\documentclass[dvipsnames,14pt,t]{beamer}
\usepackage{proof}
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\usepackage{isabelle}
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\usepackage[absolute,overlay]{textpos}
\usepackage{ifthen}
\usepackage{tikz}
\usepackage{courier}
\usepackage{listings}
\usetikzlibrary{arrows}
\usetikzlibrary{positioning}
\usetikzlibrary{calc}
\usetikzlibrary{shapes}
\usepackage{graphicx} 
\setmonofont[Scale=MatchLowercase]{Consolas}

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\newcommand{\isaliteral}[1]{}
\newcommand{\isactrlisub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}



\definecolor{javared}{rgb}{0.6,0,0} % for strings
\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments
\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords
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	basicstyle=\ttfamily,
	keywordstyle=\color{javapurple}\bfseries,
	stringstyle=\color{javagreen},
	commentstyle=\color{javagreen},
	morecomment=[s][\color{javadocblue}]{/**}{*/},
	numbers=left,
	numberstyle=\tiny\color{black},
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	numbers=left,
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%sudoku
\newcounter{row}
\newcounter{col}

\newcommand\setrow[9]{
        \setcounter{col}{1}
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            \edef\y{9.5 - \value{row}}
            \node[anchor=center] at (\x, \y) {\n};
            \stepcounter{col}
        }
        \stepcounter{row}
}

\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions

% beamer stuff 
\renewcommand{\slidecaption}{APP 06, King's College London, 12 November 2013}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[t]
\frametitle{%
  \begin{tabular}{@ {}c@ {}}
  \\
  \LARGE Access Control and \\[-3mm] 
  \LARGE Privacy Policies (6)\\[-6mm] 
  \end{tabular}}\bigskip\bigskip\bigskip

  %\begin{center}
  %\includegraphics[scale=1.3]{pics/barrier.jpg}
  %\end{center}

\normalsize
  \begin{center}
  \begin{tabular}{ll}
  Email:  & christian.urban at kcl.ac.uk\\
  Office: & S1.27 (1st floor Strand Building)\\
  Slides: & KEATS (also homework is there)\\
  \end{tabular}
  \end{center}


\end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[t]
  \frametitle{\Large\begin{tabular}{@ {}c@ {}}Access Control Logic\end{tabular}}
  
  Formulas
  
  \begin{itemize}
  \item[]
  
  \begin{center}\color{blue}
  \begin{tabular}[t]{rcl@ {\hspace{10mm}}l}
  \isa{F} & \isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}} & \isa{true} \\
            & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{false} \\
            & \isa{{\isaliteral{7C}{\isacharbar}}}   & \isa{F\ {\isaliteral{5C3C616E643E}{\isasymand}}\ F} \\
            & \isa{{\isaliteral{7C}{\isacharbar}}}   & \isa{F\ {\isaliteral{5C3C6F723E}{\isasymor}}\ F} \\
            & \isa{{\isaliteral{7C}{\isacharbar}}}   & \isa{F\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F}\\
            & \isa{{\isaliteral{7C}{\isacharbar}}}   & \isa{p\ {\isaliteral{28}{\isacharparenleft}}t\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}t\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}} \\  
            & \isa{{\isaliteral{7C}{\isacharbar}}}   & \alert{\isa{P\ says\ F}} & \textcolor{black}{``saying predicate''}\\ 
  \end{tabular}
  \end{center}
  
  \end{itemize}
  
Judgements

\begin{itemize}
\item[] \mbox{\hspace{9mm}}\bl{$\Gamma \vdash \text{F}$}
\end{itemize}
  
  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Judgements}

\begin{center}
\begin{tikzpicture}[scale=1]
  
  \draw (0.0,0.0) node {\LARGE \bl{$\Gamma \vdash F$}};
  \onslide<2->{
  \draw (-1,-0.3) node (X) {};
  \draw (-2.0,-2.0) node (Y) {};
  \draw (0.7,-3) node {\begin{tabular}{l}Gamma\\stands for a collection of formulas\\(``assumptions'')\end{tabular}};
  \draw[red, ->, line width = 2mm] (Y) -- (X);
 
  \draw (1.2,-0.1) node (X1) {};
  \draw (2.8,-0.1) node (Y1) {};
  \draw (4.5,-0.1) node {\begin{tabular}{l}a single formula\end{tabular}};
  \draw[red, ->, line width = 2mm] (Y1) -- (X1);

  \draw (-0.1,0.1) node (X2) {};
  \draw (0.5,1.5) node (Y2) {};
  \draw (1,1.8) node {\begin{tabular}{l}entails sign\end{tabular}};
  \draw[red, ->, line width = 2mm] (Y2) -- (X2);}
  
  \end{tikzpicture}
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Inference Rules}

\begin{center}
\begin{tikzpicture}[scale=1]
  
  \draw (0.0,0.0) node 
  {\Large\bl{\infer{\Gamma \vdash F_1 \wedge F_2}{\Gamma \vdash F_1 & \Gamma \vdash F_2}}};
 
  \draw (-0.1,-0.7) node (X) {};
  \draw (-0.1,-1.9) node (Y) {};
  \draw (-0.2,-2) node {\begin{tabular}{l}conclusion\end{tabular}};
  \draw[red, ->, line width = 2mm] (Y) -- (X);
 
  \draw (-1,0.6) node (X2) {};
  \draw (0.0,1.6) node (Y2) {};
  \draw (0,1.8) node {\begin{tabular}{l}premisses\end{tabular}};
  \draw[red, ->, line width = 2mm] (Y2) -- (X2);
   \draw (1,0.6) node (X3) {};
  \draw (0.0,1.6) node (Y3) {};
  \draw[red, ->, line width = 2mm] (Y3) -- (X3);
  \end{tikzpicture}
\end{center}

\only<2>{
\begin{textblock}{11}(1,13)
\small
\bl{$P \,\text{says}\, F \vdash Q\,\text{says}\, F\wedge P \,\text{says}\, G $}
\end{textblock}}
\only<3>{
\begin{textblock}{11}(1,13)
\small
\bl{$\underbrace{P \,\text{says}\, F}_{\Gamma} \vdash \underbrace{Q\,\text{says}\, F}_{F_1} \,\wedge
        \underbrace{P \,\text{says}\, G}_{F_2} $}
\end{textblock}}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Inference Rules}

\begin{center}
\bl{\infer{\Gamma, F\vdash F}{}}\bigskip\\

\bl{\infer{\Gamma \vdash F_2}{\Gamma \vdash F_1 \Rightarrow F_2 \quad \Gamma \vdash F_1}}
\qquad
\bl{\infer{\Gamma \vdash F_1 \Rightarrow F_2}{F_1, \Gamma \vdash F_2}}\bigskip\\

\bl{\infer{\Gamma \vdash P\,\text{says}\, F}{\Gamma \vdash F}}\medskip\\

\bl{\infer{\Gamma \vdash P \,\text{says}\, F_2}
              {\Gamma \vdash P \,\text{says}\, (F_1\Rightarrow F_2) \quad 
               \Gamma \vdash P \,\text{says}\, F_1}}
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Sending Messages}

  \begin{itemize}
  \item Alice sends a message \bl{$m$}
  \begin{center}
  \bl{Alice says $m$}
  \end{center}\medskip\pause

  \item Alice sends an encrypted message \bl{$m$} with key \bl{$K$} 
  (\bl{$\{m\}_K \dn K \Rightarrow m$})
  \begin{center}
  \bl{Alice says $\{m\}_K$}
  \end{center}\medskip\pause

  \item Decryption of Alice's message\smallskip
  \begin{center}
  \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}
              {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_K & \Gamma \vdash \text{Alice}\,\text{says}\,K}}}
  \end{center}
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Proofs}

\begin{center}
\bl{
\infer{\Gamma \vdash F}
         {\infer{\hspace{1cm}:\hspace{1cm}}
             {\infer{\hspace{1cm}:\hspace{1cm}}{:}
               &
              \infer{\hspace{1cm}:\hspace{1cm}}{:\quad :}
           }}
}
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{The Access Control Problem}


\begin{center}
  \begin{tikzpicture}[scale=1]
  
  \draw[line width=1mm] (-.3, -0.5) rectangle (1.5,2);
  \draw (-2.7,1) node {\begin{tabular}{l}access\\request\\ (\bl{$F$})\end{tabular}};
  \draw (4.2,1) node {\begin{tabular}{l}provable/\\not provable\end{tabular}};
  \draw (0.6,0.8) node {\footnotesize \begin{tabular}{l}AC-\\ Checker:\\ applies\\ inference\\ rules\end{tabular}};
 
  \draw[red, ->, line width = 2mm] (1.7,1) -- (2.7,1); 
  \draw[red,<-, line width = 2mm] (-0.6,1) -- (-1.6,1);
  \draw[red, <-, line width = 3mm] (0.6,2.2) -- (0.6,3.2); 
  
  \draw (0.6,4) node {\begin{tabular}{l}\large Access Policy (\bl{$\Gamma$})\end{tabular}};

  \end{tikzpicture}
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Proofs}

\begin{center}
\includegraphics[scale=0.4]{pics/river-stones.jpg}
\end{center}

\begin{textblock}{5}(11.7,5)
goal
\end{textblock}

\begin{textblock}{5}(11.7,14)
start
\end{textblock}

\begin{textblock}{5}(0,7)
\begin{center}
\bl{\infer[\small\textcolor{black}{\text{axiom}}]{\quad\vdash\quad}{}}\\[8mm]
\bl{\infer{\vdash}{\quad\vdash\quad}}\\[8mm]
\bl{\infer{\vdash}{\quad\vdash\qquad\vdash\quad}}
\end{center}
\end{textblock}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Sudoku}

\begin{tikzpicture}[scale=.5]
  \begin{scope}
    \draw (0, 0) grid (9, 9);
    \draw[very thick, scale=3] (0, 0) grid (3, 3);

    \setcounter{row}{1}
    \setrow { }{2}{ }  {5}{ }{1}  { }{9}{ }
    \setrow {8}{ }{ }  {2}{ }{3}  { }{ }{6}
    \setrow { }{3}{ }  { }{6}{ }  { }{7}{ }

    \setrow { }{ }{1}  { }{ }{ }  {6}{ }{ }
    \setrow {5}{4}{ }  { }{ }{ }  { }{1}{9}
    \setrow { }{ }{2}  { }{ }{ }  {7}{ }{ }

    \setrow { }{9}{ }  { }{3}{ }  { }{8}{ }
    \setrow {2}{ }{ }  {8}{ }{4}  { }{ }{7}
    \setrow { }{1}{ }  {9}{ }{7}  { }{6}{ }

    \fill[red, fill opacity=0.4] (4,0) rectangle (5,9);
    \fill[red, fill opacity=0.4] (0,5) rectangle (9,6);
    \fill[red!50, fill opacity=0.4] (3,3) rectangle (4,5);
    \fill[red!50, fill opacity=0.4] (5,3) rectangle (6,5);  
    \node[gray, anchor=center] at (4.5, -0.5) {columns};
    \node[gray, rotate=90, anchor=center] at (-0.6, 4.5, -0.5) {rows};
    \node[gray, anchor=center] at (4.5, 4.5) {box};
  \end{scope}
  \end{tikzpicture}

\small
\begin{textblock}{7}(9,3)
\begin{enumerate}
\item {\bf Row-Column:} each cell, must contain exactly one number
\item {\bf Row-Number:} each row must contain each number exactly once
\item {\bf Column-Number:} each column must contain each number exactly once
\item {\bf Box-Number:} each box must contain each number exactly once
\end{enumerate}
\end{textblock}


\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Solving Sudokus}

\begin{tikzpicture}[scale=.5]
  \begin{scope}
    \draw (0, 0) grid (9, 9);
    \draw[very thick, scale=3] (0, 0) grid (3, 3);

    \setcounter{row}{1}
    \setrow { }{ }{ }  {7}{ }{ }  { }{5}{8}
    \setrow {}{5}{6}  {2}{1}{8}  {7}{9}{3}
    \setrow { }{ }{ }  { }{ }{ }  {1}{ }{ }

    \setrow { }{ }{ }  { }{ }{ }  { }{8}{1}
    \setrow { }{ }{ }  {3}{7}{6}  { }{ }{ }
    \setrow {9}{6}{ }  { }{ }{ }  { }{ }{ }

    \setrow { }{ }{5}  { }{ }{ }  { }{ }{ }
    \setrow { }{ }{4}  { }{2}{1}  {8}{3}{ }
    \setrow {8}{7}{ }  { }{ }{3}  { }{ }{ }

    \fill[red, fill opacity=0.4] (0,7) rectangle (1,8);

  \end{scope}
  \end{tikzpicture}

\small
\begin{textblock}{6}(9,6)
{\bf single position rules}\\
\begin{center}
\bl{\infer{4\;\text{in empty position}}{\{1..9\} - \{4\}\;\text{in one row}}}
\end{center}

\onslide<2->{
\begin{center}
\bl{\infer{x\;\text{in empty position}}{\{1..9\} - \{x\}\;\text{in one column}}}\medskip\\
\bl{\infer{x\;\text{in empty position}}{\{1..9\} - \{x\}\;\text{in one box}}}
\end{center}}
\end{textblock}


\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Solving Sudokus}

\begin{tikzpicture}[scale=.5]
  \begin{scope}
    \draw (0, 0) grid (9, 9);
    \draw[very thick, scale=3] (0, 0) grid (3, 3);

    \setcounter{row}{1}
    \setrow { }{ }{ }  {7}{ }{ }  {\alert{\footnotesize 2}}{5}{8}
    \setrow {}{5}{6}  {2}{1}{8}  {7}{9}{3}
    \setrow { }{ }{ }  { }{ }{ }  {1}{\alert{\footnotesize 2}}{\alert{\footnotesize 2}}

    \setrow { }{ }{ }  { }{ }{ }  { }{8}{1}
    \setrow { }{ }{ }  {3}{7}{6}  { }{ }{ }
    \setrow {9}{6}{ }  { }{ }{ }  { }{ }{ }

    \setrow { }{ }{5}  { }{ }{ }  { }{ }{ }
    \setrow { }{ }{4}  { }{2}{1}  {8}{3}{ }
    \setrow {8}{7}{ }  { }{ }{3}  { }{ }{ }

  \end{scope}
  \end{tikzpicture}

\small
\begin{textblock}{6}(7.5,6)
{\bf candidate rules}\\
\begin{center}
\bl{\infer{x\;\text{candidate in empty positions}}{X - \{x\}\;\text{in one box} & X \subseteq \{1..9\}}}
\end{center}
\end{textblock}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Solving Sudokus}

\begin{tikzpicture}[scale=.5]
  \begin{scope}
    \draw (0, 0) grid (9, 9);
    \draw[very thick, scale=3] (0, 0) grid (3, 3);

    \setcounter{row}{1}
    \setrow { }{ }{ }  {7}{ }{ }  {\alert{\footnotesize 2}}{5}{8}
    \setrow {\alert{4}}{5}{6}  {2}{1}{8}  {7}{9}{3}
    \setrow { }{ }{ }  { }{ }{ }  {1}{\alert{\footnotesize 2}}{\alert{\footnotesize 2}}

    \setrow { }{ }{ }  { }{ }{ }  { }{8}{1}
    \setrow { }{ }{ }  {3}{7}{6}  { }{ }{ }
    \setrow {9}{6}{ }  { }{ }{ }  { }{ }{ }

    \setrow { }{ }{5}  { }{ }{ }  { }{ }{ }
    \setrow { }{ }{4}  { }{2}{1}  {8}{3}{ }
    \setrow {8}{7}{ }  { }{ }{3}  { }{ }{ }

  \end{scope}
  \end{tikzpicture}

\small
\begin{textblock}{6}(7.5,6)
\begin{center}
\bl{\infer{4\;\text{in empty position}}{\{1..9\} - \{4\}\;\text{in one row}}}\bigskip\\
\bl{\infer{2\;\text{candidate in empty positions}}{X - \{2\}\;\text{in one box} & X \subseteq \{1..9\}}}
\end{center}
\end{textblock}


\begin{textblock}{3}(13.5,6.8)
  \begin{tikzpicture}
  \onslide<1>{\node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{\mbox{\alert{a}}};}
  \onslide<2>{\node at (0,0) [single arrow, shape border rotate=90, fill=red,text=white]{\mbox{\alert{a}}};}
  \end{tikzpicture}
\end{textblock}

\begin{textblock}{3}(14.5,9.3)
  \begin{tikzpicture}
  \onslide<1>{\node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{\mbox{\alert{a}}};}
  \onslide<2>{\node at (0,0) [single arrow, shape border rotate=90, fill=red,text=white]{\mbox{\alert{a}}};}
  \end{tikzpicture}
\end{textblock}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Solving Sudokus}

\begin{tikzpicture}[scale=.5]
  \begin{scope}
    \draw (0, 0) grid (9, 9);
    \draw[very thick, scale=3] (0, 0) grid (3, 3);

    \setcounter{row}{1}
    \setrow { }{ }{ }  {7}{ }{ }  { }{5}{8}
    \setrow { }{5}{6}  {2}{1}{8}  {7}{9}{3}
    \setrow { }{ }{ }  { }{ }{ }  {1}{ }{ }

    \setrow { }{ }{ }  { }{ }{ }  { }{8}{1}
    \setrow { }{ }{ }  {3}{7}{6}  { }{ }{ }
    \setrow {9}{6}{ }  { }{ }{ }  { }{ }{ \alert{2}}

    \setrow { }{ }{5}  { }{ }{ }  { }{ }{ }
    \setrow { }{ }{4}  { }{2}{1}  {8}{3}{ }
    \setrow {8}{7}{ }  { }{ }{3}  { }{ }{ }

  \end{scope}
  \end{tikzpicture}

\small
\begin{textblock}{6}(7.5,6)
\begin{center}
\bl{\infer{2\;\text{candidate}}{X - \{2\}\;\text{in one box} & X \subseteq \{1..9\}}}
\end{center}
\end{textblock}

\begin{textblock}{3}(14.5,8.3)
  \begin{tikzpicture}
  \onslide<1>{\node at (0,0) [single arrow, shape border rotate=90, fill=red,text=white]{\mbox{\alert{a}}};}
  \end{tikzpicture}
\end{textblock}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{BTW}

Are there sudokus that cannot be solved?\pause

\begin{center}
\begin{tikzpicture}[scale=.5]
  \begin{scope}
    \draw (0, 0) grid (9, 9);
    \draw[very thick, scale=3] (0, 0) grid (3, 3);

    \setcounter{row}{1}
    \setrow {1}{2}{3}  {4}{5}{6}  {7}{8}{ }
    \setrow { }{ }{ }  { }{ }{ }  { }{ }{2}
    \setrow { }{ }{ }  { }{ }{ }  { }{ }{3}

    \setrow { }{ }{ }  { }{ }{ }  { }{ }{4}
    \setrow { }{ }{ }  { }{ }{ }  { }{ }{5}
    \setrow { }{ }{ }  { }{ }{ }  { }{ }{6}

    \setrow { }{ }{ }  { }{ }{ }  { }{ }{7}
    \setrow { }{ }{ }  { }{ }{ }  { }{ }{8}
    \setrow { }{ }{ }  { }{ }{ }  { }{ }{9}

  \end{scope}
  \end{tikzpicture}
\end{center}

Sometimes no rules apply at all....unsolvable sudoku.


\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Example Proof}

\begin{center}
\bl{\infer{P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2 \wedge P\;\text{says}\;F_1}
         {\raisebox{2mm}{\text{\LARGE $?$}}}}
\end{center}


\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Example Proof}

\begin{tabular}{@{\hspace{-6mm}}l}
\begin{minipage}{1.1\textwidth}
We have (by axiom)

\begin{center}
\begin{tabular}{@{}ll@{}}
(1) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2$}
\end{tabular}
\end{center}

From (1) we get

\begin{center}
\begin{tabular}{@{}ll@{}}
(2) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash P\;\text{says}\;F_1$}\\
(3) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2$}\\
\end{tabular}
\end{center}

From (3) and (2) we get

\begin{center}
\begin{tabular}{@{}ll@{}}
\bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2 \wedge P\;\text{says}\;F_1$}
\end{tabular}
\end{center}

Done.
\end{minipage}
\end{tabular}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Other Direction}

\begin{tabular}{@{\hspace{-6mm}}l}
\begin{minipage}{1.1\textwidth}
We want to prove

\begin{center}
\begin{tabular}{@{}ll@{}}
\bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2 \wedge P\;\text{says}\;F_1$}
\end{tabular}
\end{center}

We better be able to prove:

\begin{center}
\begin{tabular}{@{}ll@{}}
(1) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2$}\\
(2) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash P\;\text{says}\;F_1$}\\
\end{tabular}
\end{center}

For (1): If we can prove

\begin{center}
\begin{tabular}{@{}ll@{}}
\bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2 \wedge P\;\text{says}\;F_1$}
\end{tabular}
\end{center}

then (1) is fine. Similarly for (2).
\end{minipage}
\end{tabular}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
     
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]

I want to prove

\begin{center}
\bl{$\Gamma \vdash \text{del\_file}$}
\end{center}\pause

There is an inference rule

\begin{center}
\bl{\infer{\Gamma \vdash P \,\text{says}\, F}{\Gamma \vdash F}}
\end{center}\pause

So I can derive \bl{$\Gamma \vdash \text{Alice} \,\text{says}\,\text{del\_file}$}.\bigskip\pause

\bl{$\Gamma$} contains already \bl{$\text{Alice} \,\text{says}\,\text{del\_file}$}. \\
So I can use the rule

\begin{center}
\bl{\infer{\Gamma, F \vdash F}{}}
\end{center}

\onslide<5>{\bf\alert{What is wrong with this?}}
\hfill{\bf Done. Qed.}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%        

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Program}

How to prove \bl{$\Gamma \vdash F$}?\bigskip\bigskip

\begin{center}
\Large \bl{\infer{\Gamma, F\vdash F}{}}
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

\begin{center}
\Large 
\bl{\infer{\Gamma \vdash F_1 \Rightarrow F_2}{F_1, \Gamma \vdash F_2}}
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

\begin{center}
\Large 
\bl{\infer{\Gamma \vdash P \,\text{says}\, F}{\Gamma \vdash F}}
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

\begin{center}
\Large 
\bl{\infer{\Gamma \vdash F_1 \vee F_2}{\Gamma \vdash F_1}}\qquad
\bl{\infer{\Gamma \vdash F_1 \vee F_2}{\Gamma \vdash F_2}}\
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

\begin{center}
\Large 
\bl{\infer{\Gamma \vdash F_1 \wedge F_2}{\Gamma \vdash F_1 \quad \Gamma \vdash F_2}}
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  


     
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{Program: \texttt{prove2}}

I want to prove \bl{$\Gamma \vdash \text{Pred}$}\bigskip\bigskip\pause

\begin{enumerate}
\item I found that \bl{$\Gamma$} contains the assumption \bl{$F_1 \Rightarrow F_2$}\bigskip\pause
\item If I can prove \bl{$\Gamma \vdash F_1$},\pause{} then I can prove
\begin{center}
\bl{$\Gamma \vdash F_2$}
\end{center}\bigskip\pause

\item So I am able to try to prove \bl{$\Gamma \vdash \text{Pred}$} with the additional assumption
\bl{$F_2$}.\bigskip
\begin{center}
\bl{$F_2, \Gamma \vdash \text{Pred}$}
\end{center}
\end{enumerate}

\only<4>{
\begin{textblock}{11}(1,10.5)
\bl{\infer{\Gamma\vdash F_2}{\Gamma\vdash F_1\Rightarrow F_2 & \Gamma\vdash F_1}}
\end{textblock}}


\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%       
     
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{}

  Recall the following scenario:

  \begin{itemize}
  \item If \textcolor{blue}{Admin} says that \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{} {}}} 
  should be deleted, then this file must be deleted.
  \item \textcolor{blue}{Admin} trusts \textcolor{blue}{Bob} to decide whether 
  \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}}} should be deleted.
  \item \textcolor{blue}{Bob} wants to delete \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}}}.
  \end{itemize}\bigskip

  \small
  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{3D}{\isacharequal}}}\small\begin{tabular}{l}
  \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}},\\
  \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}},\\
  \isa{Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}}\\
  \end{tabular}}\medskip

  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}}}
  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

\begin{itemize}
\item \bl{$P \,\text{says}\, F$} means \bl{$P$} can send a ``signal'' \bl{$F$} through a wire, or
can make a statement \bl{$F$}\bigskip

\item \bl{$P$} is entitled to do \bl{$F$}\smallskip\\ 
\bl{$P \,\text{controls}\, F \,\dn\, (P\,\text{says}\, F) \Rightarrow F$}\medskip

\begin{center}
\bl{\infer{\Gamma \vdash F}{\Gamma \vdash P\,\text{controls}\, F & \Gamma \vdash P\,\text{says}\,F}}
\end{center}


\end{itemize}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{Trusted Third Party}

Simple protocol for establishing a secure connection via a mutually
trusted 3rd party (server):

\begin{center}
\begin{tabular}{@ {\hspace{-7mm}}l@{\hspace{2mm}}r@ {\hspace{1mm}}l}
Message 1 & \bl{$A \rightarrow S :$} & \bl{$A, B$}\\
Message 2 & \bl{$S \rightarrow A :$} & \bl{$\{K_{AB}\}_{K_{AS}}$} and \bl{$\{\{K_{AB}\}_{K_{BS}} \}_{K_{AS}}$}\\
Message 3 & \bl{$A \rightarrow B :$} & \bl{$\{K_{AB}\}_{K_{BS}} $}\\
Message 4 & \bl{$A \rightarrow B :$} & \bl{$\{m\}_{K_{AB}}$}\\
\end{tabular}
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Sending Rule}

  \bl{\begin{center}
  \mbox{\infer{\Gamma \vdash Q \;\text{says}\; F}
              {\Gamma \vdash P \;\text{says}\; F & \Gamma \vdash P \;\text{sends}\; Q : F}}
  \end{center}}\bigskip\pause
  
  \bl{$P \,\text{sends}\, Q : F \dn$}\\
  \hspace{6mm}\bl{$(P \,\text{says}\, F) \Rightarrow (Q \,\text{says}\, F)$}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Trusted Third Party}

  \begin{center}
  \bl{\begin{tabular}{l}
  $A$ sends $S$ : $\text{Connect}(A,B)$\\  
  \bl{$S \,\text{says}\, (\text{Connect}(A,B) \Rightarrow$}\\ 
  \hspace{2.5cm}\bl{$\{K_{AB}\}_{K_{AS}} \wedge 
  \{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}})$}\\
 $S$ sends $A$ : $\{K_{AB}\}_{K_{AS}}$ \bl{$\wedge$} $\{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}}$\\
  $A$ sends $B$ : $\{K_{AB}\}_{K_{BS}}$\\
  $A$ sends $B$ : $\{m\}_{K_{AB}}$
  \end{tabular}}
  \end{center}\bigskip\pause
  
  
  \bl{$\Gamma \vdash B \,\text{says} \, m$}?
  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      

   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Public/Private Keys}

  \begin{itemize}
  \item Bob has a private and public key: \bl{$K_{Bob}^{pub}$}, \bl{$K_{Bob}^{priv}$}\bigskip
  \begin{center}
  \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}
              {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_{K_{Bob}^{pub}} & 
               \Gamma \vdash K_{Bob}^{priv}}}}
  \end{center}\bigskip\pause

  \item this is {\bf not} a derived rule! 
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Security Levels}
  \small

  \begin{itemize}
  \item Top secret (\bl{$T\!S$})
  \item Secret (\bl{$S$})
  \item Public (\bl{$P$})
  \end{itemize}

  \begin{center}
  \bl{$slev(P) < slev(S) < slev(T\!S)$}\pause
  \end{center}

  \begin{itemize}
  \item Bob has a clearance for ``secret''
  \item Bob can read documents that are public or sectret, but not top secret
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Reading a File}

  \bl{\begin{center}
  \begin{tabular}{c}
  \begin{tabular}{@ {}l@ {}}
  \only<2->{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$}}\\ 
  \only<2->{\hspace{3cm}}Bob controls Permitted $($File, read$)$\\
  Bob says Permitted $($File, read$)$\only<2->{\\}
  \only<2>{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$}}%
  \only<3>{\textcolor{red}{$slev($File$)$ $=$ $P$}\\}%
  \only<3>{\textcolor{red}{$slev($Bob$)$ $=$ $S$}\\}%
  \only<3>{\textcolor{red}{$slev(P)$ $<$ $slev(S)$}\\}%
  \end{tabular}\\
  \hline
  Permitted $($File, read$)$
  \end{tabular}
  \end{center}}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Substitution Rule}
  \small
  
  \bl{\begin{center}
  \begin{tabular}{c}
  $\Gamma \vdash slev(P) = l_1$ \hspace{4mm} $\Gamma \vdash slev(Q) = l_2$
  \hspace{4mm} $\Gamma \vdash l_1 < l_2$\\\hline
  $\Gamma \vdash slev(P) < slev(Q)$
  \end{tabular}
  \end{center}}\bigskip\pause

  \begin{itemize}
  \item \bl{$slev($Bob$)$ $=$ $S$}
  \item \bl{$slev($File$)$ $=$ $P$}
  \item \bl{$slev(P) < slev(S)$}
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Reading a File}

  \bl{\begin{center}
  \begin{tabular}{c}
  \begin{tabular}{@ {}l@ {}}
  $slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$\\ 
  \hspace{3cm}Bob controls Permitted $($File, read$)$\\
  Bob says Permitted $($File, read$)$\\
  $slev($File$)$ $=$ $P$\\
  $slev($Bob$)$ $=$ $T\!S$\\
  \only<1>{\textcolor{red}{$?$}}%
  \only<2>{\textcolor{red}{$slev(P) < slev(S)$}\\}%
  \only<2>{\textcolor{red}{$slev(S) < slev(T\!S)$}}%
  \end{tabular}\\
  \hline
  Permitted $($File, read$)$
  \end{tabular}
  \end{center}}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Transitivity Rule}
  \small
  
  \bl{\begin{center}
  \begin{tabular}{c}
  $\Gamma \vdash l_1 < l_2$ 
  \hspace{4mm} $\Gamma \vdash l_2 < l_3$\\\hline
  $\Gamma \vdash l_1 < l_3$
  \end{tabular}
  \end{center}}\bigskip

  \begin{itemize}
  \item \bl{$slev(P) < slev (S)$}
  \item \bl{$slev(S) < slev (T\!S)$}
  \item[] \bl{$slev(P) < slev (T\!S)$}
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Reading Files}

  \begin{itemize}
  \item Access policy for reading
  \end{itemize}

  \bl{\begin{center}
  \begin{tabular}{c}
  \begin{tabular}{@ {}l@ {}}
  $\forall f.\;slev(f)$ \only<1>{$<$}\only<2>{\textcolor{red}{$\le$}} $slev($Bob$)$ $\Rightarrow$\\ 
  \hspace{3cm}Bob controls Permitted $(f$, read$)$\\
  Bob says Permitted $($File, read$)$\\
  $slev($File$)$ $=$ \only<1>{$P$}\only<2>{\textcolor{red}{$T\!S$}}\\
  $slev($Bob$)$ $=$ $T\!S$\\
  $slev(P) < slev(S)$\\
  $slev(S) < slev(T\!S)$
  \end{tabular}\\
  \hline
  Permitted $($File, read$)$
  \end{tabular}
  \end{center}}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Writing Files}

  \begin{itemize}
  \item Access policy for \underline{writing}
  \end{itemize}

  \bl{\begin{center}
  \begin{tabular}{c}
  \begin{tabular}{@ {}l@ {}}
  $\forall f.\;slev($Bob$)$ $\le$ $slev(f)$ $\Rightarrow$\\ 
  \hspace{3cm}Bob controls Permitted $(f$, write$)$\\
  Bob says Permitted $($File, write$)$\\
  $slev($File$)$ $=$ $T\!S$\\
  $slev($Bob$)$ $=$ $S$\\
  $slev(P) < slev(S)$\\
  $slev(S) < slev(T\!S)$
  \end{tabular}\\
  \hline
  Permitted $($File, write$)$
  \end{tabular}
  \end{center}}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

  
\end{document}
  
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Encryption}

  \begin{itemize}
  \item Encryption of a message\smallskip
  \begin{center}
  \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_K}
              {\Gamma \vdash \text{Alice}\;\text{says}\;m & \Gamma \vdash \text{Alice}\,\text{says}\,K}}}
  \end{center}
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
  
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Public/Private Keys}

  \begin{itemize}
  \item Bob has a private and public key: \bl{$K_{Bob}^{pub}$}, \bl{$K_{Bob}^{priv}$}\bigskip
  \begin{center}
  \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}
              {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_{K_{Bob}^{pub}} & 
               \Gamma \vdash K_{Bob}^{priv}}}}
  \end{center}\bigskip\pause

  \item this is {\bf not} a derived rule! 
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
  
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Trusted Third Party}

  \begin{itemize}
  \item Alice calls Sam for a key to communicate with Bob
  \item Sam responds with a key that Alice can read and a key Bob can read (pre-shared)
  \item Alice sends the message encrypted with the key and the second key it recieved
  \end{itemize}\bigskip

  \begin{center}
  \bl{\begin{tabular}{lcl}
  $A$ sends $S$ &:& $\textit{Connect}(A,B)$\\
  $S$ sends $A$ &:& $\{K_{AB}\}_{K_{AS}}$ \textcolor{black}{and} $\{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}}$\\
  $A$ sends $B$ &:& $\{K_{AB}\}_{K_{BS}}$\\
  $A$ sends $B$ &:& $\{m\}_{K_{AB}}$
  \end{tabular}}
  \end{center}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
  
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Controls}
  \small
  
  \begin{itemize}
  \item \bl{\isa{P\ controls\ F\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{28}{\isacharparenleft}}P\ says\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F}}

  \item its meaning ``\bl{P} is entitled to do \bl{F}''
  \item if \bl{P controls F} and \bl{P says F} then \bl{F}\pause

  \begin{center}
  \bl{\mbox{
  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}}
        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ controls\ F}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F}}}
  }}
  \end{center}\pause

  \begin{center}
  \bl{\mbox{
  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}}
        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{28}{\isacharparenleft}}P\ says\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F}}}
  }}
  \end{center}
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Security Levels}
  \small

  \begin{itemize}
  \item Top secret (\bl{$T\!S$})
  \item Secret (\bl{$S$})
  \item Public (\bl{$P$})
  \end{itemize}

  \begin{center}
  \bl{$slev(P) < slev(S) < slev(T\!S)$}\pause
  \end{center}

  \begin{itemize}
  \item Bob has a clearance for ``secret''
  \item Bob can read documents that are public or sectret, but not top secret
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Reading a File}

  \bl{\begin{center}
  \begin{tabular}{c}
  \begin{tabular}{@ {}l@ {}}
  \only<2->{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$}}\\ 
  \only<2->{\hspace{3cm}}Bob controls Permitted $($File, read$)$\\
  Bob says Permitted $($File, read$)$\only<2->{\\}
  \only<2>{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$}}%
  \only<3>{\textcolor{red}{$slev($File$)$ $=$ $P$}\\}%
  \only<3>{\textcolor{red}{$slev($Bob$)$ $=$ $S$}\\}%
  \only<3>{\textcolor{red}{$slev(P)$ $<$ $slev(S)$}\\}%
  \end{tabular}\\
  \hline
  Permitted $($File, read$)$
  \end{tabular}
  \end{center}}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Substitution Rule}
  \small
  
  \bl{\begin{center}
  \begin{tabular}{c}
  $\Gamma \vdash slev(P) = l_1$ \hspace{4mm} $\Gamma \vdash slev(Q) = l_2$
  \hspace{4mm} $\Gamma \vdash l_1 < l_2$\\\hline
  $\Gamma \vdash slev(P) < slev(Q)$
  \end{tabular}
  \end{center}}\bigskip\pause

  \begin{itemize}
  \item \bl{$slev($Bob$)$ $=$ $S$}
  \item \bl{$slev($File$)$ $=$ $P$}
  \item \bl{$slev(P) < slev(S)$}
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Reading a File}

  \bl{\begin{center}
  \begin{tabular}{c}
  \begin{tabular}{@ {}l@ {}}
  $slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$\\ 
  \hspace{3cm}Bob controls Permitted $($File, read$)$\\
  Bob says Permitted $($File, read$)$\\
  $slev($File$)$ $=$ $P$\\
  $slev($Bob$)$ $=$ $T\!S$\\
  \only<1>{\textcolor{red}{$?$}}%
  \only<2>{\textcolor{red}{$slev(P) < slev(S)$}\\}%
  \only<2>{\textcolor{red}{$slev(S) < slev(T\!S)$}}%
  \end{tabular}\\
  \hline
  Permitted $($File, read$)$
  \end{tabular}
  \end{center}}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Transitivity Rule}
  \small
  
  \bl{\begin{center}
  \begin{tabular}{c}
  $\Gamma \vdash l_1 < l_2$ 
  \hspace{4mm} $\Gamma \vdash l_2 < l_3$\\\hline
  $\Gamma \vdash l_1 < l_3$
  \end{tabular}
  \end{center}}\bigskip

  \begin{itemize}
  \item \bl{$slev(P) < slev (S)$}
  \item \bl{$slev(S) < slev (T\!S)$}
  \item[] \bl{$slev(P) < slev (T\!S)$}
  \end{itemize}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Reading Files}

  \begin{itemize}
  \item Access policy for reading
  \end{itemize}

  \bl{\begin{center}
  \begin{tabular}{c}
  \begin{tabular}{@ {}l@ {}}
  $\forall f.\;slev(f)$ \only<1>{$<$}\only<2>{\textcolor{red}{$\le$}} $slev($Bob$)$ $\Rightarrow$\\ 
  \hspace{3cm}Bob controls Permitted $(f$, read$)$\\
  Bob says Permitted $($File, read$)$\\
  $slev($File$)$ $=$ \only<1>{$P$}\only<2>{\textcolor{red}{$T\!S$}}\\
  $slev($Bob$)$ $=$ $T\!S$\\
  $slev(P) < slev(S)$\\
  $slev(S) < slev(T\!S)$
  \end{tabular}\\
  \hline
  Permitted $($File, read$)$
  \end{tabular}
  \end{center}}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Writing Files}

  \begin{itemize}
  \item Access policy for \underline{writing}
  \end{itemize}

  \bl{\begin{center}
  \begin{tabular}{c}
  \begin{tabular}{@ {}l@ {}}
  $\forall f.\;slev($Bob$)$ $\le$ $slev(f)$ $\Rightarrow$\\ 
  \hspace{3cm}Bob controls Permitted $(f$, write$)$\\
  Bob says Permitted $($File, write$)$\\
  $slev($File$)$ $=$ $T\!S$\\
  $slev($Bob$)$ $=$ $S$\\
  $slev(P) < slev(S)$\\
  $slev(S) < slev(T\!S)$
  \end{tabular}\\
  \hline
  Permitted $($File, write$)$
  \end{tabular}
  \end{center}}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

  
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Sending Rule}

  \bl{\begin{center}
  \mbox{\infer{\Gamma \vdash Q \;\textit{says}\; F}
              {\Gamma \vdash P \;\textit{says}\; F & \Gamma \vdash P \;\textit{sends}\; Q : F}}
  \end{center}}\bigskip\pause
  
  \bl{$P \,\text{sends}\, Q : F \dn$}\\
  \hspace{6mm}\bl{$(P \,\text{says}\, F) \Rightarrow (Q \,\text{says}\, F)$}

  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mode<presentation>{
  \begin{frame}[c]
  \frametitle{Trusted Third Party}

  \begin{center}
  \bl{\begin{tabular}{l}
  $A$ sends $S$ : $\textit{Connect}(A,B)$\\  
  \bl{$S \,\text{says}\, (\textit{Connect}(A,B) \Rightarrow$}\\ 
  \hspace{2.5cm}\bl{$\{K_{AB}\}_{K_{AS}} \wedge 
  \{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}})$}\\
 $S$ sends $A$ : $\{K_{AB}\}_{K_{AS}}$ \bl{$\wedge$} $\{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}}$\\
  $A$ sends $B$ : $\{K_{AB}\}_{K_{BS}}$\\
  $A$ sends $B$ : $\{m\}_{K_{AB}}$
  \end{tabular}}
  \end{center}\bigskip\pause
  
  
  \bl{$\Gamma \vdash B \,\text{says} \, m$}?
  \end{frame}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
\end{document}

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