| author | Christian Urban <christian.urban@kcl.ac.uk> |
| Mon, 30 Sep 2024 10:47:49 +0100 | |
| changeset 965 | e7dbebf43ac3 |
| parent 957 | 03c5a8987141 |
| child 974 | 06148fc63273 |
| permissions | -rw-r--r-- |
| 701 | 1 |
% !TEX program = xelatex |
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\documentclass[dvipsnames,14pt,t,xelatex,aspectratio=169,xcolor={table}]{beamer}
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\usepackage{../slides}
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\usepackage{../langs}
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\usepackage{../data}
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\usepackage{../graphicss}
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\usepackage{../grammar}
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%\usepackage{soul}
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%\usepackage{mathpartir}
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\usetikzlibrary{shapes,arrows,shadows}
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\usepackage[most]{tcolorbox}
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\newtcbox{\hl}[1][]{%
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size=fbox, |
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tcbox raise base, nobeforeafter, |
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enhanced, colframe=gray, |
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colback=gray!30, boxrule=1pt, |
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fontupper=\ttfamily, |
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#1} |
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% beamer stuff |
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\renewcommand{\slidecaption}{CFL 09, King's College London}
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\newcommand{\bl}[1]{\textcolor{blue}{#1}}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[t]
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\\[-3mm] |
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\LARGE Compilers and \\[-2mm] |
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\LARGE Formal Languages\\[-3mm] |
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\end{tabular}}
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\normalsize |
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\begin{center}
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\begin{tabular}{ll}
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Email: & christian.urban at kcl.ac.uk\\ |
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Office Hour: & Thurdays 15 -- 16\\ |
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Location: & N7.07 (North Wing, Bush House)\\ |
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Slides \& Progs: & KEATS\\ |
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Pollev: & \texttt{\alert{https://pollev.com/cfltutoratki576}}\\
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\end{tabular}
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\end{center}
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\begin{center}
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53 |
\begin{tikzpicture}
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\node[drop shadow,fill=white,inner sep=0pt] |
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{\footnotesize\rowcolors{1}{capri!10}{white}
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\begin{tabular}{|p{4.8cm}|p{4.8cm}|}\hline
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1 Introduction, Languages & 6 While-Language \\ |
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2 Regular Expressions, Derivatives & 7 Compilation, JVM \\ |
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3 Automata, Regular Languages & 8 Compiling Functional Languages \\ |
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4 Lexing, Tokenising & \cellcolor{blue!50} 9 Optimisations \\
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5 Grammars, Parsing & 10 LLVM \\ \hline |
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\end{tabular}%
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}; |
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\end{tikzpicture}
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\end{center}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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||
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[c,fragile]
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\frametitle{The Fun Language}
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\footnotesize |
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\begin{textblock}{13}(0.9,3)
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\begin{lstlisting}[numbers=none]
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def fib(n) = if n == 0 then 0 |
77 |
else if n == 1 then 1 |
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else fib(n - 1) + fib(n - 2); |
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def fact(n) = if n == 0 then 1 else n * fact(n - 1); |
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def ack(m, n) = if m == 0 then n + 1 |
83 |
else if n == 0 then ack(m - 1, 1) |
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else ack(m - 1, ack(m, n - 1)); |
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def gcd(a, b) = if b == 0 then a else gcd(b, a % b); |
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\end{lstlisting}
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\end{textblock}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%\begin{frame}[c, fragile]
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%\frametitle{Stack Estimation}
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%\small |
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%\mbox{}\\[-15mm]%
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% |
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%\bl{\begin{center}
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%\begin{tabular}{@{\hspace{-4mm}}l@{\hspace{2mm}}c@{\hspace{2mm}}l@{}}
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%$\textit{estimate}(n)$ & $\dn$ & $1$\\
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%$\textit{estimate}(x)$ & $\dn$ & $1$\\
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%$\textit{estimate}(a_1\;aop\;a_2)$ & $\dn$ &
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%$\textit{estimate}(a_1) + \textit{estimate}(a_2)$\\
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%$\textit{estimate}(\pcode{if}\;b\;\pcode{then}\;e_1\;\pcode{else}\;e_2)$ & $\dn$ &
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%$\textit{estimate}(b) +$\\
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%& & $\quad max(\textit{estimate}(e_1), \textit{estimate}(e_2))$\\
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%$\textit{estimate}(\pcode{write}(e))$ & $\dn$ &
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%$\textit{estimate}(e) + 1$\\
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%$\textit{estimate}(e_1 ; e_2)$ & $\dn$ &
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%$max(\textit{estimate}(e_1), \textit{estimate}(e_2))$\\
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%$\textit{estimate}(f(e_1, ..., e_n))$ & $\dn$ &
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%$\sum_{i = 1..n}\;estimate(e_i)$\medskip\\
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%$\textit{estimate}(a_1\;bop\;a_2)$ & $\dn$ &
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%$\textit{estimate}(a_1) + \textit{estimate}(a_2)$\\
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%\end{tabular}
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%\end{center}}
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%\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[c,fragile]
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\frametitle{\mbox{}\hspace{5cm}Factorial}
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\begin{textblock}{7}(0,1.6)\footnotesize
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\begin{minipage}{6cm}
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\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
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.method public static fact(I)I |
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.limit locals 1 |
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.limit stack 6 |
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iload 0 |
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ldc 0 |
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if_icmpne If_else_0 |
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ldc 1 |
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goto If_end_1 |
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If_else_0: |
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iload 0 |
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iload 0 |
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ldc 1 |
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isub |
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invokestatic fact/fact/fact(I)I |
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imul |
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If_end_1: |
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ireturn |
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.end method |
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\end{lstlisting}
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\end{minipage}
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\end{textblock}
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\begin{textblock}{7}(6,7)
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\begin{bubble}[7cm]\small
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\begin{lstlisting}[language=Lisp,
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basicstyle=\ttfamily, |
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numbers=none, |
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xleftmargin=1mm,linebackgroundcolor=\color{cream}]
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def fact(n) = |
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if n == 0 then 1 |
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else n * fact(n - 1) |
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\end{lstlisting}
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\end{bubble}
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\end{textblock}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 901 | 168 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[c,fragile]
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\frametitle{\mbox{}\hspace{5cm}Factorial}
|
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176 |
||
177 |
\begin{textblock}{7}(0,1.0)\footnotesize
|
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\begin{minipage}{6cm}
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\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
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.method public static facT(II)I |
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.limit locals 2 |
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.limit stack 6 |
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iload 0 |
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ldc 0 |
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if_icmpne If_else_2 |
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iload 1 |
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goto If_end_3 |
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If_else_2: |
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iload 0 |
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ldc 1 |
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isub |
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iload 0 |
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iload 1 |
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imul |
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invokestatic fact/fact/facT(II)I |
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If_end_3: |
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ireturn |
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.end method |
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\end{lstlisting}
|
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200 |
\end{minipage}
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\end{textblock}
|
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202 |
||
203 |
\begin{textblock}{7}(6,7)
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\begin{bubble}[7cm]\small
|
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205 |
\begin{lstlisting}[language=Lisp,
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basicstyle=\ttfamily, |
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207 |
numbers=none, |
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208 |
xleftmargin=1mm,linebackgroundcolor=\color{cream}]
|
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def facT(n, acc) = |
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if n == 0 then acc |
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else facT(n - 1, n * acc) |
| 853 | 212 |
\end{lstlisting}
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213 |
\end{bubble}
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\end{textblock}
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215 |
||
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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218 |
||
219 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[fragile]
|
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221 |
||
| 957 | 222 |
\begin{textblock}{7}(1,-0.38)\footnotesize
|
| 853 | 223 |
\begin{minipage}{6cm}
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\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none, escapeinside={(*@}{@*)}]
|
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.method public static facT(II)I |
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.limit locals 2 |
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.limit stack 6 |
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(*@\hl{facT\_Start:} @*)
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iload 0 |
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ldc 0 |
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if_icmpne If_else_2 |
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iload 1 |
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goto If_end_3 |
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If_else_2: |
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iload 0 |
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ldc 1 |
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isub |
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iload 0 |
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iload 1 |
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imul |
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(*@\hl{istore 1} @*)
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(*@\hl{istore 0} @*)
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(*@\hl{goto facT\_Start} @*)
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If_end_3: |
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ireturn |
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.end method |
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\end{lstlisting}
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248 |
\end{minipage}
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249 |
\end{textblock}
|
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250 |
||
251 |
\begin{textblock}{7}(6,7)
|
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252 |
\begin{bubble}[7cm]\small
|
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253 |
\begin{lstlisting}[language=Lisp,
|
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254 |
basicstyle=\ttfamily, |
|
255 |
numbers=none, |
|
256 |
xleftmargin=1mm,linebackgroundcolor=\color{cream}]
|
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257 |
def facT(n, acc) = |
|
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if n == 0 then acc |
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else facT(n - 1, n * acc); |
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260 |
\end{lstlisting}
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261 |
\end{bubble}
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262 |
\end{textblock}
|
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263 |
||
264 |
\end{frame}
|
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265 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
266 |
||
267 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
268 |
\begin{frame}[t, fragile]
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\frametitle{Tail Recursion}
|
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270 |
||
271 |
A call to \texttt{f(args)} is usually compiled as\medskip
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272 |
||
273 |
{\small\begin{lstlisting}[basicstyle=\ttfamily, numbers=none]
|
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274 |
args onto stack |
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275 |
invokestatic .../f |
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276 |
\end{lstlisting}}\pause
|
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277 |
||
278 |
||
279 |
A call is in tail position provided:\medskip |
|
280 |
||
281 |
{\small\begin{itemize}
|
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282 |
\item \texttt{if Bexp then \hl{Exp} else \hl{Exp}}
|
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283 |
\item \texttt{Exp ; \hl{Exp}}
|
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284 |
\item \texttt{Exp op Exp}
|
|
285 |
\end{itemize}}\medskip
|
|
286 |
||
287 |
then a call \texttt{f(args)} can be compiled as\medskip\small
|
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288 |
||
289 |
\begin{lstlisting}[numbers=none]
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290 |
prepare environment |
|
291 |
jump to start of function |
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292 |
\end{lstlisting}
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293 |
||
294 |
\end{frame}
|
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295 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
296 |
||
297 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
298 |
\begin{frame}[c, fragile]
|
|
299 |
\frametitle{Tail Recursive Call}
|
|
300 |
\footnotesize |
|
301 |
||
302 |
\begin{textblock}{13}(-0.3,3)
|
|
303 |
\begin{lstlisting}[language=Scala, numbers=none]
|
|
304 |
def compile_expT(a: Exp, env: Mem, name: String): Instrs = |
|
305 |
... |
|
306 |
case Call(n, args) => if (name == n) |
|
307 |
{
|
|
308 |
val stores = |
|
309 |
args.zipWithIndex.map { case (x, y) => i"istore $y" }
|
|
310 |
||
311 |
args.map(a => compile_expT(a, env, "")).mkString ++ |
|
312 |
stores.reverse.mkString ++ |
|
313 |
i"goto ${n}_Start"
|
|
314 |
} else {
|
|
315 |
val is = "I" * args.length |
|
316 |
args.map(a => compile_expT(a, env, "")).mkString ++ |
|
317 |
i"invokestatic XXX/XXX/${n}(${is})I"
|
|
318 |
} |
|
319 |
\end{lstlisting}
|
|
320 |
\end{textblock}
|
|
321 |
||
322 |
\end{frame}
|
|
323 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 609 | 324 |
|
325 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 957 | 326 |
\begin{frame}[c]
|
327 |
\frametitle{???}
|
|
328 |
||
329 |
\small |
|
330 |
\begin{tabular}{cc}
|
|
331 |
\includegraphics[scale=0.2]{basic-code.jpg} &
|
|
332 |
\includegraphics[scale=0.2]{machine-code.jpg}
|
|
333 |
\end{tabular}
|
|
334 |
||
335 |
||
336 |
\end{frame}
|
|
337 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
338 |
||
339 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
340 |
\begin{frame}[t]
|
|
341 |
\frametitle{Opcodes}
|
|
342 |
||
343 |
\small |
|
344 |
\begin{tabular}{cc}
|
|
345 |
\includegraphics[scale=0.3]{machine-code-large.png}
|
|
346 |
\end{tabular}
|
|
347 |
||
348 |
||
349 |
\end{frame}
|
|
350 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
351 |
||
352 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
353 |
\begin{frame}<1-2>[t]
|
|
354 |
||
355 |
\end{frame}
|
|
356 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
357 |
||
358 |
||
359 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 609 | 360 |
\begin{frame}[c,fragile]
|
| 901 | 361 |
\frametitle{Peephole Optimisations}
|
| 609 | 362 |
|
| 901 | 363 |
\begin{center}
|
364 |
\begin{tabular}{ll}
|
|
365 |
\texttt{ldc}: & \texttt{iconst\_0} \ldots \texttt{iconst\_5}\\
|
|
366 |
& \texttt{bipush} $n$ where $-128 < n <= 128$\bigskip\\
|
|
367 |
\texttt{iload}: & \texttt{iload\_0} \ldots \texttt{iload\_3}\bigskip\\
|
|
368 |
\texttt{istore}: & \texttt{istore\_0} \ldots \texttt{istore\_3}\\
|
|
369 |
\end{tabular}
|
|
370 |
\end{center}
|
|
| 701 | 371 |
|
372 |
\end{frame}
|
|
373 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
374 |
||
375 |
||
376 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 901 | 377 |
%\begin{frame}[c,fragile]
|
378 |
%\frametitle{Factorial Funct.~on the JVM}
|
|
379 |
% |
|
380 |
%\begin{textblock}{7}(1,1.8)\footnotesize
|
|
381 |
%\begin{minipage}{6cm}
|
|
382 |
%\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
|
|
383 |
%.method public static facT(II)I |
|
384 |
%.limit locals 2 |
|
385 |
%.limit stack 6 |
|
386 |
% iload 0 |
|
387 |
% ldc 0 |
|
388 |
% if_icmpne If_else_2 |
|
389 |
% iload 1 |
|
390 |
% goto If_end_3 |
|
391 |
%If_else_2: |
|
392 |
% iload 0 |
|
393 |
% ldc 1 |
|
394 |
% isub |
|
395 |
% iload 0 |
|
396 |
% iload 1 |
|
397 |
% imul |
|
398 |
% invokestatic fact/fact/facT(II)I |
|
399 |
%If_end_3: |
|
400 |
% ireturn |
|
401 |
%.end method |
|
402 |
%\end{lstlisting}
|
|
403 |
%\end{minipage}
|
|
404 |
%\end{textblock}
|
|
405 |
% |
|
406 |
%\begin{textblock}{7}(6,7)
|
|
407 |
%\begin{bubble}[7cm]\small
|
|
408 |
%\begin{lstlisting}[language=Lisp,
|
|
409 |
% basicstyle=\ttfamily, |
|
410 |
% numbers=none, |
|
411 |
% xleftmargin=1mm,linebackgroundcolor=\color{cream}]
|
|
412 |
%def facT(n, acc) = |
|
413 |
% if n == 0 then acc |
|
414 |
% else facT(n - 1, n * acc); |
|
415 |
%\end{lstlisting}
|
|
416 |
%\end{bubble}
|
|
417 |
%\end{textblock}
|
|
418 |
% |
|
419 |
%\end{frame}
|
|
420 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
421 |
||
422 |
||
423 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 701 | 424 |
\begin{frame}[fragile,c]
|
425 |
\frametitle{LLVM}
|
|
| 609 | 426 |
|
427 |
\begin{itemize}
|
|
| 701 | 428 |
\item Chris Lattner, Vikram Adve (started in 2000) |
429 |
\item Apple hired Lattner in 2006 |
|
430 |
\item modular architecture, LLVM-IR |
|
431 |
\item \texttt{lli} and \texttt{llc}
|
|
432 |
\end{itemize}
|
|
| 609 | 433 |
|
434 |
\end{frame}
|
|
| 701 | 435 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
436 |
||
437 |
||
438 |
||
439 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
440 |
\tikzstyle{sensor}=[draw, fill=blue!20, text width=3.8em, line width=1mm,
|
|
441 |
text centered, minimum height=2em,drop shadow] |
|
442 |
\tikzstyle{ann} = [above, text width=4em, text centered]
|
|
443 |
\tikzstyle{sc} = [sensor, text width=7em, fill=red!20,
|
|
444 |
minimum height=6em, rounded corners, drop shadow,line width=1mm] |
|
445 |
||
446 |
\begin{frame}[fragile,c]
|
|
447 |
\frametitle{LLVM: Overview}
|
|
448 |
||
449 |
\begin{tikzpicture}
|
|
450 |
% Validation Layer is the same except that there are a set of nodes and links which are added |
|
451 |
||
452 |
\path (0,0) node (IR) [sc] {\textbf{LLVM-IR}\\ Optimisations};
|
|
453 |
\path (IR.west)+(-2.2,1.7) node (sou1) [sensor] {C++};
|
|
454 |
\path (IR.west)+(-2.2,0.5) node (sou2)[sensor] {C};
|
|
455 |
\path (IR.west)+(-2.2,-1.0) node (dots)[ann] {$\vdots$};
|
|
456 |
\path (IR.west)+(-2.2,-1.8) node (sou3)[sensor] {Haskell};
|
|
457 |
||
458 |
\path [draw,->,line width=1mm] (sou1.east) -- node [above] {} (IR.160);
|
|
459 |
\path [draw,->,line width=1mm] (sou2.east) -- node [above] {} (IR.180);
|
|
460 |
\path [draw,->,line width=1mm] (sou3.east) -- node [above] {} (IR.200);
|
|
461 |
||
462 |
\path (IR.east)+(2.2,2.0) node (tar1)[sensor] {x86};
|
|
463 |
\path (IR.east)+(2.2,0.8) node (tar2)[sensor] {ARM};
|
|
464 |
\path (IR.east)+(2.2,-0.4) node (tar3)[sensor] {MIPS};
|
|
465 |
\path (IR.east)+(2.2,-1.6) node (tar4)[sensor] {RISC};
|
|
466 |
\path (IR.east)+(2.2,-2.8) node (tar5)[sensor] {Power PC};
|
|
467 |
\path (IR.east)+(2.2,-4.2) node (dots2)[ann] {$\vdots$};
|
|
468 |
||
469 |
\path [draw,<-,line width=1mm] (tar1.west) -- node [above] {} (IR.10);
|
|
470 |
\path [draw,<-,line width=1mm] (tar2.west) -- node [above] {} (IR.5);
|
|
471 |
\path [draw,<-,line width=1mm] (tar3.west) -- node [above] {} (IR.0);
|
|
472 |
\path [draw,<-,line width=1mm] (tar4.west) -- node [above] {} (IR.-5);
|
|
473 |
\path [draw,<-,line width=1mm] (tar5.west) -- node [above] {} (IR.-10);
|
|
474 |
||
475 |
\end{tikzpicture}
|
|
476 |
\end{frame}
|
|
477 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
478 |
||
| 609 | 479 |
|
480 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 701 | 481 |
\begin{frame}[fragile,t]
|
482 |
\frametitle{LLVM-IR}
|
|
483 |
\small |
|
484 |
||
485 |
\begin{textblock}{7.7}(8,11.4)
|
|
486 |
\begin{bubble}[5cm]\small
|
|
487 |
\begin{lstlisting}[language=Lisp,
|
|
488 |
numbers=none, |
|
489 |
xleftmargin=1mm,linebackgroundcolor=\color{cream}]
|
|
490 |
def fact(n) = |
|
491 |
if n == 0 then 1 |
|
492 |
else n * fact(n - 1) |
|
493 |
\end{lstlisting}
|
|
494 |
\end{bubble}
|
|
495 |
\end{textblock}
|
|
|
310
d384fe01d0e8
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
309
diff
changeset
|
496 |
|
| 701 | 497 |
\begin{lstlisting}[language=LLVM,xleftmargin=-7mm]
|
498 |
define i32 @fact (i32 %n) {
|
|
499 |
%tmp_19 = icmp eq i32 %n, 0 |
|
500 |
br i1 %tmp_19, label %if_br_23, label %else_br_24 |
|
501 |
||
502 |
if_br_23: |
|
503 |
ret i32 1 |
|
504 |
||
505 |
else_br_24: |
|
506 |
%tmp_21 = sub i32 %n, 1 |
|
507 |
%tmp_22 = call i32 @fact (i32 %tmp_21) |
|
508 |
%tmp_20 = mul i32 %n, %tmp_22 |
|
509 |
ret i32 %tmp_20 |
|
510 |
} |
|
511 |
\end{lstlisting}
|
|
| 609 | 512 |
|
513 |
\end{frame}
|
|
| 701 | 514 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
515 |
||
516 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
517 |
\begin{frame}[fragile,c]
|
|
518 |
\frametitle{LLVM Types}
|
|
519 |
||
520 |
\tt |
|
521 |
\begin{center}
|
|
522 |
\begin{tabular}{ll}
|
|
523 |
boolean & i1 \\ |
|
524 |
byte & i8 \\ |
|
525 |
short & i16\\ |
|
526 |
char & i16\\ |
|
527 |
integer & i32\\ |
|
528 |
long & i64\\ |
|
529 |
float & float\\ |
|
530 |
double & double\\ |
|
531 |
*\_ & pointer to \\ |
|
532 |
**\_ & pointer to a pointer to\\ |
|
533 |
\mbox{}[\_] & arrays of\\
|
|
534 |
\end{tabular}
|
|
535 |
\end{center}
|
|
536 |
||
537 |
\end{frame}
|
|
538 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
539 |
||
540 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
541 |
\begin{frame}[fragile,c]
|
|
| 702 | 542 |
\frametitle{LLVM-IR Instructions}
|
| 701 | 543 |
\small |
544 |
||
| 702 | 545 |
\begin{lstlisting}[language=LLVM,xleftmargin=0mm]
|
| 701 | 546 |
br i1 %var, label %if_br, label %else_br |
547 |
||
548 |
icmp eq i32 %x, %y ; for equal |
|
549 |
icmp sle i32 %x, %y ; signed less or equal |
|
550 |
icmp slt i32 %x, %y ; signed less than |
|
551 |
icmp ult i32 %x, %y ; unsigned less than |
|
552 |
||
553 |
%var = call i32 @foo(...args...) |
|
554 |
\end{lstlisting}
|
|
555 |
||
556 |
\end{frame}
|
|
557 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
558 |
||
559 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
560 |
\begin{frame}[fragile,c]
|
|
561 |
\frametitle{SSA Format}
|
|
562 |
||
563 |
\bl{$(1 + a) + (3 + (b * 5))$}\bigskip\bigskip
|
|
564 |
||
565 |
\begin{lstlisting}[language=LLVM]
|
|
| 702 | 566 |
tmp0 = add 1 a |
567 |
tmp1 = mul b 5 |
|
568 |
tmp2 = add 3 tmp1 |
|
569 |
tmp3 = add tmp0 tmp2 |
|
570 |
\end{lstlisting}\bigskip\bigskip
|
|
571 |
||
572 |
\hfill Static Single Assignment |
|
| 701 | 573 |
|
574 |
\end{frame}
|
|
575 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 609 | 576 |
|
577 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 701 | 578 |
\begin{frame}[fragile,c]
|
579 |
\frametitle{Abstract Syntax Trees}
|
|
580 |
\footnotesize |
|
581 |
||
582 |
\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=-3mm]
|
|
583 |
// Fun language (expressions) |
|
584 |
abstract class Exp |
|
585 |
abstract class BExp |
|
586 |
||
587 |
case class Call(name: String, args: List[Exp]) extends Exp |
|
588 |
case class If(a: BExp, e1: Exp, e2: Exp) extends Exp |
|
589 |
case class Write(e: Exp) extends Exp |
|
590 |
case class Var(s: String) extends Exp |
|
591 |
case class Num(i: Int) extends Exp |
|
592 |
case class Aop(o: String, a1: Exp, a2: Exp) extends Exp |
|
593 |
case class Sequence(e1: Exp, e2: Exp) extends Exp |
|
594 |
case class Bop(o: String, a1: Exp, a2: Exp) extends BExp |
|
595 |
\end{lstlisting}
|
|
| 609 | 596 |
|
| 701 | 597 |
\end{frame}
|
598 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
599 |
||
600 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
601 |
\begin{frame}[fragile,c]
|
|
602 |
\frametitle{K-(Intermediate)Language}
|
|
603 |
\footnotesize |
|
| 609 | 604 |
|
| 701 | 605 |
\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=-3mm]
|
606 |
abstract class KExp |
|
607 |
abstract class KVal |
|
608 |
||
| 702 | 609 |
// K-Values |
| 701 | 610 |
case class KVar(s: String) extends KVal |
611 |
case class KNum(i: Int) extends KVal |
|
612 |
case class Kop(o: String, v1: KVal, v2: KVal) extends KVal |
|
613 |
case class KCall(o: String, vrs: List[KVal]) extends KVal |
|
614 |
case class KWrite(v: KVal) extends KVal |
|
615 |
||
| 702 | 616 |
// K-Expressions |
| 701 | 617 |
case class KIf(x1: String, e1: KExp, e2: KExp) extends KExp |
618 |
case class KLet(x: String, v: KVal, e: KExp) extends KExp |
|
619 |
case class KReturn(v: KVal) extends KExp |
|
| 609 | 620 |
\end{lstlisting}
|
621 |
||
| 701 | 622 |
\end{frame}
|
623 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
624 |
||
625 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
626 |
\begin{frame}[fragile,c]
|
|
| 819 | 627 |
\frametitle{KLet}
|
628 |
||
629 |
\begin{lstlisting}[language=LLVM]
|
|
630 |
tmp0 = add 1 a |
|
631 |
tmp1 = mul b 5 |
|
632 |
tmp2 = add 3 tmp1 |
|
633 |
tmp3 = add tmp0 tmp2 |
|
634 |
\end{lstlisting}
|
|
635 |
||
636 |
\begin{lstlisting}[language=LLVMIR]
|
|
637 |
KLet tmp0 , add 1 a in |
|
638 |
KLet tmp1 , mul b 5 in |
|
639 |
KLet tmp2 , add 3 tmp1 in |
|
640 |
KLet tmp3 , add tmp0 tmp2 in |
|
641 |
... |
|
642 |
\end{lstlisting}
|
|
643 |
||
644 |
\begin{lstlisting}[language=Scala,numbers=none]
|
|
645 |
case class KLet(x: String, e1: KVal, e2: KExp) |
|
646 |
\end{lstlisting}
|
|
647 |
||
648 |
\end{frame}
|
|
649 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
650 |
||
651 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
652 |
\begin{frame}[fragile,c]
|
|
653 |
\frametitle{KLet}
|
|
654 |
||
655 |
\begin{lstlisting}[language=LLVM]
|
|
656 |
tmp0 = add 1 a |
|
657 |
tmp1 = mul b 5 |
|
658 |
tmp2 = add 3 tmp1 |
|
659 |
tmp3 = add tmp0 tmp2 |
|
660 |
\end{lstlisting}
|
|
661 |
||
662 |
\begin{lstlisting}[language=LLVMIR]
|
|
663 |
let tmp0 = add 1 a in |
|
664 |
let tmp1 = mul b 5 in |
|
665 |
let tmp2 = add 3 tmp1 in |
|
666 |
let tmp3 = add tmp0 tmp2 in |
|
667 |
... |
|
668 |
\end{lstlisting}
|
|
669 |
||
670 |
\begin{lstlisting}[language=Scala,numbers=none]
|
|
671 |
case class KLet(x: String, e1: KVal, e2: KExp) |
|
672 |
\end{lstlisting}
|
|
673 |
||
674 |
\end{frame}
|
|
675 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
676 |
||
677 |
||
678 |
||
679 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
680 |
\begin{frame}[fragile,c]
|
|
| 701 | 681 |
\frametitle{CPS-Translation}
|
682 |
\small |
|
683 |
||
684 |
\begin{lstlisting}[language=Scala,numbers=none]
|
|
685 |
def CPS(e: Exp)(k: KVal => KExp) : KExp = |
|
686 |
e match { ... }
|
|
687 |
\end{lstlisting}
|
|
688 |
\bigskip\bigskip |
|
689 |
||
| 702 | 690 |
the continuation \texttt{k} can be thought of:\medskip
|
691 |
||
| 701 | 692 |
\small |
693 |
\begin{lstlisting}[language=LLVMIR,numbers=none,xleftmargin=30mm,escapeinside={(*@}{@*)}]
|
|
694 |
let tmp0 = add 1 a in |
|
695 |
let tmp1 = mul (*@$\Box$@*) 5 in |
|
696 |
let tmp2 = add 3 tmp1 in |
|
697 |
let tmp3 = add tmp0 tmp2 in |
|
698 |
KReturn tmp3 |
|
699 |
\end{lstlisting}
|
|
| 609 | 700 |
|
701 |
\end{frame}
|
|
| 701 | 702 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
703 |
||
704 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
705 |
\begin{frame}[fragile,c]
|
|
706 |
\frametitle{CPS-Translation}
|
|
707 |
\small |
|
708 |
||
709 |
\begin{lstlisting}[language=Scala,numbers=none]
|
|
710 |
def CPS(e: Exp)(k: KVal => KExp) : KExp = |
|
711 |
e match {
|
|
712 |
case Var(s) => k(KVar(s)) |
|
713 |
case Num(i) => k(KNum(i)) |
|
714 |
... |
|
715 |
} |
|
716 |
\end{lstlisting}
|
|
717 |
\bigskip\bigskip |
|
718 |
||
719 |
\small |
|
720 |
\begin{lstlisting}[language=LLVMIR,numbers=none,xleftmargin=30mm,escapeinside={(*@}{@*)}]
|
|
721 |
let tmp0 = add 1 a in |
|
722 |
let tmp1 = mul (*@$\Box$@*) 5 in |
|
723 |
let tmp2 = add 3 tmp1 in |
|
724 |
let tmp3 = add tmp0 tmp2 in |
|
725 |
KReturn tmp3 |
|
726 |
\end{lstlisting}
|
|
727 |
||
728 |
\end{frame}
|
|
729 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 609 | 730 |
|
731 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 701 | 732 |
\begin{frame}[fragile,c]
|
733 |
\frametitle{CPS-Translation}
|
|
734 |
\small |
|
| 609 | 735 |
|
| 701 | 736 |
\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=-5mm]
|
737 |
def CPS(e: Exp)(k: KVal => KExp) : KExp = e match {
|
|
738 |
case Aop(o, e1, e2) => {
|
|
739 |
val z = Fresh("tmp")
|
|
740 |
CPS(e1)(y1 => |
|
741 |
CPS(e2)(y2 => |
|
742 |
KLet(z, Kop(o, y1, y2), k(KVar(z))))) |
|
743 |
} ... |
|
744 |
} |
|
| 609 | 745 |
\end{lstlisting}
|
| 702 | 746 |
\mbox{}\\[-8mm]%
|
747 |
% |
|
| 701 | 748 |
\begin{lstlisting}[language=LLVMIR,numbers=none,xleftmargin=30mm,escapeinside={(*@}{@*)}]
|
749 |
... |
|
750 |
let z = op (*@$\Box_{y_1}$@*) (*@$\Box_{y_2}$@*)
|
|
751 |
let tmp0 = add 1 a in |
|
| 702 | 752 |
let tmp1 = mul (*@$\Box\!\!\!\!\raisebox{0.6mm}{\alert{\texttt{z}}}$@*) 5 in
|
| 701 | 753 |
let tmp2 = add 3 tmp1 in |
754 |
let tmp3 = add tmp0 tmp2 in |
|
755 |
KReturn tmp3 |
|
756 |
\end{lstlisting}
|
|
| 609 | 757 |
|
758 |
\end{frame}
|
|
| 701 | 759 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 609 | 760 |
|
761 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 701 | 762 |
\begin{frame}[fragile,c]
|
763 |
\frametitle{CPS-Translation}
|
|
764 |
\small |
|
| 609 | 765 |
|
| 701 | 766 |
\begin{lstlisting}[language=Scala,numbers=none]
|
767 |
def CPS(e: Exp)(k: KVal => KExp) : KExp = |
|
768 |
e match {
|
|
769 |
case Sequence(e1, e2) => |
|
770 |
CPS(e1)(_ => CPS(e2)(y2 => k(y2))) |
|
771 |
... |
|
772 |
} |
|
773 |
\end{lstlisting}
|
|
| 702 | 774 |
\bigskip |
| 701 | 775 |
|
776 |
\small |
|
777 |
\begin{lstlisting}[language=LLVMIR,numbers=none,xleftmargin=30mm,escapeinside={(*@}{@*)}]
|
|
778 |
let tmp0 = add 1 a in |
|
779 |
let tmp1 = mul (*@$\Box$@*) 5 in |
|
780 |
let tmp2 = add 3 tmp1 in |
|
781 |
let tmp3 = add tmp0 tmp2 in |
|
782 |
KReturn tmp3 |
|
783 |
\end{lstlisting}
|
|
| 609 | 784 |
|
785 |
\end{frame}
|
|
| 701 | 786 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
310
d384fe01d0e8
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
309
diff
changeset
|
787 |
|
| 610 | 788 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 701 | 789 |
\begin{frame}[fragile,c]
|
790 |
\frametitle{CPS-Translation}
|
|
791 |
\small |
|
792 |
||
793 |
\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=-3mm]
|
|
794 |
def CPS(e: Exp)(k: KVal => KExp) : KExp = |
|
795 |
e match {
|
|
796 |
... |
|
797 |
case If(Bop(o, b1, b2), e1, e2) => {
|
|
798 |
val z = Fresh("tmp")
|
|
799 |
CPS(b1)(y1 => |
|
800 |
CPS(b2)(y2 => |
|
801 |
KLet(z, Kop(o, y1, y2), |
|
802 |
KIf(z, CPS(e1)(k), CPS(e2)(k))))) |
|
803 |
} |
|
804 |
... |
|
805 |
} |
|
806 |
\end{lstlisting}
|
|
807 |
||
808 |
\end{frame}
|
|
809 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
810 |
||
| 702 | 811 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
812 |
\begin{frame}[c,fragile]
|
|
813 |
\frametitle{The Basic Language, 1980+}
|
|
814 |
||
815 |
\begin{lstlisting}[language={[Visual]Basic},numbers=none]
|
|
816 |
5 LET S = 0 |
|
817 |
10 INPUT V |
|
818 |
20 PRINT "Input number" |
|
819 |
30 IF N = 0 THEN GOTO 99 |
|
820 |
40 FOR I = 1 TO N |
|
821 |
45 LET S = S + V(I) |
|
822 |
50 NEXT I |
|
823 |
60 PRINT S/N |
|
824 |
70 GOTO 5 |
|
825 |
99 END |
|
826 |
\end{lstlisting}
|
|
827 |
||
828 |
\hfill ``Spaghetti Code'' |
|
829 |
\end{frame}
|
|
830 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
831 |
||
832 |
||
833 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
834 |
\begin{frame}[c,fragile]
|
|
835 |
\frametitle{Target Specific ASM}
|
|
836 |
||
837 |
\begin{lstlisting}[language={},numbers=none]
|
|
838 |
llc -march=x86-64 fact.ll |
|
839 |
llc -march=arm fact.ll |
|
840 |
||
841 |
||
842 |
Intel: xorl %eax, %eax |
|
843 |
ARM: mov pc, lr |
|
844 |
\end{lstlisting}
|
|
845 |
||
846 |
\end{frame}
|
|
847 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 701 | 848 |
|
849 |
||
850 |
||
851 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 610 | 852 |
\begin{frame}[c]
|
853 |
||
854 |
\large\bf |
|
| 701 | 855 |
Using a compiler, \\ |
856 |
how can you mount the\\ |
|
857 |
perfect attack against a system? |
|
| 610 | 858 |
|
| 701 | 859 |
\end{frame}
|
| 610 | 860 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
861 |
||
862 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
863 |
\mode<presentation>{
|
|
864 |
\begin{frame}[c]
|
|
865 |
||
866 |
{\large\bf
|
|
867 |
What is a \alert{perfect} attack?}\bigskip
|
|
868 |
||
869 |
\begin{enumerate}
|
|
870 |
\item you can potentially completely take over a target system |
|
871 |
\item your attack is (nearly) undetectable |
|
872 |
\item the victim has (almost) no chance to recover |
|
873 |
\end{enumerate}
|
|
874 |
||
875 |
\end{frame}}
|
|
876 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
877 |
||
878 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
879 |
\mode<presentation>{
|
|
880 |
\begin{frame}[c]
|
|
881 |
||
882 |
||
883 |
\begin{center}
|
|
884 |
\begin{tikzpicture}[scale=1]
|
|
885 |
||
886 |
\onslide<1->{
|
|
887 |
\node (A) at (0,0) [draw=black, rectangle, very thick, minimum height=18mm, minimum width=17mm] {};
|
|
888 |
\node [below right] at (A.north west) {\footnotesize\begin{tabular}{@{}l@{}}
|
|
889 |
\only<1,2>{clean}\only<3->{\alert{hacked}}\\compiler\end{tabular}};}
|
|
890 |
||
891 |
||
892 |
\onslide<2->{
|
|
893 |
\node (B) at (-2,2) [draw=black, rectangle, very thick, minimum height=10mm, minimum width=12mm] {};
|
|
894 |
\node [below right] at (B.north west) {\footnotesize\begin{tabular}{@{}l@{}}login\\(src)\end{tabular}};
|
|
895 |
||
896 |
\node (C) at (2,2) [draw=black, rectangle, very thick, minimum height=10mm, minimum width=12mm] {};
|
|
897 |
\node [below right] at (C.north west) {\footnotesize\begin{tabular}{@{}l@{}}login\\(bin)\end{tabular}};
|
|
898 |
||
899 |
\draw[->, line width=2mm] (B) -- (C); |
|
900 |
} |
|
901 |
||
902 |
\onslide<3->{\node [above left=-1.5mm] at (C.south east) {\footnotesize \alert{$\blacksquare$}};}
|
|
903 |
||
904 |
\end{tikzpicture}
|
|
905 |
\end{center}
|
|
906 |
||
907 |
\end{frame}}
|
|
908 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
909 |
||
910 |
||
911 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
912 |
\mode<presentation>{
|
|
913 |
\begin{frame}[c]
|
|
914 |
||
915 |
\begin{center}
|
|
916 |
\begin{tikzpicture}[scale=1]
|
|
917 |
||
918 |
\onslide<1->{
|
|
919 |
\node (A) at (0,0) [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};
|
|
920 |
\node [below right] at (A.north west) {\small V0.01};
|
|
921 |
\node [below right] (A1) at (A.south west) {\small Scala};
|
|
922 |
\node [below right] (A1) at (A1.south west) {\small\textcolor{gray}{host language}};
|
|
923 |
\node [above right] at (A.north west) {my compiler (src)};}
|
|
924 |
||
925 |
\onslide<2->{
|
|
926 |
\node (B) at (1.8,0) [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};
|
|
927 |
\node [below right] at (B.north west) {\small V0.02};
|
|
928 |
\node [below right] at (B.south west) {\small Scala};
|
|
929 |
\node at (3,0) {\ldots};
|
|
930 |
||
931 |
\node (C) at (5,0) [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};
|
|
932 |
\node [below right] at (C.north west) {\small V1.00};
|
|
933 |
\node [below right] at (C.south west) {\small Scala};}
|
|
934 |
||
935 |
\onslide<3->{
|
|
936 |
\node (D) at (6.8,0) [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};
|
|
937 |
\node [below right] at (D.north west) {\small V1.00};
|
|
938 |
||
939 |
\node (E) at (6.8,2) [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};
|
|
940 |
\node [below right] at (E.north west) {\small V1.01};}
|
|
941 |
||
942 |
\onslide<4->{
|
|
943 |
\node (F) at (8.6,0) [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};
|
|
944 |
\node [below right] at (F.north west) {\small V1.01};
|
|
945 |
||
946 |
\node (G) at (8.6,2) [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};
|
|
947 |
\node [below right] at (G.north west) {\small V1.02};
|
|
948 |
\node at (9.8,0) {\ldots};
|
|
949 |
\node at (9.8,2) {\ldots};
|
|
950 |
\node at (8,-2) {\textcolor{gray}{\begin{tabular}{@{}l@{}}no host language\\needed\end{tabular}}};
|
|
951 |
} |
|
952 |
||
953 |
\end{tikzpicture}
|
|
954 |
\end{center}
|
|
955 |
||
956 |
\end{frame}}
|
|
957 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
958 |
||
959 |
||
960 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
961 |
\mode<presentation>{
|
|
962 |
\begin{frame}<1-3>
|
|
963 |
\frametitle{\LARGE\begin{tabular}{c}Hacking Compilers
|
|
964 |
\end{tabular}}
|
|
965 |
||
966 |
%Why is it so paramount to have a small trusted code base (TCB)? |
|
967 |
\bigskip\bigskip |
|
968 |
||
969 |
\begin{columns}
|
|
970 |
\begin{column}{2.7cm}
|
|
971 |
\begin{minipage}{2.5cm}%
|
|
972 |
\begin{tabular}{c@ {}}
|
|
973 |
\includegraphics[scale=0.2]{../pics/ken-thompson.jpg}\\[-1.8mm]
|
|
974 |
\footnotesize Ken Thompson\\[-1.8mm] |
|
975 |
\footnotesize Turing Award, 1983\\ |
|
976 |
\end{tabular}
|
|
977 |
\end{minipage}
|
|
978 |
\end{column}
|
|
979 |
\begin{column}{9cm}
|
|
980 |
\begin{tabular}{l@ {\hspace{1mm}}p{8cm}}
|
|
981 |
||
982 |
& Ken Thompson showed how to hide a Trojan Horse in a |
|
983 |
compiler \textcolor{red}{without} leaving any traces in the source code.\\[2mm]
|
|
984 |
||
985 |
& No amount of source level verification will protect |
|
986 |
you from such Thompson-hacks.\\[2mm] |
|
987 |
||
988 |
\end{tabular}
|
|
989 |
\end{column}
|
|
990 |
\end{columns}
|
|
991 |
||
992 |
\only<2>{
|
|
993 |
\begin{textblock}{6}(4,2)
|
|
994 |
\begin{tikzpicture}
|
|
995 |
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
|
996 |
{\normalsize
|
|
997 |
\begin{minipage}{8cm}
|
|
998 |
\begin{quote}
|
|
999 |
\includegraphics[scale=0.05]{../pics/evil.png}
|
|
1000 |
\begin{enumerate}
|
|
1001 |
\item[1)] Assume you ship the compiler as binary and also with sources. |
|
1002 |
\item[2)] Make the compiler aware when it compiles itself. |
|
1003 |
\item[3)] Add the Trojan horse. |
|
1004 |
\item[4)] Compile. |
|
1005 |
\item[5)] Delete Trojan horse from the sources of the compiler. |
|
1006 |
\item[6)] Go on holiday for the rest of your life. ;o)\\[-7mm]\mbox{}
|
|
1007 |
\end{enumerate}
|
|
1008 |
\end{quote}
|
|
1009 |
\end{minipage}};
|
|
1010 |
\end{tikzpicture}
|
|
1011 |
\end{textblock}}
|
|
1012 |
||
1013 |
\end{frame}}
|
|
1014 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1015 |
||
| 701 | 1016 |
|
1017 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1018 |
\begin{frame}[c]
|
|
1019 |
\frametitle{Dijkstra on Testing}
|
|
1020 |
||
1021 |
\begin{bubble}[10cm]
|
|
1022 |
``Program testing can be a very effective way to show the |
|
1023 |
presence of bugs, but it is hopelessly inadequate for showing |
|
1024 |
their absence.'' |
|
1025 |
\end{bubble}\bigskip
|
|
1026 |
||
1027 |
\small |
|
1028 |
What is good about compilers: the either seem to work, |
|
1029 |
or go horribly wrong (most of the time). |
|
1030 |
||
1031 |
\end{frame}
|
|
1032 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1033 |
||
1034 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1035 |
\begin{frame}[c]
|
|
1036 |
\frametitle{\Large Proving Programs to be Correct}
|
|
1037 |
||
1038 |
\begin{bubble}[10cm]
|
|
1039 |
\small |
|
1040 |
{\bf Theorem:} There are infinitely many prime
|
|
1041 |
numbers.\medskip\\ |
|
1042 |
||
1043 |
{\bf Proof} \ldots\\
|
|
1044 |
\end{bubble}\bigskip
|
|
1045 |
||
1046 |
||
1047 |
similarly\bigskip |
|
1048 |
||
1049 |
\begin{bubble}[10cm]
|
|
1050 |
\small |
|
1051 |
{\bf Theorem:} The program is doing what
|
|
1052 |
it is supposed to be doing.\medskip |
|
1053 |
||
1054 |
{\bf Long, long proof} \ldots\\
|
|
1055 |
\end{bubble}\bigskip\medskip
|
|
1056 |
||
1057 |
\small This can be a gigantic proof. The only hope is to have |
|
1058 |
help from the computer. `Program' is here to be understood to be |
|
1059 |
quite general (compiler, OS, \ldots). |
|
1060 |
||
1061 |
\end{frame}
|
|
1062 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1063 |
||
1064 |
||
1065 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1066 |
||
1067 |
\begin{frame}[c]
|
|
1068 |
\frametitle{Can This Be Done?}
|
|
1069 |
||
1070 |
\begin{itemize}
|
|
1071 |
\item in 2008, verification of a small C-compiler |
|
1072 |
\begin{itemize}
|
|
1073 |
\item ``if my input program has a certain behaviour, then the compiled machine code has the same behaviour'' |
|
1074 |
\item is as good as \texttt{gcc -O1}, but much, much less buggy
|
|
1075 |
\end{itemize}
|
|
1076 |
\end{itemize}
|
|
1077 |
||
1078 |
\begin{center}
|
|
1079 |
\includegraphics[scale=0.12]{../pics/compcert.png}
|
|
1080 |
\end{center}
|
|
1081 |
||
1082 |
\end{frame}
|
|
1083 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1084 |
||
1085 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1086 |
\begin{frame}[t]
|
|
1087 |
\frametitle{Fuzzy Testing C-Compilers}
|
|
1088 |
||
1089 |
\begin{itemize}
|
|
1090 |
\item tested GCC, LLVM and others by randomly generating |
|
1091 |
C-programs |
|
1092 |
\item found more than 300 bugs in GCC and also |
|
1093 |
many in LLVM (some of them highest-level critical)\bigskip |
|
1094 |
\item about CompCert: |
|
1095 |
||
1096 |
\begin{bubble}[10.7cm]\small ``The striking thing about our CompCert
|
|
1097 |
results is that the middle-end bugs we found in all other |
|
1098 |
compilers are absent. As of early 2011, the under-development |
|
1099 |
version of CompCert is the only compiler we have tested for |
|
1100 |
which Csmith cannot find wrong-code errors. This is not for |
|
1101 |
lack of trying: we have devoted about six CPU-years to the |
|
1102 |
task.'' |
|
1103 |
\end{bubble}
|
|
1104 |
\end{itemize}
|
|
1105 |
||
1106 |
\end{frame}
|
|
1107 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1108 |
||
1109 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1110 |
||
1111 |
\begin{frame}[c]
|
|
1112 |
\frametitle{Next Week}
|
|
1113 |
||
1114 |
\begin{itemize}
|
|
1115 |
\item Revision Lecture\medskip |
|
1116 |
\item How many strings are in $\bl{L(a^*)}$?\pause\medskip
|
|
1117 |
\item How many strings are in $\bl{L((a + b)^*)}$?\\ Are there more than
|
|
1118 |
in $\bl{L(a^*)}$?
|
|
1119 |
\end{itemize}
|
|
1120 |
||
1121 |
||
1122 |
||
1123 |
\end{frame}
|
|
1124 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1125 |
||
| 822 | 1126 |
\begin{frame}<1-20>[c]
|
1127 |
\end{frame}
|
|
| 701 | 1128 |
|
1129 |
\end{document}
|
|
1130 |
||
1131 |
%%% Local Variables: |
|
1132 |
%%% mode: latex |
|
1133 |
%%% TeX-master: t |
|
1134 |
%%% End: |
|
1135 |
||
1136 |
||
1137 |
||
1138 |
||
1139 |
||
| 610 | 1140 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1141 |
\begin{frame}[c]
|
|
1142 |
||
1143 |
\begin{center}
|
|
1144 |
\includegraphics[scale=0.6]{../pics/bridge-limits.png}
|
|
1145 |
\end{center}
|
|
1146 |
||
1147 |
\end{frame}
|
|
1148 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1149 |
||
1150 |
||
1151 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1152 |
\begin{frame}[c]
|
|
1153 |
\frametitle{Compilers \& Boeings 777}
|
|
1154 |
||
1155 |
First flight in 1994. They want to achieve triple redundancy in hardware |
|
1156 |
faults.\bigskip |
|
1157 |
||
1158 |
They compile 1 Ada program to\medskip |
|
1159 |
||
1160 |
\begin{itemize}
|
|
1161 |
\item Intel 80486 |
|
1162 |
\item Motorola 68040 (old Macintosh's) |
|
1163 |
\item AMD 29050 (RISC chips used often in laser printers) |
|
1164 |
\end{itemize}\medskip
|
|
1165 |
||
1166 |
using 3 independent compilers.\bigskip\pause |
|
1167 |
||
1168 |
\small Airbus uses C and static analysers. Recently started using CompCert. |
|
1169 |
||
1170 |
\end{frame}
|
|
1171 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1172 |
||
1173 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1174 |
\begin{frame}[c]
|
|
1175 |
\frametitle{Goal}
|
|
1176 |
||
1177 |
Remember the Bridges example? |
|
1178 |
||
1179 |
\begin{itemize}
|
|
1180 |
\item Can we look at our programs and somehow ensure |
|
1181 |
they are bug free/correct?\pause\bigskip |
|
1182 |
||
1183 |
\item Very hard: Anything interesting about programs is equivalent |
|
1184 |
to the Halting Problem, which is undecidable.\pause\bigskip |
|
1185 |
||
1186 |
\item \alert{Solution:} We avoid this ``minor'' obstacle by
|
|
1187 |
being as close as possible of deciding the halting |
|
1188 |
problem, without actually deciding the halting problem. |
|
1189 |
\small$\quad\Rightarrow$ yes, no, don't know (static analysis) |
|
1190 |
\end{itemize}
|
|
1191 |
||
1192 |
\end{frame}
|
|
1193 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1194 |
||
1195 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1196 |
\begin{frame}[c]
|
|
1197 |
\frametitle{What is Static Analysis?}
|
|
1198 |
||
1199 |
\begin{center}
|
|
1200 |
\includegraphics[scale=0.4]{../pics/state.png}
|
|
1201 |
\end{center}
|
|
1202 |
||
1203 |
\begin{itemize}
|
|
1204 |
\item depending on some initial input, a program |
|
1205 |
(behaviour) will ``develop'' over time. |
|
1206 |
\end{itemize}
|
|
1207 |
||
1208 |
\end{frame}
|
|
1209 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1210 |
||
1211 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1212 |
\begin{frame}[c]
|
|
1213 |
\frametitle{What is Static Analysis?}
|
|
1214 |
||
1215 |
\begin{center}
|
|
1216 |
\includegraphics[scale=0.4]{../pics/state2.png}
|
|
1217 |
\end{center}
|
|
1218 |
||
1219 |
\end{frame}
|
|
1220 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1221 |
||
1222 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1223 |
\begin{frame}[c]
|
|
1224 |
\frametitle{What is Static Analysis?}
|
|
1225 |
||
1226 |
\begin{center}
|
|
1227 |
\includegraphics[scale=0.4]{../pics/state3.jpg}
|
|
1228 |
\end{center}
|
|
1229 |
||
1230 |
\end{frame}
|
|
1231 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1232 |
||
1233 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1234 |
\begin{frame}[c]
|
|
1235 |
\frametitle{What is Static Analysis?}
|
|
1236 |
||
1237 |
\begin{center}
|
|
1238 |
\includegraphics[scale=0.4]{../pics/state4.jpg}
|
|
1239 |
\end{center}
|
|
1240 |
||
1241 |
\begin{itemize}
|
|
1242 |
\item to be avoided |
|
1243 |
\end{itemize}
|
|
1244 |
||
1245 |
\end{frame}
|
|
1246 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1247 |
||
1248 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1249 |
\begin{frame}[c]
|
|
1250 |
\frametitle{What is Static Analysis?}
|
|
1251 |
||
1252 |
\begin{center}
|
|
1253 |
\includegraphics[scale=0.4]{../pics/state5.png}
|
|
1254 |
\end{center}
|
|
1255 |
||
1256 |
\begin{itemize}
|
|
1257 |
\item this needs more work |
|
1258 |
\end{itemize}
|
|
1259 |
||
1260 |
\end{frame}
|
|
1261 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1262 |
||
1263 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1264 |
\begin{frame}[c]
|
|
1265 |
\frametitle{What is Static Analysis?}
|
|
1266 |
||
1267 |
\begin{center}
|
|
1268 |
\includegraphics[scale=0.4]{../pics/state6.png}
|
|
1269 |
\end{center}
|
|
1270 |
||
1271 |
\end{frame}
|
|
1272 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1273 |
||
1274 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1275 |
\begin{frame}[c,fragile]
|
|
1276 |
\frametitle{\Large\begin{tabular}{c}Concrete Example:\\[-1mm]
|
|
1277 |
Are Vars Definitely Initialised?\end{tabular}}
|
|
1278 |
||
1279 |
Assuming \texttt{x} is initialised, what about \texttt{y}?\bigskip
|
|
1280 |
||
1281 |
Prog.~1:\\ |
|
1282 |
\begin{lstlisting}[numbers=none,
|
|
1283 |
basicstyle=\ttfamily, |
|
1284 |
language=While,xleftmargin=3mm] |
|
1285 |
if x < 1 then y := x else y := x + 1; |
|
1286 |
y := y + 1 |
|
1287 |
\end{lstlisting}\medskip
|
|
1288 |
||
1289 |
Prog.~2:\\ |
|
1290 |
\begin{lstlisting}[numbers=none,
|
|
1291 |
basicstyle=\ttfamily, |
|
1292 |
language=While,xleftmargin=3mm] |
|
1293 |
if x < x then y := y + 1 else y := x; |
|
1294 |
y := y + 1 |
|
1295 |
\end{lstlisting}
|
|
1296 |
||
1297 |
\end{frame}
|
|
1298 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1299 |
||
1300 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1301 |
\begin{frame}[c,fragile]
|
|
1302 |
\frametitle{\Large\begin{tabular}{c}Concrete Example:\\[-1mm]
|
|
1303 |
Are Vars Definitely Initialised?\end{tabular}}
|
|
1304 |
||
1305 |
What should the rules be for deciding when a |
|
1306 |
variable is initialised?\bigskip\pause |
|
1307 |
||
1308 |
\begin{itemize}
|
|
1309 |
\item variable \texttt{x} is definitely initialized after
|
|
1310 |
\texttt{skip}\\
|
|
1311 |
iff \texttt{x} is definitely initialized before \texttt{skip}.
|
|
1312 |
\end{itemize}
|
|
1313 |
||
1314 |
\end{frame}
|
|
1315 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1316 |
||
1317 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1318 |
\begin{frame}[c,fragile]
|
|
1319 |
% \frametitle{\Large\begin{tabular}{c}Concrete Example:\\[-1mm]
|
|
1320 |
% Are Vars Definitely Initialised?\end{tabular}}
|
|
1321 |
||
1322 |
$\bl{A}$ is the set of definitely defined variables:
|
|
1323 |
||
1324 |
\begin{center}
|
|
1325 |
\begin{tabular}{c}
|
|
| 612 | 1326 |
\bl{\infer{\mbox{}}{A\triangleright\texttt{skip}\triangleright{}A}}\qquad
|
1327 |
\bl{\infer{vars(a) \subseteq A}{A\triangleright
|
|
1328 |
(\texttt{x\,:=\,a})\triangleright\{x\}\cup A}}
|
|
| 610 | 1329 |
\medskip\\\pause |
1330 |
||
| 612 | 1331 |
\bl{\infer{A_1\triangleright{}s_1\triangleright{}A_2
|
1332 |
\quad A_2\triangleright{}s_2\triangleright{}A_3}
|
|
1333 |
{A_1\triangleright{}(s_1 ; s_2)\triangleright{}A_3}}
|
|
| 610 | 1334 |
\medskip\\\pause |
1335 |
||
| 612 | 1336 |
\bl{\infer{vars(b)\subseteq A\quad A\triangleright{}s_1\triangleright{}A_1
|
1337 |
\quad A\triangleright{}s_2\triangleright{}A_2}
|
|
1338 |
{A\triangleright(\texttt{if}\;b\;\texttt{then}\;s_1\;\texttt{else}\;s_2)\triangleright{}A_1\cap A_2}}
|
|
| 610 | 1339 |
\medskip\\\pause |
1340 |
||
| 612 | 1341 |
\bl{\infer{vars(b)\subseteq A\quad A\triangleright{}s\triangleright{}A'}
|
1342 |
{A\triangleright(\texttt{while}\;b\;\texttt{do}\;s)\triangleright{}A}}\pause
|
|
| 610 | 1343 |
\end{tabular}
|
1344 |
\end{center}
|
|
1345 |
||
1346 |
\hfill we start with $\bl{A = \{\}}$
|
|
1347 |
\end{frame}
|
|
1348 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1349 |