| author | Christian Urban <christian.urban@kcl.ac.uk> |
| Sun, 02 Oct 2022 08:42:01 +0100 | |
| changeset 881 | 3b2f76950473 |
| parent 871 | 94b84d880c2b |
| child 904 | d97283992d4f |
| permissions | -rw-r--r-- |
| 704 | 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{soul}
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\tikzset{onslide/.code args={<#1>#2}{%
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\only<#1>{\pgfkeysalso{#2}} % \pgfkeysalso doesn't change the path
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}} |
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\makeatletter |
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\newenvironment<>{btHighlight}[1][]
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{\begin{onlyenv}#2\begingroup\tikzset{bt@Highlight@par/.style={#1}}\begin{lrbox}{\@tempboxa}}
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{\end{lrbox}\bt@HL@box[bt@Highlight@par]{\@tempboxa}\endgroup\end{onlyenv}}
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\newcommand<>\btHL[1][]{%
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\only#2{\begin{btHighlight}[#1]\bgroup\aftergroup\bt@HL@endenv}%
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} |
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\def\bt@HL@endenv{%
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\end{btHighlight}%
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\egroup |
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} |
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\newcommand{\bt@HL@box}[2][]{%
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\tikz[#1]{%
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\pgfpathrectangle{\pgfpoint{1pt}{0pt}}{\pgfpoint{\wd #2}{\ht #2}}%
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\pgfusepath{use as bounding box}%
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\node[anchor=base west, fill=orange!30,outer sep=0pt,inner xsep=1pt, inner ysep=0pt, rounded corners=3pt, minimum height=\ht\strutbox+1pt,#1]{\raisebox{1pt}{\strut}\strut\usebox{#2}};
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}% |
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} |
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\makeatother |
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% beamer stuff |
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\renewcommand{\slidecaption}{CFL 10, 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 Hours: & Thursdays 12 -- 14\\ |
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%Location: & N7.07 (North Wing, Bush House)\\ |
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Slides \& Progs: & KEATS (also homework is there)\\ |
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\end{tabular}
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\end{center}
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\begin{center}
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\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 & 9 Optimisations \\ |
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5 Grammars, Parsing & \cellcolor{blue!50}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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\tikzstyle{sensor}=[draw, fill=blue!20, text width=3.8em, line width=1mm,
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text centered, minimum height=2em,drop shadow] |
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\tikzstyle{ann} = [above, text width=4em, text centered]
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\tikzstyle{sc} = [sensor, text width=7em, fill=red!20,
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minimum height=6em, rounded corners, drop shadow,line width=1mm] |
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\begin{frame}[fragile,c]
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\frametitle{LLVM: Overview}
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\begin{tikzpicture}
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% Validation Layer is the same except that there are a set of nodes and links which are added |
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\path (0,0) node (IR) [sc] {\textbf{LLVM-IR}\\ Optimisations};
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\path (IR.west)+(-2.2,1.7) node (sou1) [sensor] {C++};
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\path (IR.west)+(-2.2,0.5) node (sou2)[sensor] {C};
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\path (IR.west)+(-2.2,-1.0) node (dots)[ann] {$\vdots$};
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\path (IR.west)+(-2.2,-1.8) node (sou3)[sensor] {Haskell};
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\path [draw,->,line width=1mm] (sou1.east) -- node [above] {} (IR.160);
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\path [draw,->,line width=1mm] (sou2.east) -- node [above] {} (IR.180);
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\path [draw,->,line width=1mm] (sou3.east) -- node [above] {} (IR.200);
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\path (IR.east)+(2.2,2.0) node (tar1)[sensor] {x86};
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\path (IR.east)+(2.2,0.8) node (tar2)[sensor] {ARM};
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\path (IR.east)+(2.2,-0.4) node (tar3)[sensor] {MIPS};
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\path (IR.east)+(2.2,-1.6) node (tar4)[sensor] {RISC};
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\path (IR.east)+(2.2,-2.8) node (tar5)[sensor] {Power PC};
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\path (IR.east)+(2.2,-4.2) node (dots2)[ann] {$\vdots$};
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\path [draw,<-,line width=1mm] (tar1.west) -- node [above] {} (IR.10);
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\path [draw,<-,line width=1mm] (tar2.west) -- node [above] {} (IR.5);
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\path [draw,<-,line width=1mm] (tar3.west) -- node [above] {} (IR.0);
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\path [draw,<-,line width=1mm] (tar4.west) -- node [above] {} (IR.-5);
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\path [draw,<-,line width=1mm] (tar5.west) -- node [above] {} (IR.-10);
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\end{tikzpicture}
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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{Static Single-Assignment}
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\bl{$(1 + a) + (3 + (b * 5))$}\bigskip\bigskip
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\begin{lstlisting}[language=LLVMIR,numbers=left]
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let tmp0 = add 1 a in |
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let tmp1 = mul b 5 in |
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let tmp2 = add 3 tmp1 in |
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let tmp3 = add tmp0 tmp2 |
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in tmp3 |
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\end{lstlisting}
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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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\small |
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\mbox{}\bigskip\bigskip\bigskip
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\begin{lstlisting}[language=llvm,numbers=left]
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define i32 @fact (i32 %n) {
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%tmp_20 = icmp eq i32 %n, 0 |
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br i1 %tmp_20, label %if_branch_24, label %else_branch_25 |
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if_branch_24: |
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ret i32 1 |
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else_branch_25: |
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%tmp_22 = sub i32 %n, 1 |
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%tmp_23 = call i32 @fact (i32 %tmp_22) |
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%tmp_21 = mul i32 %n, %tmp_23 |
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ret i32 %tmp_21 |
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} |
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\end{lstlisting}
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\begin{lstlisting}[language={},numbers=none]
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def fact(n) = if n == 0 then 1 else n * fact(n - 1) |
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\end{lstlisting}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[fragile,c]
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\frametitle{LLVM Types}
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\tt |
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\begin{center}
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\begin{tabular}{ll}
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boolean & i1 \\ |
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byte & i8 \\ |
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short & i16\\ |
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char & i16\\ |
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integer & i32\\ |
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long & i64\\ |
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float & float\\ |
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double & double\\ |
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*\_ & pointer to \\ |
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**\_ & pointer to a pointer to\\ |
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\mbox{}[\_] & arrays of\\
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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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\begin{lstlisting}[language=LLVM,numbers=none]
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br i1 %var, label %if_br, label %else_br |
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icmp eq i32 %x, %y ; for equal |
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icmp sle i32 %x, %y ; signed less or equal |
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icmp slt i32 %x, %y ; signed less than |
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icmp ult i32 %x, %y ; unsigned less than |
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%var = call i32 @foo(...args...) |
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\end{lstlisting}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[fragile,c]
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\frametitle{Abstract Syntax Trees}
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\footnotesize |
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\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=-3mm]
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// Fun language (expressions) |
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abstract class Exp |
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abstract class BExp |
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case class Call(name: String, args: List[Exp]) extends Exp |
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case class If(a: BExp, e1: Exp, e2: Exp) extends Exp |
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case class Write(e: Exp) extends Exp |
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case class Var(s: String) extends Exp |
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case class Num(i: Int) extends Exp |
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case class Aop(o: String, a1: Exp, a2: Exp) extends Exp |
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case class Sequence(e1: Exp, e2: Exp) extends Exp |
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case class Bop(o: String, a1: Exp, a2: Exp) extends BExp |
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\end{lstlisting}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[fragile,c]
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\frametitle{K-(Intermediate)Language}
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\footnotesize |
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\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=-3mm]
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abstract class KExp |
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abstract class KVal |
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// K-Values |
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case class KVar(s: String) extends KVal |
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case class KNum(i: Int) extends KVal |
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case class Kop(o: String, v1: KVal, v2: KVal) extends KVal |
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case class KCall(o: String, vrs: List[KVal]) extends KVal |
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case class KWrite(v: KVal) extends KVal |
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// K-Expressions |
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case class KIf(x1: String, e1: KExp, e2: KExp) extends KExp |
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case class KLet(x: String, v: KVal, e: KExp) extends KExp |
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case class KReturn(v: KVal) extends KExp |
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\end{lstlisting}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[fragile,c]
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\frametitle{KLet}
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\begin{lstlisting}[language=LLVM]
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tmp0 = add 1 a |
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tmp1 = mul b 5 |
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tmp2 = add 3 tmp1 |
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tmp3 = add tmp0 tmp2 |
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\end{lstlisting}
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\begin{lstlisting}[language=LLVMIR]
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KLet tmp0 , add 1 a in |
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KLet tmp1 , mul b 5 in |
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KLet tmp2 , add 3 tmp1 in |
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KLet tmp3 , add tmp0 tmp2 in |
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... |
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\end{lstlisting}
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\begin{lstlisting}[language=Scala,numbers=none]
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case class KLet(x: String, e1: KVal, e2: KExp) |
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\end{lstlisting}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[fragile,c]
|
|
282 |
\frametitle{KLet}
|
|
283 |
||
284 |
\begin{lstlisting}[language=LLVM]
|
|
285 |
tmp0 = add 1 a |
|
286 |
tmp1 = mul b 5 |
|
287 |
tmp2 = add 3 tmp1 |
|
288 |
tmp3 = add tmp0 tmp2 |
|
289 |
\end{lstlisting}
|
|
290 |
||
291 |
\begin{lstlisting}[language=LLVMIR]
|
|
292 |
let tmp0 = add 1 a in |
|
293 |
let tmp1 = mul b 5 in |
|
294 |
let tmp2 = add 3 tmp1 in |
|
295 |
let tmp3 = add tmp0 tmp2 in |
|
296 |
... |
|
297 |
\end{lstlisting}
|
|
298 |
||
299 |
\begin{lstlisting}[language=Scala,numbers=none]
|
|
300 |
case class KLet(x: String, e1: KVal, e2: KExp) |
|
301 |
\end{lstlisting}
|
|
302 |
||
303 |
\end{frame}
|
|
304 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
305 |
||
306 |
||
307 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
308 |
\begin{frame}[fragile,c]
|
|
309 |
\frametitle{CPS-Translation}
|
|
310 |
\small |
|
311 |
||
312 |
\begin{lstlisting}[language=Scala,numbers=none]
|
|
313 |
def CPS(e: Exp)(k: KVal => KExp) : KExp = |
|
314 |
e match { ... }
|
|
315 |
\end{lstlisting}
|
|
316 |
\bigskip\bigskip |
|
317 |
||
318 |
the continuation \texttt{k} can be thought of:\medskip
|
|
319 |
||
320 |
\small |
|
321 |
\begin{lstlisting}[language=LLVMIR,numbers=none,xleftmargin=30mm,escapeinside={(*@}{@*)}]
|
|
322 |
let tmp0 = add 1 a in |
|
323 |
let tmp1 = mul (*@$\Box$@*) 5 in |
|
324 |
let tmp2 = add 3 tmp1 in |
|
325 |
let tmp3 = add tmp0 tmp2 in |
|
326 |
KReturn tmp3 |
|
327 |
\end{lstlisting}
|
|
328 |
||
329 |
\end{frame}
|
|
330 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
331 |
||
332 |
||
333 |
||
| 818 | 334 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
335 |
\begin{frame}[c,fragile]
|
|
| 704 | 336 |
|
337 |
||
338 |
\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-6mm]
|
|
339 |
def fact(n: Int) : Int = {
|
|
340 |
if (n == 0) 1 else n * fact(n - 1) |
|
341 |
} |
|
342 |
||
343 |
||
344 |
def factC(n: Int, ret: Int => Int) : Int = {
|
|
345 |
if (n == 0) ret(1) |
|
346 |
else factC(n - 1, x => ret(n * x)) |
|
347 |
} |
|
348 |
||
349 |
fact(10) |
|
350 |
factC(10, identity) |
|
351 |
\end{lstlisting}
|
|
352 |
\end{frame}
|
|
353 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
354 |
||
355 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
356 |
\begin{frame}[c,fragile]
|
|
357 |
||
358 |
||
359 |
\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-6mm]
|
|
360 |
def fibC(n: Int, ret: Int => Int) : Int = {
|
|
361 |
if (n == 0 || n == 1) ret(1) else |
|
362 |
fibC(n - 1, |
|
363 |
r1 => fibC(n - 2, |
|
364 |
r2 => ret(r1 + r2))) |
|
365 |
} |
|
366 |
||
367 |
fibC(10, identity) |
|
368 |
\end{lstlisting}
|
|
369 |
\end{frame}
|
|
370 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
371 |
||
372 |
||
373 |
||
374 |
||
| 543 | 375 |
|
376 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
377 |
\begin{frame}[c]
|
|
378 |
||
| 704 | 379 |
\Large\bf Are there more strings in \\ \hfill\bl{$L(a^*)$} or
|
| 617 | 380 |
\bl{$L((a + b)^*)$}?
|
| 543 | 381 |
|
382 |
\end{frame}
|
|
383 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
384 |
||
385 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
386 |
\begin{frame}[c]
|
|
| 704 | 387 |
\frametitle{Can you remember this HW?}
|
388 |
||
389 |
\begin{itemize}
|
|
390 |
\item (1) How many basic regular expressions are there to match |
|
391 |
the string \bl{$abcd$}?
|
|
392 |
\item (2) How many if they cannot include |
|
393 |
\bl{$\ONE$} and \bl{$\ZERO$}?
|
|
394 |
\item (3) How many if they are also not |
|
395 |
allowed to contain stars? |
|
396 |
\item (4) How many if they are also |
|
397 |
not allowed to contain \bl{$\_ + \_$}?
|
|
398 |
\end{itemize}
|
|
399 |
||
400 |
\end{frame}
|
|
401 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
402 |
||
403 |
||
404 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
405 |
\begin{frame}[c]
|
|
| 543 | 406 |
|
407 |
\Large\bf There are more problems, than there are |
|
408 |
programs.\bigskip\bigskip\pause\\ |
|
409 |
||
410 |
There must be a problem for which there is no program. |
|
411 |
\end{frame}
|
|
412 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
413 |
||
414 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
415 |
\begin{frame}[c]
|
|
416 |
\frametitle{Subsets}
|
|
417 |
||
418 |
\Large |
|
419 |
If \bl{$A \subseteq B$} then \bl{$A$} has fewer or equal elements
|
|
420 |
than \bl{$B$}\bigskip\bigskip
|
|
421 |
||
422 |
\Large |
|
423 |
\bl{$A \subseteq B$} and \bl{$B \subseteq A$}\bigskip
|
|
424 |
||
425 |
then \bl{$A = B$}
|
|
426 |
\end{frame}
|
|
427 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
428 |
||
429 |
||
430 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
431 |
\begin{frame}[c]
|
|
432 |
||
433 |
\begin{center}
|
|
434 |
\begin{tikzpicture}
|
|
435 |
||
436 |
\draw (-4,2.5) node [scale=2.5] (X) |
|
437 |
{\begin{tabular}{l}
|
|
438 |
$\{$
|
|
439 |
\!\includegraphics[scale=0.02]{../pics/o1.jpg},
|
|
440 |
\includegraphics[scale=0.02]{../pics/o2.jpg},
|
|
441 |
\!\includegraphics[scale=0.02]{../pics/o3.jpg},
|
|
442 |
\includegraphics[scale=0.02]{../pics/o4.jpg},
|
|
443 |
\!\includegraphics[scale=0.027]{../pics/o5.jpg}
|
|
444 |
$\}$ |
|
445 |
\end{tabular}};
|
|
446 |
||
447 |
\draw (-5.6,-2.5) node [scale=2.5] (Y) |
|
448 |
{\begin{tabular}{l}
|
|
449 |
$\{$
|
|
450 |
\!\includegraphics[scale=0.059]{../pics/a1.jpg},
|
|
451 |
\includegraphics[scale=0.048]{../pics/a2.jpg},
|
|
452 |
\includegraphics[scale=0.02]{../pics/a3.jpg}
|
|
453 |
$\}$ |
|
454 |
\end{tabular}};
|
|
455 |
||
456 |
\draw (0,1.5) node (X1) {5 elements};
|
|
457 |
\draw (0,-3.5) node (y1) {3 elements};
|
|
458 |
\end{tikzpicture}
|
|
459 |
\end{center}
|
|
460 |
||
461 |
\end{frame}
|
|
462 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
463 |
||
464 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
465 |
\begin{frame}[c]
|
|
466 |
\frametitle{Newton vs Feynman}
|
|
467 |
||
468 |
\begin{center}
|
|
469 |
\begin{tabular}{cc}
|
|
470 |
\includegraphics[scale=0.2]{../pics/newton.jpg} &
|
|
471 |
\includegraphics[scale=0.2]{../pics/feynman.jpg}\\
|
|
472 |
classical physics & quantum physics |
|
473 |
\end{tabular}
|
|
474 |
\end{center}
|
|
475 |
\end{frame}
|
|
476 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
477 |
||
478 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
479 |
\begin{frame}[c]
|
|
480 |
\frametitle{The Goal of the Talk}
|
|
481 |
\large |
|
482 |
\begin{itemize}
|
|
483 |
\item show you that something very unintuitive happens with very large sets |
|
484 |
\bigskip\bigskip |
|
485 |
||
486 |
\item convince you that there are more {\bf problems} than {\bf programs}
|
|
487 |
\end{itemize}
|
|
488 |
\end{frame}
|
|
489 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
490 |
||
491 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
492 |
\begin{frame}[t]
|
|
493 |
% |
|
494 |
\begin{center}
|
|
495 |
\begin{tikzpicture}
|
|
496 |
||
497 |
\draw (-5,2.5) node [scale=2.3] (X) |
|
498 |
{\begin{tabular}{@ {\hspace{-3mm}}l}
|
|
499 |
\bl{$B$ $=$ $\{$
|
|
500 |
\!\includegraphics[scale=0.02]{../pics/o1.jpg},
|
|
501 |
\includegraphics[scale=0.02]{../pics/o2.jpg},
|
|
502 |
\!\includegraphics[scale=0.02]{../pics/o3.jpg},
|
|
503 |
\includegraphics[scale=0.02]{../pics/o4.jpg},
|
|
504 |
\!\includegraphics[scale=0.027]{../pics/o5.jpg}
|
|
505 |
$\}$} |
|
506 |
\end{tabular}};
|
|
507 |
||
508 |
\draw (-6.6,-0.5) node [scale=2.3] (Y) |
|
509 |
{\begin{tabular}{@ {\hspace{-3mm}}l}
|
|
510 |
\bl{$A$ $=$ $\{$
|
|
511 |
\!\includegraphics[scale=0.059]{../pics/a1.jpg},
|
|
512 |
\includegraphics[scale=0.048]{../pics/a2.jpg},
|
|
513 |
\includegraphics[scale=0.02]{../pics/a3.jpg}
|
|
514 |
$\}$} |
|
515 |
\end{tabular}};
|
|
516 |
||
517 |
\only<1>{\draw (-5, -3) node[scale=2]
|
|
518 |
{\bl{$|A|$ $=$ $5$}, \bl{$|B|$ $=$ $3$}};}
|
|
519 |
\only<2>{
|
|
520 |
\draw [->, line width=1mm, red] (-7.4, 0.2) -- (-6.1, 2.1); |
|
521 |
\draw [->, line width=1mm, red] (-5.8, 0.2) -- (-3.1, 2.1); |
|
522 |
\draw [->, line width=1mm, red] (-4.5, 0.2) -- (-7.6, 2.1); |
|
523 |
\draw (-5, -3) node[scale=2] {then \bl{$|A|$ $\le$ $|B|$}};
|
|
524 |
} |
|
525 |
\only<3>{
|
|
526 |
\draw [<-, line width=1mm, red] (-7.5, 0.2) -- (-6.1, 2.1); |
|
527 |
\draw [<-, line width=1mm, red] (-7.3, 0.2) -- (-3.1, 2.1); |
|
528 |
\draw [<-, line width=1mm, red] (-6, 0.2) -- (-7.5, 2.1); |
|
529 |
\draw [<-, line width=1mm, red] (-4.5, 0.2) -- (-4.5, 2.1); |
|
530 |
\draw [<-, line width=1mm, red] (-4.3, 0.2) -- (-1.3, 2.1); |
|
531 |
||
532 |
\draw (-5, -3) node[scale=1.5] {\small{}for \bl{$=$}
|
|
533 |
has to be a {\bf one-to-one} mapping};
|
|
534 |
} |
|
535 |
||
536 |
||
537 |
\end{tikzpicture}
|
|
538 |
\end{center}
|
|
539 |
||
540 |
\end{frame}
|
|
541 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
542 |
||
543 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
544 |
\begin{frame}[c]
|
|
545 |
\frametitle{Cardinality}
|
|
546 |
||
547 |
\Large |
|
548 |
\bl{$|A|$} $\dn$ ``how many elements''\bigskip\\
|
|
549 |
||
550 |
\bl{$A \subseteq B \Rightarrow |A| \leq |B|$}\bigskip\\\pause
|
|
551 |
||
552 |
if there is an injective function \bl{$f: A \rightarrow B$} then \bl{$|A| \leq |B|$}\
|
|
553 |
||
554 |
\begin{center}
|
|
555 |
\bl{\large$\forall x y.\; f(x) = f(y) \Rightarrow x = y$}
|
|
556 |
\end{center}
|
|
557 |
\end{frame}
|
|
558 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
559 |
||
560 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
561 |
\begin{frame}[t]
|
|
562 |
||
563 |
\begin{center}
|
|
564 |
\begin{tikzpicture}
|
|
565 |
||
566 |
\draw (-6.6,2.5) node [scale=2.3] (X) |
|
567 |
{\begin{tabular}{@ {\hspace{-3mm}}l}
|
|
568 |
$A$ $=$ $\{$
|
|
569 |
\!\includegraphics[scale=0.02]{../pics/o1.jpg},
|
|
570 |
\includegraphics[scale=0.02]{../pics/o2.jpg},
|
|
571 |
\!\includegraphics[scale=0.02]{../pics/o3.jpg}
|
|
572 |
$\}$ |
|
573 |
\end{tabular}};
|
|
574 |
||
575 |
\draw (-6.6,-0.5) node [scale=2.3] (Y) |
|
576 |
{\begin{tabular}{@ {\hspace{-3mm}}l}
|
|
577 |
$B$ $=$ $\{$
|
|
578 |
\!\includegraphics[scale=0.059]{../pics/a1.jpg},
|
|
579 |
\includegraphics[scale=0.048]{../pics/a2.jpg},
|
|
580 |
\includegraphics[scale=0.02]{../pics/a3.jpg}
|
|
581 |
$\}$ |
|
582 |
\end{tabular}};
|
|
583 |
\onslide<3->{\draw (-7, -3) node[scale=1.5]
|
|
584 |
{then \bl{$|A|$ \alert{$=$} $|B|$}};}
|
|
585 |
\only<1>{
|
|
586 |
\draw [->, line width=1mm, red] (-7.4, 0.2) -- (-6.1, 2.1); |
|
587 |
\draw [->, line width=1mm, red] (-5.8, 0.2) -- (-4.3, 2.1); |
|
588 |
\draw [->, line width=1mm, red] (-4.5, 0.2) -- (-7.6, 2.1); |
|
589 |
} |
|
590 |
\only<2->{
|
|
591 |
\draw [<-, line width=1mm, blue] (-7.5, 0.2) -- (-7.5, 2.1); |
|
592 |
\draw [<-, line width=1mm, blue] (-5.8, 0.2) -- (-4.3, 2.1); |
|
593 |
\draw [<-, line width=1mm, blue] (-4.5, 0.2) -- (-6.1, 2.1); |
|
594 |
} |
|
595 |
||
596 |
||
597 |
\end{tikzpicture}
|
|
598 |
\end{center}
|
|
599 |
||
600 |
\end{frame}
|
|
601 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
|
315
470922b46a63
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
224
diff
changeset
|
602 |
|
|
86
6a7fe83820c8
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
603 |
|
| 543 | 604 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
605 |
\begin{frame}[c]
|
|
606 |
\frametitle{Natural Numbers}
|
|
607 |
||
608 |
\Large |
|
609 |
\bl{$\mathbb{N}$} \bl{$\dn$} \bl{$\{0, 1, 2, 3, .......\}$}\bigskip\pause
|
|
610 |
||
611 |
\bl{$A$} is \alert{countable} iff \bl{$|A| \leq |\mathbb{N}|$}
|
|
612 |
||
613 |
\end{frame}
|
|
614 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
615 |
||
616 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
617 |
\begin{frame}[c]
|
|
618 |
\frametitle{First Question}
|
|
619 |
||
620 |
\Large |
|
621 |
\bl{$|\mathbb{N} - \{0\}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\bigskip
|
|
622 |
||
623 |
\large |
|
624 |
\bl{$\geq$} or \bl{$\leq$} or \bl{$=$} ?
|
|
625 |
\bigskip\bigskip\bigskip\pause |
|
626 |
||
627 |
\bl{$x$ $\mapsto$ $x + 1$},\\ \bl{$|\mathbb{N} - \{0\}|$ $=$
|
|
628 |
$|\mathbb{N}|$}
|
|
629 |
\end{frame}
|
|
630 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
631 |
||
632 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
633 |
\mode<presentation>{
|
|
634 |
\begin{frame}[c]
|
|
635 |
||
636 |
\Large |
|
637 |
\bl{$|\mathbb{N} - \{0, 1\}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\pause
|
|
638 |
||
639 |
\bl{$|\mathbb{N} - \mathbb{O}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\bigskip
|
|
640 |
||
641 |
\normalsize |
|
642 |
\bl{$\mathbb{O}$} $\dn$ odd numbers\quad \bl{$\{1,3,5......\}$}\\ \pause
|
|
643 |
\bl{$\mathbb{E}$} $\dn$ even numbers\quad \bl{$\{0,2,4......\}$}\\
|
|
644 |
\end{frame}}
|
|
645 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
646 |
||
647 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
648 |
\mode<presentation>{
|
|
649 |
\begin{frame}[c]
|
|
650 |
||
651 |
\Large |
|
652 |
\bl{$|\mathbb{N} \cup \mathbb{-N}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\bigskip
|
|
653 |
||
654 |
||
655 |
\normalsize |
|
656 |
\bl{$\mathbb{\phantom{-}N}$} $\dn$ positive numbers\quad \bl{$\{0,1,2,3,......\}$}\\
|
|
657 |
\bl{$\mathbb{-N}$} $\dn$ negative numbers\quad \bl{$\{0,-1,-2,-3,......\}$}\\
|
|
658 |
\end{frame}}
|
|
659 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
660 |
||
661 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
662 |
\mode<presentation>{
|
|
663 |
\begin{frame}[c]
|
|
664 |
||
665 |
\Large |
|
666 |
\bl{$A$} is \alert{countable} if there exists an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip
|
|
667 |
||
668 |
\bl{$A$} is \alert{uncountable} if there does not exist an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip\bigskip
|
|
669 |
||
670 |
||
671 |
countable: \bl{$|A| \leq |\mathbb{N}|$}\\
|
|
672 |
uncountable: \bl{$|A| > |\mathbb{N}|$}\pause\bigskip
|
|
673 |
||
674 |
||
675 |
Does there exist such an \bl{$A$} ?
|
|
676 |
||
677 |
\end{frame}}
|
|
678 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
679 |
||
680 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
681 |
\mode<presentation>{
|
|
682 |
\begin{frame}[c]
|
|
683 |
\frametitle{Hilbert's Hotel}
|
|
684 |
||
685 |
\begin{center}
|
|
686 |
\includegraphics[scale=0.8]{../pics/hilberts_hotel.jpg}
|
|
687 |
\end{center}
|
|
688 |
||
689 |
\begin{itemize}
|
|
690 |
\item \ldots has as many rooms as there are natural numbers |
|
691 |
\end{itemize}
|
|
692 |
||
693 |
\end{frame}}
|
|
694 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
695 |
||
696 |
||
697 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
698 |
\begin{frame}[t]
|
|
699 |
\frametitle{\begin{tabular}{c}Real Numbers between\\[-2mm] 0 and 1\end{tabular}}
|
|
700 |
||
701 |
\begin{center}
|
|
702 |
\begin{tikzpicture}
|
|
703 |
\draw [fill, color=black!50] (1,4) rectangle (2, 3); |
|
704 |
\draw [fill, color=black!50] (2,3) rectangle (3, 2); |
|
705 |
\draw [fill, color=black!50] (3,2) rectangle (4, 1); |
|
706 |
\draw [fill, color=black!50] (4,1) rectangle (5, 0); |
|
707 |
\draw (0, 0) grid (8, 5); |
|
708 |
\draw [line width = 1.mm] (1,0) -- (1, 5); |
|
709 |
\draw [line width = 1.mm] (0, 4) -- (8, 4); |
|
710 |
\draw (0.5,3.5) node {$1$};
|
|
711 |
\draw (0.5,2.5) node {$2$};
|
|
712 |
\draw (0.5,1.5) node {$3$};
|
|
713 |
\draw (0.5,0.5) node {$4$};
|
|
714 |
||
715 |
\draw (1.5,3.5) node {\only<1>{$3$}\only<2->{$4$}};
|
|
716 |
\draw (2.5,3.5) node {$3$};
|
|
717 |
\draw (3.5,3.5) node {$3$};
|
|
718 |
\draw (4.5,3.5) node {$3$};
|
|
719 |
\draw (5.5,3.5) node {$3$};
|
|
720 |
\draw (6.5,3.5) node {$3$};
|
|
721 |
\draw (7.5,3.5) node {$\ldots$};
|
|
722 |
||
723 |
\draw (1.5,2.5) node {$1$};
|
|
724 |
\draw (2.5,2.5) node {\only<1-2>{$2$}\only<3->{$3$}};
|
|
725 |
\draw (3.5,2.5) node {$3$};
|
|
726 |
\draw (4.5,2.5) node {$4$};
|
|
727 |
\draw (5.5,2.5) node {$5$};
|
|
728 |
\draw (6.5,2.5) node {$6$};
|
|
729 |
\draw (7.5,2.5) node {$7$};
|
|
730 |
||
731 |
\draw (1.5,1.5) node {$0$};
|
|
732 |
\draw (2.5,1.5) node {$1$};
|
|
733 |
\draw (3.5,1.5) node {\only<1-3>{$0$}\only<4->{$1$}};
|
|
734 |
\draw (4.5,1.5) node {$1$};
|
|
735 |
\draw (5.5,1.5) node {$0$};
|
|
736 |
\draw (6.5,1.5) node {$\ldots$};
|
|
737 |
||
738 |
\draw (1.5,0.5) node {$7$};
|
|
739 |
\draw (2.5,0.5) node {$8$};
|
|
740 |
\draw (3.5,0.5) node {$5$};
|
|
741 |
\draw (4.5,0.5) node {\only<1-4>{$3$}\only<5->{$4$}};
|
|
742 |
\draw (5.5,0.5) node {$9$};
|
|
743 |
\draw (6.5,0.5) node {$\ldots$};
|
|
744 |
||
745 |
\draw (1.5,-0.5) node {$\ldots$};
|
|
746 |
\draw (8.5,3.5) node {$\ldots$};
|
|
747 |
\end{tikzpicture}
|
|
748 |
\end{center}
|
|
749 |
\mbox{}\\[-20mm]\mbox{}
|
|
750 |
||
751 |
\onslide<6->{
|
|
752 |
\begin{center}
|
|
753 |
\Large\bl{$|\mathbb{N}| < |R|$}
|
|
754 |
\end{center}
|
|
755 |
} |
|
756 |
||
757 |
\end{frame}
|
|
758 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
759 |
||
760 |
||
761 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
762 |
\mode<presentation>{
|
|
763 |
\begin{frame}[t]
|
|
764 |
\frametitle{The Set of Problems}
|
|
765 |
||
766 |
$\aleph_0$ |
|
767 |
||
768 |
\begin{center}
|
|
769 |
\begin{tikzpicture}
|
|
770 |
\draw [fill, color=black!50] (1,4) rectangle (2, 3); |
|
771 |
\draw [fill, color=black!50] (2,3) rectangle (3, 2); |
|
772 |
\draw [fill, color=black!50] (3,2) rectangle (4, 1); |
|
773 |
\draw [fill, color=black!50] (4,1) rectangle (5, 0); |
|
774 |
\draw (0, 0) grid (8, 5); |
|
775 |
\draw [line width = 1.mm] (1,0) -- (1, 5); |
|
776 |
\draw [line width = 1.mm] (0, 4) -- (8, 4); |
|
777 |
\draw (0.5,3.5) node {$1$};
|
|
778 |
\draw (0.5,2.5) node {$2$};
|
|
779 |
\draw (0.5,1.5) node {$3$};
|
|
780 |
\draw (0.5,0.5) node {$4$};
|
|
781 |
||
782 |
\draw (1.5,4.5) node {$0$};
|
|
783 |
\draw (2.5,4.5) node {$1$};
|
|
784 |
\draw (3.5,4.5) node {$2$};
|
|
785 |
\draw (4.5,4.5) node {$3$};
|
|
786 |
\draw (5.5,4.5) node {$4$};
|
|
787 |
\draw (6.5,4.5) node {$5$};
|
|
788 |
\draw (7.5,4.5) node {$\ldots$};
|
|
789 |
||
790 |
\draw (1.5,3.5) node {$0$};
|
|
791 |
\draw (2.5,3.5) node {$1$};
|
|
792 |
\draw (3.5,3.5) node {$0$};
|
|
793 |
\draw (4.5,3.5) node {$1$};
|
|
794 |
\draw (5.5,3.5) node {$0$};
|
|
795 |
\draw (6.5,3.5) node {$1$};
|
|
796 |
\draw (7.5,3.5) node {$\ldots$};
|
|
797 |
||
798 |
\draw (1.5,2.5) node {$0$};
|
|
799 |
\draw (2.5,2.5) node {$0$};
|
|
800 |
\draw (3.5,2.5) node {$0$};
|
|
801 |
\draw (4.5,2.5) node {$1$};
|
|
802 |
\draw (5.5,2.5) node {$1$};
|
|
803 |
\draw (6.5,2.5) node {$0$};
|
|
804 |
\draw (7.5,2.5) node {$0$};
|
|
805 |
||
806 |
\draw (1.5,1.5) node {$0$};
|
|
807 |
\draw (2.5,1.5) node {$0$};
|
|
808 |
\draw (3.5,1.5) node {$0$};
|
|
809 |
\draw (4.5,1.5) node {$0$};
|
|
810 |
\draw (5.5,1.5) node {$0$};
|
|
811 |
\draw (6.5,1.5) node {$\ldots$};
|
|
812 |
||
813 |
\draw (1.5,0.5) node {$1$};
|
|
814 |
\draw (2.5,0.5) node {$1$};
|
|
815 |
\draw (3.5,0.5) node {$0$};
|
|
816 |
\draw (4.5,0.5) node {$1$};
|
|
817 |
\draw (5.5,0.5) node {$1$};
|
|
818 |
\draw (6.5,0.5) node {$\ldots$};
|
|
819 |
||
820 |
\draw (1.5,-0.5) node {$\ldots$};
|
|
821 |
\draw (8.5,3.5) node {$\ldots$};
|
|
822 |
||
823 |
\end{tikzpicture}
|
|
824 |
\end{center}
|
|
825 |
||
826 |
||
827 |
\onslide<2>{
|
|
828 |
\begin{center}
|
|
829 |
\large \bl{|Progs| $=$ $|\mathbb{N}|$ $<$ |Probs|}
|
|
830 |
\end{center}
|
|
831 |
} |
|
832 |
||
833 |
||
834 |
\end{frame}}
|
|
835 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
836 |
||
837 |
||
838 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
839 |
\mode<presentation>{
|
|
840 |
\begin{frame}[c]
|
|
841 |
\frametitle{Halting Problem}
|
|
842 |
||
843 |
\large |
|
844 |
Assume a program \bl{$H$} that decides for all programs \bl{$A$} and all
|
|
845 |
input data \bl{$D$} whether\bigskip
|
|
846 |
||
847 |
\begin{itemize}
|
|
848 |
\item \bl{$H(A, D) \dn 1$} iff \bl{$A(D)$} terminates
|
|
849 |
\item \bl{$H(A, D) \dn 0$} otherwise
|
|
850 |
\end{itemize}
|
|
851 |
||
852 |
\end{frame}}
|
|
853 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
854 |
||
855 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
856 |
\mode<presentation>{
|
|
857 |
\begin{frame}[c]
|
|
858 |
\frametitle{Halting Problem (2)}
|
|
859 |
||
860 |
\large |
|
861 |
Given such a program \bl{$H$} define the following program \bl{$C$}:
|
|
862 |
for all programs \bl{$A$}\bigskip
|
|
863 |
||
864 |
\begin{itemize}
|
|
865 |
\item \bl{$C(A) \dn 0$} iff \bl{$H(A, A) = 0$}
|
|
866 |
\item \bl{$C(A) \dn$ loops} otherwise
|
|
867 |
\end{itemize}
|
|
868 |
||
869 |
\end{frame}}
|
|
870 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
871 |
||
872 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
873 |
\mode<presentation>{
|
|
874 |
\begin{frame}[c]
|
|
875 |
\frametitle{Contradiction}
|
|
876 |
||
877 |
||
878 |
\bl{$H(C, C)$} is either \bl{$0$} or \bl{$1$}.
|
|
879 |
||
880 |
\begin{itemize}
|
|
881 |
\item \bl{$H(C, C) = 1$} $\stackrel{\text{def}\,H}{\Rightarrow}$ \bl{$C(C)\downarrow$} $\stackrel{\text{def}\,C}{\Rightarrow}$ \bl{$H(C, C)=0$}
|
|
882 |
\item \bl{$H(C, C) = 0$} $\stackrel{\text{def}\,H}{\Rightarrow}$ \bl{$C(C)$} loops $\stackrel{\text{def}\,C}{\Rightarrow}$\\
|
|
883 |
\hspace{7cm}\bl{$H(C, C)=1$}
|
|
884 |
\end{itemize}
|
|
885 |
||
886 |
Contradiction in both cases. So \bl{$H$} cannot exist.
|
|
887 |
||
888 |
\end{frame}}
|
|
889 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
890 |
||
891 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
892 |
\mode<presentation>{
|
|
893 |
\begin{frame}[c]
|
|
894 |
\frametitle{Take Home Points}
|
|
895 |
\large |
|
896 |
||
897 |
\begin{itemize}
|
|
898 |
\item there are sets that are more infinite than others\bigskip |
|
899 |
\item even with the most powerful computer we can imagine, there |
|
900 |
are problems that cannot be solved by any program\bigskip\bigskip |
|
901 |
||
902 |
\item in CS we actually hit quite often such problems (halting problem) |
|
903 |
\end{itemize}
|
|
904 |
||
905 |
\end{frame}}
|
|
906 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 864 | 907 |
|
908 |
\begin{frame}<1-20>[c]
|
|
909 |
\end{frame}
|
|
910 |
||
911 |
||
|
86
6a7fe83820c8
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
912 |
\end{document}
|
|
6a7fe83820c8
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
913 |
|
|
6a7fe83820c8
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
914 |
%%% Local Variables: |
|
6a7fe83820c8
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
915 |
%%% mode: latex |
|
6a7fe83820c8
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
916 |
%%% TeX-master: t |
|
6a7fe83820c8
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
917 |
%%% End: |
|
6a7fe83820c8
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
918 |