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\begin{document}

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\begin{frame}[t]

\frametitle{%

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

  \\[-3mm]

  \LARGE Compilers and \\[-2mm] 

  \LARGE Formal Languages (10)\\[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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\mode<presentation>{

\begin{frame}[c]

\large\bf

Using a compiler, \\how can you mount the\\ perfect attack against a system?

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

{\large\bf

What is a \alert{perfect} attack?}\bigskip

\begin{enumerate}

\item you can potentially completely take over a target system

\item your attack is (nearly) undetectable

\item the victim has (almost) no chance to recover

\end{enumerate}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

  \begin{center}

  \begin{tikzpicture}[scale=1]

  \onslide<1->{

  \node (A) at (0,0)  [draw=black, rectangle, very thick, minimum height=18mm, minimum width=17mm] {};

  \node [below right] at (A.north west) {\footnotesize\begin{tabular}{@{}l@{}}

  \only<1,2>{clean}\only<3->{\alert{hacked}}\\compiler\end{tabular}};}

  \onslide<2->{

  \node (B) at (-2,2)  [draw=black, rectangle, very thick, minimum height=10mm, minimum width=12mm] {};

  \node [below right] at (B.north west) {\footnotesize\begin{tabular}{@{}l@{}}login\\(src)\end{tabular}};

  \node (C) at (2,2)  [draw=black, rectangle, very thick, minimum height=10mm, minimum width=12mm] {};

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 \onslide<3->{\node [above left=-1.5mm] at (C.south east) {\footnotesize \alert{$\blacksquare$}};}

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  \end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

  \begin{center}

  \begin{tikzpicture}[scale=1]

  \onslide<1->{

  \node (A) at (0,0)  [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};

  \node [below right] at (A.north west) {\small V0.01};

  \node [below right] (A1) at (A.south west) {\small Scala};

  \node [below right] (A1) at (A1.south west) {\small\textcolor{gray}{host language}};

  \node [above right] at (A.north west) {my compiler (src)};}

  \onslide<2->{

  \node (B) at (1.8,0)  [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};

  \node [below right] at (B.north west) {\small V0.02};

  \node [below right] at (B.south west) {\small Scala};

  \node at (3,0) {\ldots};

  \node (C) at (5,0)  [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};

  \node [below right] at (C.north west) {\small V1.00};

  \node [below right] at (C.south west) {\small Scala};}

  \onslide<3->{

  \node (D) at (6.8,0)  [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};

  \node [below right] at (D.north west) {\small V1.00};

  \node (E) at (6.8,2)  [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};

  \node [below right] at (E.north west) {\small V1.01};}

  \onslide<4->{

  \node (F) at (8.6,0)  [draw=black, rectangle, very thick, minimum height=18mm, minimum width=14mm] {};

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  \node at (9.8,0) {\ldots};

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  \node at (8,-2) {\textcolor{gray}{\begin{tabular}{@{}l@{}}no host language\\needed\end{tabular}}};

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  \end{center}

\end{frame}}

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

  \mode<presentation>{

  \begin{frame}<1-3>

  \frametitle{\LARGE\begin{tabular}{c}Hacking Compilers 

  \end{tabular}}

  %Why is it so paramount to have a small trusted code base (TCB)?

  \bigskip\bigskip

  \begin{columns}

  \begin{column}{2.7cm}

  \begin{minipage}{2.5cm}%

  \begin{tabular}{c@ {}}

  \includegraphics[scale=0.2]{../pics/ken-thompson.jpg}\\[-1.8mm]

  \footnotesize Ken Thompson\\[-1.8mm]

  \footnotesize Turing Award, 1983\\

  \end{tabular}

  \end{minipage}

  \end{column}

  \begin{column}{9cm}

  \begin{tabular}{l@ {\hspace{1mm}}p{8cm}}

  & Ken Thompson showed how to hide a Trojan Horse in a 

  compiler \textcolor{red}{without} leaving any traces in the source code.\\[2mm]

  & No amount of source level verification will protect 

  you from such Thompson-hacks.\\[2mm]

  \end{tabular}

  \end{column}

  \end{columns}

  \only<2>{

  \begin{textblock}{6}(4,2)

  \begin{tikzpicture}

  \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

  {\normalsize

  \begin{minipage}{8cm}

  \begin{quote}

  \includegraphics[scale=0.05]{../pics/evil.png}

  \begin{enumerate}

  \item[1)] Assume you ship the compiler as binary and also with sources.

  \item[2)] Make the compiler aware when it compiles itself.

  \item[3)] Add the Trojan horse.

  \item[4)] Compile.

  \item[5)] Delete Trojan horse from the sources of the compiler.

  \item[6)] Go on holiday for the rest of your life. ;o)\\[-7mm]\mbox{}

  \end{enumerate}

  \end{quote}

  \end{minipage}};

  \end{tikzpicture}

  \end{textblock}}

  \end{frame}}

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

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\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}

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\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}

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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 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}

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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}

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    {\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}};

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

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       \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);

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       \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]