\documentclass[dvipsnames,14pt,t]{beamer}

\usepackage{../slides}

\usepackage{../graphics}

\usepackage{../langs}

\usepackage{../data}

\usepackage{../grammar}

% beamer stuff 

\renewcommand{\slidecaption}{CFL, King's College London}

\newcommand{\bl}[1]{\textcolor{blue}{#1}}       

\begin{document}

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\begin{frame}[t]

\frametitle{%

  \begin{tabular}{@ {}c@ {}}

  \\[-3mm]

  \LARGE Compilers and \\[-2mm] 

  \LARGE Formal Languages\\[3mm] 

  \end{tabular}}

  \normalsize

  \begin{center}

  \begin{tabular}{ll}

  Email:  & christian.urban at kcl.ac.uk\\

  Office: & N7.07 (North Wing, Bush House)\\

  Slides: & KEATS (also home work is there)\\

  \end{tabular}

  \end{center}

\end{frame}

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\begin{frame}[c]

\Large\bf There are more problems, than there are

programs.\bigskip\bigskip\pause\\

There must be a problem for which there is no program.

\end{frame}

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\begin{frame}[c]

\frametitle{Subsets}

\Large

If \bl{$A \subseteq B$} then \bl{$A$} has fewer elements

than \bl{$B$}\bigskip\bigskip

\Large

\bl{$A \subseteq B$} and \bl{$B \subseteq A$}\bigskip

then \bl{$A = B$}

\end{frame}

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  \begin{frame}[c]

  \begin{center}

  \begin{tikzpicture}

  \draw (-4,2.5) node [scale=2.5] (X) 

    {\begin{tabular}{l}

     $\{$  

     \!\includegraphics[scale=0.02]{../pics/o1.jpg},

     \includegraphics[scale=0.02]{../pics/o2.jpg},

     \!\includegraphics[scale=0.02]{../pics/o3.jpg},

     \includegraphics[scale=0.02]{../pics/o4.jpg},

     \!\includegraphics[scale=0.027]{../pics/o5.jpg}

     $\}$

    \end{tabular}};

    \draw (-5.6,-2.5) node [scale=2.5] (Y) 

    {\begin{tabular}{l}

     $\{$  

     \!\includegraphics[scale=0.059]{../pics/a1.jpg},

     \includegraphics[scale=0.048]{../pics/a2.jpg},

     \includegraphics[scale=0.02]{../pics/a3.jpg}

     $\}$

    \end{tabular}};

     \draw (0,1.5) node (X1) {5 elements};

     \draw (0,-3.5) node (y1) {3 elements};

  \end{tikzpicture}

  \end{center}

  \end{frame}

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  \begin{frame}[c]

  \frametitle{Newton vs Feynman}

  \begin{center}

  \begin{tabular}{cc}

  \includegraphics[scale=0.2]{../pics/newton.jpg} &

  \includegraphics[scale=0.2]{../pics/feynman.jpg}\\

  classical physics & quantum physics

  \end{tabular}

  \end{center}

  \end{frame}

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  \begin{frame}[c]

  \frametitle{The Goal of the Talk}

 \large

  \begin{itemize}

  \item show you that something very unintuitive happens with very large sets	

  \bigskip\bigskip

  \item convince you that there are more {\bf problems} than {\bf programs}

  \end{itemize}	

  \end{frame}

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\begin{frame}[t]

%

  \begin{center}

  \begin{tikzpicture}

  \draw (-5,2.5) node [scale=2.3] (X) 

    {\begin{tabular}{@ {\hspace{-3mm}}l}

     \bl{$B$ $=$ $\{$  

     \!\includegraphics[scale=0.02]{../pics/o1.jpg},

     \includegraphics[scale=0.02]{../pics/o2.jpg},

     \!\includegraphics[scale=0.02]{../pics/o3.jpg},

     \includegraphics[scale=0.02]{../pics/o4.jpg},

     \!\includegraphics[scale=0.027]{../pics/o5.jpg}

     $\}$}

    \end{tabular}};

    \draw (-6.6,-0.5) node [scale=2.3] (Y) 

    {\begin{tabular}{@ {\hspace{-3mm}}l}

     \bl{$A$ $=$ $\{$  

     \!\includegraphics[scale=0.059]{../pics/a1.jpg},

     \includegraphics[scale=0.048]{../pics/a2.jpg},

     \includegraphics[scale=0.02]{../pics/a3.jpg}

     $\}$}

     \end{tabular}};

     \only<1>{\draw (-5, -3) node[scale=2] 

       {\bl{$|A|$ $=$ $5$}, \bl{$|B|$ $=$ $3$}};}

     \only<2>{

       \draw [->, line width=1mm, red] (-7.4, 0.2) -- (-6.1, 2.1);

       \draw [->, line width=1mm, red] (-5.8, 0.2) -- (-3.1, 2.1);

       \draw [->, line width=1mm, red] (-4.5, 0.2) -- (-7.6, 2.1);

       \draw (-5, -3) node[scale=2] {then \bl{$|A|$ $\le$ $|B|$}};

       }

    \only<3>{

       \draw [<-, line width=1mm, red] (-7.5, 0.2) -- (-6.1, 2.1);

       \draw [<-, line width=1mm, red] (-7.3, 0.2) -- (-3.1, 2.1);

       \draw [<-, line width=1mm, red] (-6, 0.2) -- (-7.5, 2.1);

       \draw [<-, line width=1mm, red] (-4.5, 0.2) -- (-4.5, 2.1);

       \draw [<-, line width=1mm, red] (-4.3, 0.2) -- (-1.3, 2.1);

       \draw (-5, -3) node[scale=1.5] {\small{}for \bl{$=$}

        has to be a {\bf one-to-one} mapping};

       }

  \end{tikzpicture}

  \end{center}

\end{frame}

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\begin{frame}[c]

\frametitle{Cardinality}

\Large

\bl{$|A|$} $\dn$ ``how many elements''\bigskip\\

\bl{$A \subseteq B  \Rightarrow |A| \leq |B|$}\bigskip\\\pause

if there is an injective function \bl{$f: A \rightarrow B$} then \bl{$|A| \leq |B|$}\

\begin{center}

\bl{\large$\forall x y.\; f(x) = f(y) \Rightarrow x = y$}

\end{center}

\end{frame}

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\begin{frame}[t]

  \begin{center}

  \begin{tikzpicture}

  \draw (-6.6,2.5) node [scale=2.3] (X) 

    {\begin{tabular}{@ {\hspace{-3mm}}l}

     $A$ $=$ $\{$  

     \!\includegraphics[scale=0.02]{../pics/o1.jpg},

     \includegraphics[scale=0.02]{../pics/o2.jpg},

     \!\includegraphics[scale=0.02]{../pics/o3.jpg}

     $\}$

    \end{tabular}};

    \draw (-6.6,-0.5) node [scale=2.3] (Y) 

    {\begin{tabular}{@ {\hspace{-3mm}}l}

     $B$ $=$ $\{$  

     \!\includegraphics[scale=0.059]{../pics/a1.jpg},

     \includegraphics[scale=0.048]{../pics/a2.jpg},

     \includegraphics[scale=0.02]{../pics/a3.jpg}

     $\}$

     \end{tabular}};

   \onslide<3->{\draw (-7, -3) node[scale=1.5] 

      {then \bl{$|A|$ \alert{$=$} $|B|$}};}

     \only<1>{

       \draw [->, line width=1mm, red] (-7.4, 0.2) -- (-6.1, 2.1);

       \draw [->, line width=1mm, red] (-5.8, 0.2) -- (-4.3, 2.1);

       \draw [->, line width=1mm, red] (-4.5, 0.2) -- (-7.6, 2.1);

     }

    \only<2->{

       \draw [<-, line width=1mm, blue] (-7.5, 0.2) -- (-7.5, 2.1);

       \draw [<-, line width=1mm, blue] (-5.8, 0.2) -- (-4.3, 2.1);

       \draw [<-, line width=1mm, blue] (-4.5, 0.2) -- (-6.1, 2.1);

       }

  \end{tikzpicture}

  \end{center}

\end{frame}

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\begin{frame}[c]

\frametitle{Natural Numbers}

\Large

\bl{$\mathbb{N}$} \bl{$\dn$} \bl{$\{0, 1, 2, 3, .......\}$}\bigskip\pause 

\bl{$A$} is \alert{countable} iff \bl{$|A| \leq |\mathbb{N}|$}

\end{frame}

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\begin{frame}[c]

\frametitle{First Question}

\Large

\bl{$|\mathbb{N} - \{0\}|   \;\;\;\alert{?}\;\;\;  |\mathbb{N}| $}\bigskip\bigskip 

\large

\bl{$\geq$} or \bl{$\leq$} or \bl{$=$} ?

\bigskip\bigskip\bigskip\pause

\bl{$x$ $\mapsto$ $x + 1$},\\  \bl{$|\mathbb{N} - \{0\}|$ $=$  

$|\mathbb{N}|$}

\end{frame}

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\mode<presentation>{

\begin{frame}[c]

\Large

\bl{$|\mathbb{N} - \{0, 1\}|   \;\;\;\alert{?}\;\;\;  |\mathbb{N}| $}\bigskip\pause 

\bl{$|\mathbb{N} - \mathbb{O}|   \;\;\;\alert{?}\;\;\;  |\mathbb{N}| $}\bigskip\bigskip

\normalsize

\bl{$\mathbb{O}$} $\dn$ odd numbers\quad   \bl{$\{1,3,5......\}$}\\ \pause

\bl{$\mathbb{E}$} $\dn$ even numbers\quad   \bl{$\{0,2,4......\}$}\\

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\Large

\bl{$|\mathbb{N} \cup \mathbb{-N}|   \;\;\;\alert{?}\;\;\;  |\mathbb{N}| $}\bigskip\bigskip

\normalsize

\bl{$\mathbb{\phantom{-}N}$} $\dn$ positive numbers\quad   \bl{$\{0,1,2,3,......\}$}\\

\bl{$\mathbb{-N}$} $\dn$ negative numbers\quad   \bl{$\{0,-1,-2,-3,......\}$}\\

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\Large

\bl{$A$} is \alert{countable} if there exists an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip

\bl{$A$} is \alert{uncountable} if there does not exist an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip\bigskip 

countable:  \bl{$|A| \leq |\mathbb{N}|$}\\

uncountable:  \bl{$|A| > |\mathbb{N}|$}\pause\bigskip

Does there exist such an \bl{$A$} ?

\end{frame}}

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  \mode<presentation>{

  \begin{frame}[c]

  \frametitle{Hilbert's Hotel}

  \begin{center}

 \includegraphics[scale=0.8]{../pics/hilberts_hotel.jpg}

  \end{center}

  \begin{itemize}

  \item \ldots has as many rooms as there are natural numbers

  \end{itemize}

  \end{frame}}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

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\begin{frame}[t]

 \frametitle{\begin{tabular}{c}Real Numbers between\\[-2mm] 0 and 1\end{tabular}}

  \begin{center}

  \begin{tikzpicture}

  \draw [fill, color=black!50] (1,4) rectangle (2, 3);

  \draw [fill, color=black!50] (2,3) rectangle (3, 2);

  \draw [fill, color=black!50] (3,2) rectangle (4, 1);

  \draw [fill, color=black!50] (4,1) rectangle (5, 0);

  \draw (0, 0) grid (8, 5);

  \draw [line width = 1.mm] (1,0) -- (1, 5);    

  \draw [line width = 1.mm] (0, 4) -- (8, 4);    

  \draw (0.5,3.5) node {$1$};

  \draw (0.5,2.5) node {$2$};

  \draw (0.5,1.5) node {$3$};

  \draw (0.5,0.5) node {$4$};

  \draw (1.5,3.5) node {\only<1>{$3$}\only<2->{$4$}};

  \draw (2.5,3.5) node {$3$};

  \draw (3.5,3.5) node {$3$};

  \draw (4.5,3.5) node {$3$};

  \draw (5.5,3.5) node {$3$};

  \draw (6.5,3.5) node {$3$};

  \draw (7.5,3.5) node {$\ldots$};

  \draw (1.5,2.5) node {$1$};

  \draw (2.5,2.5) node {\only<1-2>{$2$}\only<3->{$3$}};

  \draw (3.5,2.5) node {$3$};

  \draw (4.5,2.5) node {$4$};

  \draw (5.5,2.5) node {$5$};

  \draw (6.5,2.5) node {$6$};

  \draw (7.5,2.5) node {$7$};

  \draw (1.5,1.5) node {$0$};

  \draw (2.5,1.5) node {$1$};

  \draw (3.5,1.5) node {\only<1-3>{$0$}\only<4->{$1$}};

  \draw (4.5,1.5) node {$1$};

  \draw (5.5,1.5) node {$0$};

  \draw (6.5,1.5) node {$\ldots$};

   \draw (1.5,0.5) node {$7$};

  \draw (2.5,0.5) node {$8$};

  \draw (3.5,0.5) node {$5$};

  \draw (4.5,0.5) node {\only<1-4>{$3$}\only<5->{$4$}};

  \draw (5.5,0.5) node {$9$};

  \draw (6.5,0.5) node {$\ldots$};

   \draw (1.5,-0.5) node {$\ldots$};

   \draw (8.5,3.5) node {$\ldots$};

  \end{tikzpicture}

  \end{center}

  \mbox{}\\[-20mm]\mbox{}

  \onslide<6->{

  \begin{center}

  \Large\bl{$|\mathbb{N}| < |R|$}

  \end{center}

  }

\end{frame}

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\mode<presentation>{

\begin{frame}[t]

 \frametitle{The Set of Problems}

  $\aleph_0$

  \begin{center}

  \begin{tikzpicture}

  \draw [fill, color=black!50] (1,4) rectangle (2, 3);

  \draw [fill, color=black!50] (2,3) rectangle (3, 2);

  \draw [fill, color=black!50] (3,2) rectangle (4, 1);

  \draw [fill, color=black!50] (4,1) rectangle (5, 0);

  \draw (0, 0) grid (8, 5);

  \draw [line width = 1.mm] (1,0) -- (1, 5);    

  \draw [line width = 1.mm] (0, 4) -- (8, 4);    

  \draw (0.5,3.5) node {$1$};

  \draw (0.5,2.5) node {$2$};

  \draw (0.5,1.5) node {$3$};

  \draw (0.5,0.5) node {$4$};

  \draw (1.5,4.5) node {$0$};

  \draw (2.5,4.5) node {$1$};

  \draw (3.5,4.5) node {$2$};

  \draw (4.5,4.5) node {$3$};

  \draw (5.5,4.5) node {$4$};

  \draw (6.5,4.5) node {$5$};

  \draw (7.5,4.5) node {$\ldots$}; 

  \draw (1.5,3.5) node {$0$};

  \draw (2.5,3.5) node {$1$};

  \draw (3.5,3.5) node {$0$};

  \draw (4.5,3.5) node {$1$};

  \draw (5.5,3.5) node {$0$};

  \draw (6.5,3.5) node {$1$};

  \draw (7.5,3.5) node {$\ldots$};

  \draw (1.5,2.5) node {$0$};

  \draw (2.5,2.5) node {$0$};

  \draw (3.5,2.5) node {$0$};

  \draw (4.5,2.5) node {$1$};

  \draw (5.5,2.5) node {$1$};

  \draw (6.5,2.5) node {$0$};

  \draw (7.5,2.5) node {$0$};

  \draw (1.5,1.5) node {$0$};

  \draw (2.5,1.5) node {$0$};

  \draw (3.5,1.5) node {$0$};

  \draw (4.5,1.5) node {$0$};

  \draw (5.5,1.5) node {$0$};

  \draw (6.5,1.5) node {$\ldots$};

   \draw (1.5,0.5) node {$1$};

  \draw (2.5,0.5) node {$1$};

  \draw (3.5,0.5) node {$0$};

  \draw (4.5,0.5) node {$1$};

  \draw (5.5,0.5) node {$1$};

  \draw (6.5,0.5) node {$\ldots$};

  \draw (1.5,-0.5) node {$\ldots$};

   \draw (8.5,3.5) node {$\ldots$};

  \end{tikzpicture}

  \end{center}

  \onslide<2>{

  \begin{center}

  \large \bl{|Progs| $=$ $|\mathbb{N}|$ $<$ |Probs|}

 \end{center}

  }

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{Halting Problem}

\large

Assume a program \bl{$H$} that decides for all programs \bl{$A$} and all 

input data \bl{$D$} whether\bigskip

\begin{itemize}

\item \bl{$H(A, D) \dn 1$} iff \bl{$A(D)$} terminates

\item \bl{$H(A, D) \dn 0$} otherwise

\end{itemize}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{Halting Problem (2)}

\large

Given such a program \bl{$H$} define the following program \bl{$C$}:

for all programs \bl{$A$}\bigskip

\begin{itemize}

\item \bl{$C(A) \dn 0$} iff \bl{$H(A, A) = 0$} 

\item \bl{$C(A) \dn$ loops} otherwise

\end{itemize}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{Contradiction}

\bl{$H(C, C)$} is either \bl{$0$} or \bl{$1$}.

\begin{itemize}

\item \bl{$H(C, C) = 1$} $\stackrel{\text{def}\,H}{\Rightarrow}$ \bl{$C(C)\downarrow$} $\stackrel{\text{def}\,C}{\Rightarrow}$ \bl{$H(C, C)=0$} 

\item \bl{$H(C, C) = 0$} $\stackrel{\text{def}\,H}{\Rightarrow}$ \bl{$C(C)$} loops $\stackrel{\text{def}\,C}{\Rightarrow}$\\ 

\hspace{7cm}\bl{$H(C, C)=1$} 

\end{itemize}

Contradiction in both cases. So \bl{$H$} cannot exist.

\end{frame}}

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  \mode<presentation>{

  \begin{frame}[c]

  \frametitle{Take Home Points}

 \large