--- a/slides/slides10.tex Tue Dec 05 13:49:47 2017 +0000
+++ b/slides/slides10.tex Wed Dec 06 00:06:37 2017 +0000
@@ -195,8 +195,6 @@
& No amount of source level verification will protect
you from such Thompson-hacks.\\[2mm]
- & Therefore in safety-critical systems it is important to rely
- on only a very small TCB.
\end{tabular}
\end{column}
\end{columns}
@@ -225,8 +223,549 @@
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{Compilers \& Boeings 777}
+
+First flight in 1994. They want to achieve triple redundancy in hardware
+faults.\bigskip
+
+They compile 1 Ada program to\medskip
+
+\begin{itemize}
+\item Intel 80486
+\item Motorola 68040 (old Macintosh's)
+\item AMD 29050 (RISC chips used often in laser printers)
+\end{itemize}\medskip
+
+using 3 independent compilers.\bigskip\pause
+
+\small Airbus uses C and static analysers. Recently started using CompCert.
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+
+{\Large\bf
+How many strings are in \bl{$L(a^*)$}?}\bigskip\pause
+
+\normalsize
+\begin{center}
+\begin{tabular}{llllll}
+ \bl{$[]$} & \bl{$a$} & \bl{$aa$} & \bl{$aaa$} & \bl{$aaaa$} & \ldots\\
+ \bl{0} & \bl{1} & \bl{2} & \bl{3} & \bl{4} & \ldots
+\end{tabular}
+\end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{Subsets}
+
+\Large
+If \bl{$A \subseteq B$} then \bl{$A$} has fewer or equal elements
+than \bl{$B$}\bigskip\bigskip
+
+\Large
+\bl{$A \subseteq B$} and \bl{$B \subseteq A$}\bigskip
+
+then \bl{$A = B$}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \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}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Take Home Points}
+ \large
+
+ \begin{itemize}
+ \item there are sets that are more infinite than others\bigskip
+ \item even with the most powerful computer we can imagine, there
+ are problems that cannot be solved by any program\bigskip\bigskip
+
+ \item in CS we actually hit quite often such problems (halting problem)
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\end{document}
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