}，将匹配到的 U 中的符号转移到 D 并加入到 E， 同时将未找到的符号加入 U。 \\
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最后搜索标准库 \texttt{libc.a}，处理完\texttt{libc.a}时，U一定是空的，D中符号唯一，否则错误。
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\vspace{-1em}
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\subsection{重定位}
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\vspace{-0.5em}
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\begin{itemize}
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\item 重定位 PC 相对引用(\texttt{R\_X86\_64\_PC32}): \\
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重定位值 \( =\texttt{ADDR(r.symbol)} - \underbrace{\left(\texttt{ADDR(.text)} + \texttt{r.offset}\right)}_{\text{重定位值的地址}} + \texttt{r.addend}\)
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在asm中表示为 \texttt{4004de: e8 \underbar{05 00 00 00} \quad callq 4004e8 <sum>}
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\item 重定位绝对引用(\texttt{R\_X86\_64\_32}): \\
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重定位值 = \(\texttt{ADDR(r.symbol)} + \texttt{r.addend}\) \\
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在asm中直接以其绝对地址表示\texttt{4004d9: bf \underbar{18 10 60 00} \quad mov \$0x601018 \%edi}
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\end{itemize}
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\newpage
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\end{document}
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