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%sudoku
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\begin{document}
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\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}}
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\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}}
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\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}};
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\draw (0.5,1.5) node (Y2) {};
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\end{tikzpicture}
\end{center}
\end{frame}}
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\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}}};
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\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} $}
\end{textblock}}
\end{frame}}
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\mode<presentation>{
\begin{frame}[c]
\frametitle{Inference Rules}
\begin{center}
\bl{\infer{\Gamma, F\vdash F}{}}\bigskip\\
\bl{\infer{\Gamma \vdash F_2}{\Gamma \vdash F_1 \Rightarrow F_2 \quad \Gamma \vdash F_1}}
\qquad
\bl{\infer{\Gamma \vdash F_1 \Rightarrow F_2}{F_1, \Gamma \vdash F_2}}\bigskip\\
\bl{\infer{\Gamma \vdash P\,\text{says}\, F}{\Gamma \vdash F}}\medskip\\
\bl{\infer{\Gamma \vdash P \,\text{says}\, F_2}
{\Gamma \vdash P \,\text{says}\, (F_1\Rightarrow F_2) \quad
\Gamma \vdash P \,\text{says}\, F_1}}
\end{center}
\end{frame}}
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\mode<presentation>{
\begin{frame}[c]
\frametitle{Sending Messages}
\begin{itemize}
\item Alice sends a message \bl{$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}}
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\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}}
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\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}};
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\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}}
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\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}}
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\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}}
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\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} { }{3}{ } { }{ }{ }
\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}}
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\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} { }{3}{ } { }{ }{ }
\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}}
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\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} { }{3}{ } { }{ }{ }
\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}}
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\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} { }{3}{ } { }{ }{ }
\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}}
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\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}}
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\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}}
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\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}}
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\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{$P$} 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}}
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%
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\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}}
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\end{document}
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