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% !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{../graphics}
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{document}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[t]
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\frametitle{%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\normalsize
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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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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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c,fragile]
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\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-6mm]
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def fact(n: Int) : Int = {
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if (n == 0) 1 else n * fact(n - 1)
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}
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def factC(n: Int, ret: Int => Int) : Int = {
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if (n == 0) ret(1)
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else factC(n - 1, x => ret(n * x))
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}
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fact(10)
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factC(10, identity)
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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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\begin{lstlisting}[language=Scala, numbers=none, xleftmargin=-6mm]
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def fibC(n: Int, ret: Int => Int) : Int = {
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if (n == 0 || n == 1) ret(1) else
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fibC(n - 1,
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r1 => fibC(n - 2,
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r2 => ret(r1 + r2)))
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}
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fibC(10, identity)
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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]
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\Large\bf Are there more strings in \\ \hfill\bl{$L(a^*)$} or
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\bl{$L((a + b)^*)$}?
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{Can you remember this HW?}
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\begin{itemize}
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\item (1) How many basic regular expressions are there to match
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the string \bl{$abcd$}?
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\item (2) How many if they cannot include
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\bl{$\ONE$} and \bl{$\ZERO$}?
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\item (3) How many if they are also not
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allowed to contain stars?
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\item (4) How many if they are also
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not allowed to contain \bl{$\_ + \_$}?
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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543
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\Large\bf There are more problems, than there are
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programs.\bigskip\bigskip\pause\\
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There must be a problem for which there is no program.
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\frametitle{Subsets}
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\Large
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If \bl{$A \subseteq B$} then \bl{$A$} has fewer or equal elements
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than \bl{$B$}\bigskip\bigskip
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\Large
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\bl{$A \subseteq B$} and \bl{$B \subseteq A$}\bigskip
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then \bl{$A = B$}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[c]
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\begin{center}
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\begin{tikzpicture}
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\draw (-4,2.5) node [scale=2.5] (X)
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{\begin{tabular}{l}
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$\{$
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\!\includegraphics[scale=0.02]{../pics/o1.jpg},
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\includegraphics[scale=0.02]{../pics/o2.jpg},
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\!\includegraphics[scale=0.02]{../pics/o3.jpg},
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\includegraphics[scale=0.02]{../pics/o4.jpg},
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\!\includegraphics[scale=0.027]{../pics/o5.jpg}
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$\}$
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\end{tabular}};
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\draw (-5.6,-2.5) node [scale=2.5] (Y)
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{\begin{tabular}{l}
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$\{$
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\!\includegraphics[scale=0.059]{../pics/a1.jpg},
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\includegraphics[scale=0.048]{../pics/a2.jpg},
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\includegraphics[scale=0.02]{../pics/a3.jpg}
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$\}$
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\end{tabular}};
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\draw (0,1.5) node (X1) {5 elements};
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\draw (0,-3.5) node (y1) {3 elements};
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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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\begin{frame}[c]
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\frametitle{Newton vs Feynman}
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\begin{center}
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\begin{tabular}{cc}
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\includegraphics[scale=0.2]{../pics/newton.jpg} &
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\includegraphics[scale=0.2]{../pics/feynman.jpg}\\
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classical physics & quantum physics
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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]
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\frametitle{The Goal of the Talk}
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\large
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\begin{itemize}
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\item show you that something very unintuitive happens with very large sets
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\bigskip\bigskip
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\item convince you that there are more {\bf problems} than {\bf programs}
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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%
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\begin{center}
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\begin{tikzpicture}
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\draw (-5,2.5) node [scale=2.3] (X)
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{\begin{tabular}{@ {\hspace{-3mm}}l}
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\bl{$B$ $=$ $\{$
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\!\includegraphics[scale=0.02]{../pics/o1.jpg},
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\includegraphics[scale=0.02]{../pics/o2.jpg},
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\!\includegraphics[scale=0.02]{../pics/o3.jpg},
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\includegraphics[scale=0.02]{../pics/o4.jpg},
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\!\includegraphics[scale=0.027]{../pics/o5.jpg}
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$\}$}
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\end{tabular}};
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\draw (-6.6,-0.5) node [scale=2.3] (Y)
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{\begin{tabular}{@ {\hspace{-3mm}}l}
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\bl{$A$ $=$ $\{$
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\!\includegraphics[scale=0.059]{../pics/a1.jpg},
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\includegraphics[scale=0.048]{../pics/a2.jpg},
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\includegraphics[scale=0.02]{../pics/a3.jpg}
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$\}$}
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\end{tabular}};
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\only<1>{\draw (-5, -3) node[scale=2]
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{\bl{$|A|$ $=$ $5$}, \bl{$|B|$ $=$ $3$}};}
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\only<2>{
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\draw [->, line width=1mm, red] (-7.4, 0.2) -- (-6.1, 2.1);
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\draw [->, line width=1mm, red] (-5.8, 0.2) -- (-3.1, 2.1);
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\draw [->, line width=1mm, red] (-4.5, 0.2) -- (-7.6, 2.1);
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\draw (-5, -3) node[scale=2] {then \bl{$|A|$ $\le$ $|B|$}};
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}
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\only<3>{
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\draw [<-, line width=1mm, red] (-7.5, 0.2) -- (-6.1, 2.1);
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\draw [<-, line width=1mm, red] (-7.3, 0.2) -- (-3.1, 2.1);
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\draw [<-, line width=1mm, red] (-6, 0.2) -- (-7.5, 2.1);
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\draw [<-, line width=1mm, red] (-4.5, 0.2) -- (-4.5, 2.1);
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\draw [<-, line width=1mm, red] (-4.3, 0.2) -- (-1.3, 2.1);
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\draw (-5, -3) node[scale=1.5] {\small{}for \bl{$=$}
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has to be a {\bf one-to-one} mapping};
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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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\begin{frame}[c]
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\frametitle{Cardinality}
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\Large
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292 |
\bl{$|A|$} $\dn$ ``how many elements''\bigskip\\
|
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293 |
|
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|
294 |
\bl{$A \subseteq B \Rightarrow |A| \leq |B|$}\bigskip\\\pause
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295 |
|
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|
296 |
if there is an injective function \bl{$f: A \rightarrow B$} then \bl{$|A| \leq |B|$}\
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297 |
|
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|
298 |
\begin{center}
|
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|
299 |
\bl{\large$\forall x y.\; f(x) = f(y) \Rightarrow x = y$}
|
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|
300 |
\end{center}
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|
301 |
\end{frame}
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|
302 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
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303 |
|
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|
304 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
305 |
\begin{frame}[t]
|
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|
306 |
|
|
|
307 |
\begin{center}
|
|
|
308 |
\begin{tikzpicture}
|
|
|
309 |
|
|
|
310 |
\draw (-6.6,2.5) node [scale=2.3] (X)
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311 |
{\begin{tabular}{@ {\hspace{-3mm}}l}
|
|
|
312 |
$A$ $=$ $\{$
|
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|
313 |
\!\includegraphics[scale=0.02]{../pics/o1.jpg},
|
|
|
314 |
\includegraphics[scale=0.02]{../pics/o2.jpg},
|
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|
315 |
\!\includegraphics[scale=0.02]{../pics/o3.jpg}
|
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|
316 |
$\}$
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|
317 |
\end{tabular}};
|
|
|
318 |
|
|
|
319 |
\draw (-6.6,-0.5) node [scale=2.3] (Y)
|
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|
320 |
{\begin{tabular}{@ {\hspace{-3mm}}l}
|
|
|
321 |
$B$ $=$ $\{$
|
|
|
322 |
\!\includegraphics[scale=0.059]{../pics/a1.jpg},
|
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|
323 |
\includegraphics[scale=0.048]{../pics/a2.jpg},
|
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|
324 |
\includegraphics[scale=0.02]{../pics/a3.jpg}
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|
325 |
$\}$
|
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|
326 |
\end{tabular}};
|
|
|
327 |
\onslide<3->{\draw (-7, -3) node[scale=1.5]
|
|
|
328 |
{then \bl{$|A|$ \alert{$=$} $|B|$}};}
|
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329 |
\only<1>{
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|
330 |
\draw [->, line width=1mm, red] (-7.4, 0.2) -- (-6.1, 2.1);
|
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331 |
\draw [->, line width=1mm, red] (-5.8, 0.2) -- (-4.3, 2.1);
|
|
|
332 |
\draw [->, line width=1mm, red] (-4.5, 0.2) -- (-7.6, 2.1);
|
|
|
333 |
}
|
|
|
334 |
\only<2->{
|
|
|
335 |
\draw [<-, line width=1mm, blue] (-7.5, 0.2) -- (-7.5, 2.1);
|
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|
336 |
\draw [<-, line width=1mm, blue] (-5.8, 0.2) -- (-4.3, 2.1);
|
|
|
337 |
\draw [<-, line width=1mm, blue] (-4.5, 0.2) -- (-6.1, 2.1);
|
|
|
338 |
}
|
|
|
339 |
|
|
|
340 |
|
|
|
341 |
\end{tikzpicture}
|
|
|
342 |
\end{center}
|
|
|
343 |
|
|
|
344 |
\end{frame}
|
|
|
345 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
315
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
346 |
|
86
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
347 |
|
|
543
|
348 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
349 |
\begin{frame}[c]
|
|
|
350 |
\frametitle{Natural Numbers}
|
|
|
351 |
|
|
|
352 |
\Large
|
|
|
353 |
\bl{$\mathbb{N}$} \bl{$\dn$} \bl{$\{0, 1, 2, 3, .......\}$}\bigskip\pause
|
|
|
354 |
|
|
|
355 |
\bl{$A$} is \alert{countable} iff \bl{$|A| \leq |\mathbb{N}|$}
|
|
|
356 |
|
|
|
357 |
\end{frame}
|
|
|
358 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
359 |
|
|
|
360 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
361 |
\begin{frame}[c]
|
|
|
362 |
\frametitle{First Question}
|
|
|
363 |
|
|
|
364 |
\Large
|
|
|
365 |
\bl{$|\mathbb{N} - \{0\}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\bigskip
|
|
|
366 |
|
|
|
367 |
\large
|
|
|
368 |
\bl{$\geq$} or \bl{$\leq$} or \bl{$=$} ?
|
|
|
369 |
\bigskip\bigskip\bigskip\pause
|
|
|
370 |
|
|
|
371 |
\bl{$x$ $\mapsto$ $x + 1$},\\ \bl{$|\mathbb{N} - \{0\}|$ $=$
|
|
|
372 |
$|\mathbb{N}|$}
|
|
|
373 |
\end{frame}
|
|
|
374 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
375 |
|
|
|
376 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
377 |
\mode<presentation>{
|
|
|
378 |
\begin{frame}[c]
|
|
|
379 |
|
|
|
380 |
\Large
|
|
|
381 |
\bl{$|\mathbb{N} - \{0, 1\}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\pause
|
|
|
382 |
|
|
|
383 |
\bl{$|\mathbb{N} - \mathbb{O}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\bigskip
|
|
|
384 |
|
|
|
385 |
\normalsize
|
|
|
386 |
\bl{$\mathbb{O}$} $\dn$ odd numbers\quad \bl{$\{1,3,5......\}$}\\ \pause
|
|
|
387 |
\bl{$\mathbb{E}$} $\dn$ even numbers\quad \bl{$\{0,2,4......\}$}\\
|
|
|
388 |
\end{frame}}
|
|
|
389 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
390 |
|
|
|
391 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
392 |
\mode<presentation>{
|
|
|
393 |
\begin{frame}[c]
|
|
|
394 |
|
|
|
395 |
\Large
|
|
|
396 |
\bl{$|\mathbb{N} \cup \mathbb{-N}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\bigskip
|
|
|
397 |
|
|
|
398 |
|
|
|
399 |
\normalsize
|
|
|
400 |
\bl{$\mathbb{\phantom{-}N}$} $\dn$ positive numbers\quad \bl{$\{0,1,2,3,......\}$}\\
|
|
|
401 |
\bl{$\mathbb{-N}$} $\dn$ negative numbers\quad \bl{$\{0,-1,-2,-3,......\}$}\\
|
|
|
402 |
\end{frame}}
|
|
|
403 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
404 |
|
|
|
405 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
406 |
\mode<presentation>{
|
|
|
407 |
\begin{frame}[c]
|
|
|
408 |
|
|
|
409 |
\Large
|
|
|
410 |
\bl{$A$} is \alert{countable} if there exists an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip
|
|
|
411 |
|
|
|
412 |
\bl{$A$} is \alert{uncountable} if there does not exist an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip\bigskip
|
|
|
413 |
|
|
|
414 |
|
|
|
415 |
countable: \bl{$|A| \leq |\mathbb{N}|$}\\
|
|
|
416 |
uncountable: \bl{$|A| > |\mathbb{N}|$}\pause\bigskip
|
|
|
417 |
|
|
|
418 |
|
|
|
419 |
Does there exist such an \bl{$A$} ?
|
|
|
420 |
|
|
|
421 |
\end{frame}}
|
|
|
422 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
423 |
|
|
|
424 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
425 |
\mode<presentation>{
|
|
|
426 |
\begin{frame}[c]
|
|
|
427 |
\frametitle{Hilbert's Hotel}
|
|
|
428 |
|
|
|
429 |
\begin{center}
|
|
|
430 |
\includegraphics[scale=0.8]{../pics/hilberts_hotel.jpg}
|
|
|
431 |
\end{center}
|
|
|
432 |
|
|
|
433 |
\begin{itemize}
|
|
|
434 |
\item \ldots has as many rooms as there are natural numbers
|
|
|
435 |
\end{itemize}
|
|
|
436 |
|
|
|
437 |
\end{frame}}
|
|
|
438 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
439 |
|
|
|
440 |
|
|
|
441 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
442 |
\begin{frame}[t]
|
|
|
443 |
\frametitle{\begin{tabular}{c}Real Numbers between\\[-2mm] 0 and 1\end{tabular}}
|
|
|
444 |
|
|
|
445 |
\begin{center}
|
|
|
446 |
\begin{tikzpicture}
|
|
|
447 |
\draw [fill, color=black!50] (1,4) rectangle (2, 3);
|
|
|
448 |
\draw [fill, color=black!50] (2,3) rectangle (3, 2);
|
|
|
449 |
\draw [fill, color=black!50] (3,2) rectangle (4, 1);
|
|
|
450 |
\draw [fill, color=black!50] (4,1) rectangle (5, 0);
|
|
|
451 |
\draw (0, 0) grid (8, 5);
|
|
|
452 |
\draw [line width = 1.mm] (1,0) -- (1, 5);
|
|
|
453 |
\draw [line width = 1.mm] (0, 4) -- (8, 4);
|
|
|
454 |
\draw (0.5,3.5) node {$1$};
|
|
|
455 |
\draw (0.5,2.5) node {$2$};
|
|
|
456 |
\draw (0.5,1.5) node {$3$};
|
|
|
457 |
\draw (0.5,0.5) node {$4$};
|
|
|
458 |
|
|
|
459 |
\draw (1.5,3.5) node {\only<1>{$3$}\only<2->{$4$}};
|
|
|
460 |
\draw (2.5,3.5) node {$3$};
|
|
|
461 |
\draw (3.5,3.5) node {$3$};
|
|
|
462 |
\draw (4.5,3.5) node {$3$};
|
|
|
463 |
\draw (5.5,3.5) node {$3$};
|
|
|
464 |
\draw (6.5,3.5) node {$3$};
|
|
|
465 |
\draw (7.5,3.5) node {$\ldots$};
|
|
|
466 |
|
|
|
467 |
\draw (1.5,2.5) node {$1$};
|
|
|
468 |
\draw (2.5,2.5) node {\only<1-2>{$2$}\only<3->{$3$}};
|
|
|
469 |
\draw (3.5,2.5) node {$3$};
|
|
|
470 |
\draw (4.5,2.5) node {$4$};
|
|
|
471 |
\draw (5.5,2.5) node {$5$};
|
|
|
472 |
\draw (6.5,2.5) node {$6$};
|
|
|
473 |
\draw (7.5,2.5) node {$7$};
|
|
|
474 |
|
|
|
475 |
\draw (1.5,1.5) node {$0$};
|
|
|
476 |
\draw (2.5,1.5) node {$1$};
|
|
|
477 |
\draw (3.5,1.5) node {\only<1-3>{$0$}\only<4->{$1$}};
|
|
|
478 |
\draw (4.5,1.5) node {$1$};
|
|
|
479 |
\draw (5.5,1.5) node {$0$};
|
|
|
480 |
\draw (6.5,1.5) node {$\ldots$};
|
|
|
481 |
|
|
|
482 |
\draw (1.5,0.5) node {$7$};
|
|
|
483 |
\draw (2.5,0.5) node {$8$};
|
|
|
484 |
\draw (3.5,0.5) node {$5$};
|
|
|
485 |
\draw (4.5,0.5) node {\only<1-4>{$3$}\only<5->{$4$}};
|
|
|
486 |
\draw (5.5,0.5) node {$9$};
|
|
|
487 |
\draw (6.5,0.5) node {$\ldots$};
|
|
|
488 |
|
|
|
489 |
\draw (1.5,-0.5) node {$\ldots$};
|
|
|
490 |
\draw (8.5,3.5) node {$\ldots$};
|
|
|
491 |
\end{tikzpicture}
|
|
|
492 |
\end{center}
|
|
|
493 |
\mbox{}\\[-20mm]\mbox{}
|
|
|
494 |
|
|
|
495 |
\onslide<6->{
|
|
|
496 |
\begin{center}
|
|
|
497 |
\Large\bl{$|\mathbb{N}| < |R|$}
|
|
|
498 |
\end{center}
|
|
|
499 |
}
|
|
|
500 |
|
|
|
501 |
\end{frame}
|
|
|
502 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
503 |
|
|
|
504 |
|
|
|
505 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
506 |
\mode<presentation>{
|
|
|
507 |
\begin{frame}[t]
|
|
|
508 |
\frametitle{The Set of Problems}
|
|
|
509 |
|
|
|
510 |
$\aleph_0$
|
|
|
511 |
|
|
|
512 |
\begin{center}
|
|
|
513 |
\begin{tikzpicture}
|
|
|
514 |
\draw [fill, color=black!50] (1,4) rectangle (2, 3);
|
|
|
515 |
\draw [fill, color=black!50] (2,3) rectangle (3, 2);
|
|
|
516 |
\draw [fill, color=black!50] (3,2) rectangle (4, 1);
|
|
|
517 |
\draw [fill, color=black!50] (4,1) rectangle (5, 0);
|
|
|
518 |
\draw (0, 0) grid (8, 5);
|
|
|
519 |
\draw [line width = 1.mm] (1,0) -- (1, 5);
|
|
|
520 |
\draw [line width = 1.mm] (0, 4) -- (8, 4);
|
|
|
521 |
\draw (0.5,3.5) node {$1$};
|
|
|
522 |
\draw (0.5,2.5) node {$2$};
|
|
|
523 |
\draw (0.5,1.5) node {$3$};
|
|
|
524 |
\draw (0.5,0.5) node {$4$};
|
|
|
525 |
|
|
|
526 |
\draw (1.5,4.5) node {$0$};
|
|
|
527 |
\draw (2.5,4.5) node {$1$};
|
|
|
528 |
\draw (3.5,4.5) node {$2$};
|
|
|
529 |
\draw (4.5,4.5) node {$3$};
|
|
|
530 |
\draw (5.5,4.5) node {$4$};
|
|
|
531 |
\draw (6.5,4.5) node {$5$};
|
|
|
532 |
\draw (7.5,4.5) node {$\ldots$};
|
|
|
533 |
|
|
|
534 |
\draw (1.5,3.5) node {$0$};
|
|
|
535 |
\draw (2.5,3.5) node {$1$};
|
|
|
536 |
\draw (3.5,3.5) node {$0$};
|
|
|
537 |
\draw (4.5,3.5) node {$1$};
|
|
|
538 |
\draw (5.5,3.5) node {$0$};
|
|
|
539 |
\draw (6.5,3.5) node {$1$};
|
|
|
540 |
\draw (7.5,3.5) node {$\ldots$};
|
|
|
541 |
|
|
|
542 |
\draw (1.5,2.5) node {$0$};
|
|
|
543 |
\draw (2.5,2.5) node {$0$};
|
|
|
544 |
\draw (3.5,2.5) node {$0$};
|
|
|
545 |
\draw (4.5,2.5) node {$1$};
|
|
|
546 |
\draw (5.5,2.5) node {$1$};
|
|
|
547 |
\draw (6.5,2.5) node {$0$};
|
|
|
548 |
\draw (7.5,2.5) node {$0$};
|
|
|
549 |
|
|
|
550 |
\draw (1.5,1.5) node {$0$};
|
|
|
551 |
\draw (2.5,1.5) node {$0$};
|
|
|
552 |
\draw (3.5,1.5) node {$0$};
|
|
|
553 |
\draw (4.5,1.5) node {$0$};
|
|
|
554 |
\draw (5.5,1.5) node {$0$};
|
|
|
555 |
\draw (6.5,1.5) node {$\ldots$};
|
|
|
556 |
|
|
|
557 |
\draw (1.5,0.5) node {$1$};
|
|
|
558 |
\draw (2.5,0.5) node {$1$};
|
|
|
559 |
\draw (3.5,0.5) node {$0$};
|
|
|
560 |
\draw (4.5,0.5) node {$1$};
|
|
|
561 |
\draw (5.5,0.5) node {$1$};
|
|
|
562 |
\draw (6.5,0.5) node {$\ldots$};
|
|
|
563 |
|
|
|
564 |
\draw (1.5,-0.5) node {$\ldots$};
|
|
|
565 |
\draw (8.5,3.5) node {$\ldots$};
|
|
|
566 |
|
|
|
567 |
\end{tikzpicture}
|
|
|
568 |
\end{center}
|
|
|
569 |
|
|
|
570 |
|
|
|
571 |
\onslide<2>{
|
|
|
572 |
\begin{center}
|
|
|
573 |
\large \bl{|Progs| $=$ $|\mathbb{N}|$ $<$ |Probs|}
|
|
|
574 |
\end{center}
|
|
|
575 |
}
|
|
|
576 |
|
|
|
577 |
|
|
|
578 |
\end{frame}}
|
|
|
579 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
580 |
|
|
|
581 |
|
|
|
582 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
583 |
\mode<presentation>{
|
|
|
584 |
\begin{frame}[c]
|
|
|
585 |
\frametitle{Halting Problem}
|
|
|
586 |
|
|
|
587 |
\large
|
|
|
588 |
Assume a program \bl{$H$} that decides for all programs \bl{$A$} and all
|
|
|
589 |
input data \bl{$D$} whether\bigskip
|
|
|
590 |
|
|
|
591 |
\begin{itemize}
|
|
|
592 |
\item \bl{$H(A, D) \dn 1$} iff \bl{$A(D)$} terminates
|
|
|
593 |
\item \bl{$H(A, D) \dn 0$} otherwise
|
|
|
594 |
\end{itemize}
|
|
|
595 |
|
|
|
596 |
\end{frame}}
|
|
|
597 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
598 |
|
|
|
599 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
600 |
\mode<presentation>{
|
|
|
601 |
\begin{frame}[c]
|
|
|
602 |
\frametitle{Halting Problem (2)}
|
|
|
603 |
|
|
|
604 |
\large
|
|
|
605 |
Given such a program \bl{$H$} define the following program \bl{$C$}:
|
|
|
606 |
for all programs \bl{$A$}\bigskip
|
|
|
607 |
|
|
|
608 |
\begin{itemize}
|
|
|
609 |
\item \bl{$C(A) \dn 0$} iff \bl{$H(A, A) = 0$}
|
|
|
610 |
\item \bl{$C(A) \dn$ loops} otherwise
|
|
|
611 |
\end{itemize}
|
|
|
612 |
|
|
|
613 |
\end{frame}}
|
|
|
614 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
615 |
|
|
|
616 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
617 |
\mode<presentation>{
|
|
|
618 |
\begin{frame}[c]
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|
619 |
\frametitle{Contradiction}
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|
620 |
|
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|
621 |
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|
622 |
\bl{$H(C, C)$} is either \bl{$0$} or \bl{$1$}.
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|
623 |
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|
624 |
\begin{itemize}
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|
625 |
\item \bl{$H(C, C) = 1$} $\stackrel{\text{def}\,H}{\Rightarrow}$ \bl{$C(C)\downarrow$} $\stackrel{\text{def}\,C}{\Rightarrow}$ \bl{$H(C, C)=0$}
|
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|
626 |
\item \bl{$H(C, C) = 0$} $\stackrel{\text{def}\,H}{\Rightarrow}$ \bl{$C(C)$} loops $\stackrel{\text{def}\,C}{\Rightarrow}$\\
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|
627 |
\hspace{7cm}\bl{$H(C, C)=1$}
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|
628 |
\end{itemize}
|
|
|
629 |
|
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|
630 |
Contradiction in both cases. So \bl{$H$} cannot exist.
|
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|
631 |
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|
632 |
\end{frame}}
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|
633 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
634 |
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|
635 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
636 |
\mode<presentation>{
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|
637 |
\begin{frame}[c]
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|
638 |
\frametitle{Take Home Points}
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|
639 |
\large
|
|
|
640 |
|
|
|
641 |
\begin{itemize}
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|
|
642 |
\item there are sets that are more infinite than others\bigskip
|
|
|
643 |
\item even with the most powerful computer we can imagine, there
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|
644 |
are problems that cannot be solved by any program\bigskip\bigskip
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|
645 |
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|
646 |
\item in CS we actually hit quite often such problems (halting problem)
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|
647 |
\end{itemize}
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|
648 |
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|
649 |
\end{frame}}
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|
650 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
651 |
|
86
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
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652 |
\end{document}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
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653 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
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654 |
%%% Local Variables:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
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655 |
%%% mode: latex
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
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656 |
%%% TeX-master: t
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
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657 |
%%% End:
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
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658 |
|