
diff -r be35ff24cccc -r d1d07f05325a slides/slides05.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/slides/slides05.tex	Sun Sep 22 15:22:11 2013 +0100
@@ -0,0 +1,1296 @@
+\documentclass[dvipsnames,14pt,t]{beamer}
+\usepackage{proof}
+\usepackage{beamerthemeplainculight}
+\usepackage[T1]{fontenc}
+\usepackage[latin1]{inputenc}
+\usepackage{mathpartir}
+\usepackage{isabelle}
+\usepackage{isabellesym}
+\usepackage[absolute,overlay]{textpos}
+\usepackage{ifthen}
+\usepackage{tikz}
+\usepackage{courier}
+\usepackage{listings}
+\usetikzlibrary{arrows}
+\usetikzlibrary{positioning}
+\usetikzlibrary{calc}
+\usepackage{graphicx} 
+
+\isabellestyle{rm}
+\renewcommand{\isastyle}{\rm}%
+\renewcommand{\isastyleminor}{\rm}%
+\renewcommand{\isastylescript}{\footnotesize\rm\slshape}%
+\renewcommand{\isatagproof}{}
+\renewcommand{\endisatagproof}{}
+\renewcommand{\isamarkupcmt}[1]{#1}
+
+% Isabelle characters
+\renewcommand{\isacharunderscore}{\_}
+\renewcommand{\isacharbar}{\isamath{\mid}}
+\renewcommand{\isasymiota}{}
+\renewcommand{\isacharbraceleft}{\{}
+\renewcommand{\isacharbraceright}{\}}
+\renewcommand{\isacharless}{$\langle$}
+\renewcommand{\isachargreater}{$\rangle$}
+\renewcommand{\isasymsharp}{\isamath{\#}}
+\renewcommand{\isasymdots}{\isamath{...}}
+\renewcommand{\isasymbullet}{\act}
+
+
+
+\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
+\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc
+
+\lstset{language=Java,
+	basicstyle=\ttfamily,
+	keywordstyle=\color{javapurple}\bfseries,
+	stringstyle=\color{javagreen},
+	commentstyle=\color{javagreen},
+	morecomment=[s][\color{javadocblue}]{/**}{*/},
+	numbers=left,
+	numberstyle=\tiny\color{black},
+	stepnumber=1,
+	numbersep=10pt,
+	tabsize=2,
+	showspaces=false,
+	showstringspaces=false}
+
+\lstdefinelanguage{scala}{
+  morekeywords={abstract,case,catch,class,def,%
+    do,else,extends,false,final,finally,%
+    for,if,implicit,import,match,mixin,%
+    new,null,object,override,package,%
+    private,protected,requires,return,sealed,%
+    super,this,throw,trait,true,try,%
+    type,val,var,while,with,yield},
+  otherkeywords={=>,<-,<\%,<:,>:,\#,@},
+  sensitive=true,
+  morecomment=[l]{//},
+  morecomment=[n]{/*}{*/},
+  morestring=[b]",
+  morestring=[b]',
+  morestring=[b]"""
+}
+
+\lstset{language=Scala,
+	basicstyle=\ttfamily,
+	keywordstyle=\color{javapurple}\bfseries,
+	stringstyle=\color{javagreen},
+	commentstyle=\color{javagreen},
+	morecomment=[s][\color{javadocblue}]{/**}{*/},
+	numbers=left,
+	numberstyle=\tiny\color{black},
+	stepnumber=1,
+	numbersep=10pt,
+	tabsize=2,
+	showspaces=false,
+	showstringspaces=false}
+
+% beamer stuff 
+\renewcommand{\slidecaption}{APP 05, King's College London, 23 October 2012}
+
+\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 (5)\\[-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\\
+  Of$\!$fice: & S1.27 (1st floor Strand Building)\\
+  Slides: & KEATS (also homework is there)\\
+  \end{tabular}
+  \end{center}
+
+
+\end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Satan's Computer}
+
+Ross Anderson and Roger Needham wrote:\bigskip
+
+\begin{quote}
+In effect, our task is to program a computer which gives 
+answers which are subtly and maliciously wrong at the most 
+inconvenient possible moment\ldots{} we hope that the lessons 
+learned from programming Satan's computer may be helpful 
+in tackling the more common problem of programming Murphy's.
+\end{quote}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Protocol Specifications}
+
+The Needham-Schroeder Protocol:
+
+\begin{center}
+\begin{tabular}{@ {\hspace{-7mm}}l@{\hspace{2mm}}r@ {\hspace{1mm}}l}
+Message 1 & \bl{$A \rightarrow S :$} & \bl{$A, B, N_A$}\\
+Message 2 & \bl{$S \rightarrow A :$} & \bl{$\{N_A, B, K_{AB},\{K_{AB}, A\}_{K_{BS}} \}_{K_{AS}}$}\\
+Message 3 & \bl{$A \rightarrow B :$} & \bl{$\{K_{AB}, A\}_{K_{BS}} $}\\
+Message 4 & \bl{$B \rightarrow A :$} & \bl{$\{N_B\}_{K_{AB}}$}\\
+Message 5 & \bl{$A \rightarrow B :$} & \bl{$\{N_B-1\}_{K_{AB}}$}\\
+\end{tabular}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Cryptographic Protocol Failures}
+
+Again Ross Anderson and Roger Needham wrote:\bigskip
+
+\begin{quote}
+\textcolor{gray}{
+A lot of the recorded frauds were the result of this kind of blunder, or from 
+management negligence pure and simple.} However, there have been a 
+significant number of cases where the designers protected the right things, 
+used cryptographic algorithms which were not broken, and yet found that their 
+systems were still successfully attacked. 
+\end{quote}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{The Access Control Problem}
+
+
+\begin{center}
+  \begin{tikzpicture}[scale=1]
+  
+  \draw[line width=1mm] (-.3, 0) rectangle (1.5,2);
+  \draw (-2.7,1) node {\begin{tabular}{l}access\\request\end{tabular}};
+  \draw (4.2,1) node {\begin{tabular}{l}granted/\\not granted\end{tabular}};
+  \draw (0.6,1.2) node {\footnotesize \begin{tabular}{l}Access\\Control\\Checker\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 some rules\\(access policy)\end{tabular}};
+
+  \end{tikzpicture}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Access Control Logic}
+
+Ross Anderson about the use of Logic:\bigskip
+
+\begin{quote}
+Formal methods can be an excellent way of finding 
+bugs in security protocol designs as they force the designer 
+to make everything explicit and thus confront dif$\!$ficult design 
+choices that might otherwise be fudged. 
+\end{quote}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+  \begin{tikzpicture}[scale=1]
+  
+  \draw[line width=1mm] (-.3, 0) rectangle (1.5,2);
+  \draw (-2.7,1) node {\begin{tabular}{l}access\\request\end{tabular}};
+  \draw (4.2,1) node {\begin{tabular}{l}granted/\\not granted\end{tabular}};
+  \draw (0.6,1.2) node {\footnotesize \begin{tabular}{l}Access\\Control\\Checker\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 (0.6,3.7) node {\begin{tabular}{l}access policy\end{tabular}};
+
+  \end{tikzpicture}
+\end{center}
+
+Assuming one file on my computer contains a virus.\smallskip\\
+\only<1>{Q: Given my access policy, can this file ``infect'' my whole computer?}%
+\only<2>{Q: Can my access policy prevent that my whole computer gets infected.}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \small
+  \begin{center}
+  \mbox{
+  \inferrule{\mbox{\begin{tabular}{l}
+         \ldots\\
+         is\_at\_library (Christian)\\ 
+         is\_student (a) $\wedge$ is\_at\_library (a) $\Rightarrow$ may\_obtain\_email (a)\\
+         is\_staff (a) $\wedge$ is\_at\_library (a) $\Rightarrow$ may\_obtain\_email (a)\medskip\\
+         \onslide<2->{HoD says is\_staff (a) $\Rightarrow$ is\_staff (a)}\\
+         \onslide<2->{HoD says is\_staff (Christian)}\medskip\\
+         \onslide<3->{may\_obtain\_email (a) $\wedge$ sending\_spam (a) $\Rightarrow$\\}
+         \onslide<3->{\hspace{6cm}$\neg$ may\_obtain\_email (a)}
+       \end{tabular}
+        }}
+        {\mbox{? may\_obtain\_email (Christian)}}}
+  \end{center}
+  \end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{}
+
+  There are two ways for tackling such problems:\medskip
+
+  \begin{itemize}
+  \item either you make up our own language in which you can describe
+  the problem,\medskip
+
+  \item or you use an existing language and represent the problem in
+  this language.
+  \end{itemize}
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{\Large\begin{tabular}{@ {}c@ {}}Logic(s)\end{tabular}}
+
+  \begin{itemize}
+  \item Formulas
+
+  \begin{center}\color{blue}
+  \begin{tabular}{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}       & \textcolor{black}{implies}\\
+            & \isa{{\isaliteral{7C}{\isacharbar}}}   & \isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ F}       & \textcolor{black}{negation}\\
+            & \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}}} & \textcolor{black}{predicates}\\
+   & \onslide<2->{\isa{{\isaliteral{7C}{\isacharbar}}}} & \onslide<2->{\isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ F}} & 
+        \onslide<2->{\textcolor{black}{forall quantification}}\\
+   & \onslide<2->{\isa{{\isaliteral{7C}{\isacharbar}}}} & \onslide<2->{\isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ F}} & 
+        \onslide<2->{\textcolor{black}{exists quantification}}\\[-7mm]
+  \end{tabular}
+  \end{center}
+
+  \end{itemize}
+
+  \begin{textblock}{12}(1,14)
+  Terms \textcolor{blue}{\isa{t\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{7C}{\isacharbar}}\ c\ {\isaliteral{5C3C646F74733E}{\isasymdots}}}}
+  \end{textblock}
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+
+{\lstset{language=Scala}\fontsize{10}{12}\selectfont
+\texttt{\lstinputlisting{programs/formulas.scala}}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Judgements}
+
+  \begin{center}
+  \LARGE
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}
+  \end{center}
+
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}}} is a collection of formulas, called the \alert{assumptions}\bigskip\pause
+
+  \begin{itemize}
+  \item Example\mbox{}\\[-8mm]
+
+  \only<2>{\begin{center}\tiny
+  \textcolor{blue}{
+  \begin{tabular}{l}
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}},\\
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}},\\
+  \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}}\\
+  \end{tabular}\isa{{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}}
+  \end{center}}
+  \only<3>{\small
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}}}
+  {\mbox{\begin{tabular}{@ {}l@ {}}
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}\\
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}\\
+  \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}}\\
+  \end{tabular}}
+  }
+  }
+  \end{center}
+  }}
+  \only<4>{\small
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}Alice{\isaliteral{29}{\isacharparenright}}}}}
+  {\mbox{\begin{tabular}{@ {}l@ {}}
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Alice{\isaliteral{29}{\isacharparenright}}}\\
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}\\
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}\\
+  \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}}\\
+  \end{tabular}}
+  }
+  }
+  \end{center}
+  }}
+
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+
+{\lstset{language=Scala}\fontsize{10}{12}\selectfont
+\texttt{\lstinputlisting{programs/judgement.scala}}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Inference Rules}
+
+  \textcolor{blue}{
+  \begin{center}
+  \Large
+  \mbox{
+  \infer{\mbox{\isa{conclusion}}}
+        {\mbox{\isa{premise\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}} & \mbox{\isa{{\isaliteral{5C3C646F74733E}{\isasymdots}}}} & \mbox{\isa{premise\isaliteral{5C3C5E697375623E}{}\isactrlisub n}}}}
+  \end{center}}
+
+  The conlusion and premises are judgements\bigskip\pause
+
+  \begin{itemize}
+  \item Examples
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}}
+  \end{center}}\pause
+
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}}}}
+  \hspace{10mm}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}}
+  \end{center}}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Implication}
+  \Large
+
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}{\isaliteral{2C}{\isacharcomma}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}}
+  \end{center}}
+
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}}}}
+  \end{center}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Universal Quantification}
+  \Large
+
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F{\isaliteral{5B}{\isacharbrackleft}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ t{\isaliteral{5D}{\isacharbrackright}}}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ F}}}}
+  \end{center}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Start Rules / Axioms}
+
+  \normalsize
+  \alert{if \textcolor{blue}{\isa{F\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C47616D6D613E}{\isasymGamma}}}}}
+  
+  \textcolor{blue}{\Large
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}}{}}
+  \end{center}}\bigskip\pause
+
+  \normalsize
+  Also written as:
+  \textcolor{blue}{\Large
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}{\isaliteral{2C}{\isacharcomma}}\ F\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}}{}}
+  \end{center}}\pause
+
+  \textcolor{blue}{\Large
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ true}}}{}}
+  \end{center}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{}
+ 
+ \begin{minipage}{1.1\textwidth}
+  Let \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{3D}{\isacharequal}}}\footnotesize\begin{tabular}{l}
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}},\\
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}},\\
+  \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}}\\
+  \end{tabular}}
+  \end{minipage}
+
+  \only<2>{
+  \begin{textblock}{12}(4,3)\footnotesize
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}}\hspace{10mm}
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}}
+  \end{textblock}}
+
+  \only<3->{
+  \begin{textblock}{12}(4,3)\footnotesize
+  \mbox{\textcolor{blue}{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}}}
+  {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}} &\hspace{10mm}
+   \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}}}
+  }}
+  \end{textblock}}
+
+  \only<4>{
+  \begin{textblock}{14}(0.5,6)\footnotesize
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}}}
+  \end{textblock}}
+
+  \only<5->{
+  \begin{textblock}{14}(0.5,6)\footnotesize
+  \textcolor{blue}{
+  \infer{\mbox{\begin{tabular}{l}\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}\\ 
+           \hspace{40mm}\isa{{\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}\end{tabular}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}}}}}
+  \end{textblock}}
+
+  \only<6->{
+  \begin{textblock}{14}(5,10)\footnotesize
+  \textcolor{blue}{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}}}
+        {\vdots & \hspace{30mm} \vdots}}
+  \end{textblock}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Access Control}
+  \Large
+
+  \textcolor{blue}{
+  \begin{center}
+  \isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}
+  \end{center}}\bigskip
+
+  \normalsize
+  \begin{itemize}
+  \item If there is a proof \isa{{\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}} yes (granted)
+  \item If there isn't \isa{{\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}} no (denied)
+  \end{itemize}\bigskip\pause
+
+\begin{minipage}{1.1\textwidth}	
+  \small
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{3D}{\isacharequal}}}\small\begin{tabular}{l}
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}},\\
+  \isa{is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}},\\
+  \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ is{\isaliteral{5F}{\isacharunderscore}}at{\isaliteral{5F}{\isacharunderscore}}library\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}}\\
+  \end{tabular}}\medskip
+  
+  \textcolor{blue}{
+  \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}} $\not\vdash$ \isa{may{\isaliteral{5F}{\isacharunderscore}}obtain{\isaliteral{5F}{\isacharunderscore}}email\ {\isaliteral{28}{\isacharparenleft}}Alice{\isaliteral{29}{\isacharparenright}}}}}
+\end{minipage}
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{The Access Control Problem}
+
+
+\begin{center}
+  \begin{tikzpicture}[scale=1]
+  
+  \draw[line width=1mm] (-.3, 0) 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}granted/\\not granted\end{tabular}};
+  \draw (0.6,1.2) node {\footnotesize \begin{tabular}{l}Access\\Control\\Checker\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{Bad News}
+  
+  \begin{itemize}
+  \item We introduced (roughly) first-order logic. \bigskip\pause
+
+  \item Judgements
+  \begin{center}
+  \textcolor{blue}
+  {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}}
+  \end{center}
+
+  are in general \alert{undecidable}.\pause\medskip\\ 
+ 
+  The problem is \alert{semi-decidable}.
+ 
+  \end{itemize}
+
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{\Large\begin{tabular}{@ {}c@ {}}Access Control Logic\end{tabular}}
+  
+  \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}
+
+  where \textcolor{blue}{\isa{P\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ Alice{\isaliteral{2C}{\isacharcomma}}\ Bob{\isaliteral{2C}{\isacharcomma}}\ Christian{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}}} (principals)\bigskip\pause
+  
+  \item \textcolor{blue}{\isa{HoD\ says\ is{\isaliteral{5F}{\isacharunderscore}}staff\ {\isaliteral{28}{\isacharparenleft}}Christian{\isaliteral{29}{\isacharparenright}}}}
+  \end{itemize}
+  
+
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+
+{\lstset{language=Scala}\fontsize{10}{12}\selectfont
+\texttt{\lstinputlisting{programs/formulas1.scala}}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Rules about Says}
+
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}}}
+  \end{center}}
+
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ {\isaliteral{28}{\isacharparenleft}}F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}} & \hspace{10mm}\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}}}}
+  \end{center}}
+
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ {\isaliteral{28}{\isacharparenleft}}P\ says\ F{\isaliteral{29}{\isacharparenright}}}}}}
+  \end{center}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{}
+
+  Consider the following scenario:
+
+  \begin{itemize}
+  \item If \textcolor{blue}{Admin} says that \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}} 
+  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}{}\isactrlisub {\isadigit{1}}}} should be deleted.
+  \item \textcolor{blue}{Bob} wants to delete \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}}.
+  \end{itemize}\bigskip\pause
+
+  \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}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}},\\
+  \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}},\\
+  \isa{Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}\\
+  \end{tabular}}\medskip\pause
+
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}}
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{}
+
+  \textcolor{blue}{
+  \begin{center}
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}}}\\\bigskip
+  \mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ {\isaliteral{28}{\isacharparenleft}}F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}} & \hspace{5mm}\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}}}}
+  \end{center}}\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}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}},\\
+  \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}},\\
+  \isa{Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}\\
+  \end{tabular}}\medskip
+
+  \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}}
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{}
+  \small
+
+  \textcolor{blue}{
+  \begin{center}
+  \only<1>{$ \underbrace{
+  \mbox{\infer{\Gamma \vdash \mbox{Admin says (Bob says del\_file)}}
+  {\infer{\Gamma \vdash \mbox{Bob says del\_file}}{}}}}_{X}$}
+  \end{center}}
+
+  \textcolor{blue}{
+  \begin{center}
+  \only<1>{
+  $ \underbrace{
+  \mbox{\infer{\Gamma \vdash \mbox{Admin says del\_file}}
+  {\infer{\Gamma \vdash \mbox{Admin says (Bob says del\_file \isa{{\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}} del\_file)}}{}
+   &
+   \deduce[$\vdots$]{X}{}
+  }}}_{Y}$}
+  \end{center}}
+
+  \textcolor{blue}{
+  \begin{center}
+  \only<1>{\mbox{\infer{\Gamma \vdash \mbox{del\_file}}
+  {\infer{\Gamma \vdash \mbox{(Admin says del\_file) \isa{{\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}} del\_file}}{}
+   &
+   \deduce[$\vdots$]{Y}{}
+  }}}
+  \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{Speaks For}
+  \small
+  
+  \begin{itemize}
+  \item \bl{\isa{P\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Q\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}F{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}P\ says\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}Q\ says\ F{\isaliteral{29}{\isacharparenright}}}}
+
+  \item its meaning ``\bl{P} speaks for \bl{Q}''
+  
+
+  \begin{center}
+  \bl{\mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q\ says\ F}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Q}} & \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}}\ P\ controls\ F}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Q}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q\ controls\ F}}}
+  }}
+  \end{center}
+
+  \begin{center}
+  \bl{\mbox{
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ R}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Q}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ R}}}
+  }}
+  \end{center}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Tickets}
+  
+  \begin{itemize}
+  \item Tickets control access to restricted objects.\bigskip
+  \end{itemize}
+  \small
+
+  Example: \bl{Permitted (Bob, enter\_flight)} ? \bigskip
+  
+  \begin{minipage}{1.1\textwidth}
+  \begin{itemize}
+  \item \bl{Bob says Permitted (Bob, enter\_flight)}\\ (access request)
+  \item \bl{Ticket says (Bob controls Permitted (Bob, enter\_flight))}
+  \item \bl{Airline controls (Bob controls Permitted (Bob, enter\_flight))} (access policy)\pause
+  \item \bl{\isa{Ticket\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Airline}}\\ 
+  (trust assumption)
+  \end{itemize}
+  \end{minipage}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Tickets}
+  \small
+
+  \begin{minipage}{1.1\textwidth}
+  \begin{enumerate}
+  \item \bl{Bob says Permitted (Bob, enter\_flight)}
+  \item \bl{Ticket says (Bob controls Permitted (Bob, enter\_flight))}
+  \item \bl{Airline controls (Bob controls Permitted (Bob, enter\_flight))}
+  \item \bl{\isa{Ticket\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Airline}}
+  \end{enumerate}
+  \end{minipage}\bigskip\bigskip
+
+  Is  \bl{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Permitted\ {\isaliteral{28}{\isacharparenleft}}Bob{\isaliteral{2C}{\isacharcomma}}\ enter{\isaliteral{5F}{\isacharunderscore}}flight{\isaliteral{29}{\isacharparenright}}}} derivable ? \bigskip
+
+
+  \small
+  \begin{minipage}{1.1\textwidth}
+  \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}}}
+  }}
+  \bl{\mbox{\hspace{6mm}
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q\ says\ F}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Q}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F}}}
+  }}
+  \end{center}
+  \end{minipage}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Tickets}
+  \small
+  
+  \begin{minipage}{1.1\textwidth}
+  \begin{itemize}
+  \item Access Request:
+  \begin{center}
+  \bl{Person says Object}
+  \end{center}
+
+  \item Ticket:
+  \begin{center}
+  \bl{Ticket says (Person controls Object)}
+  \end{center}
+
+  \item Access policy:
+  \begin{center}
+  \bl{Authority controls (Person controls Object)}
+  \end{center}
+
+  \item Trust assumption:
+  \begin{center}
+  \bl{\isa{Ticket\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Authority}}
+  \end{center}
+  \end{itemize}
+  \end{minipage}
+
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{\LARGE Derived Rule for Tickets}
+  \small
+  \mbox{}\\[2cm]
+  
+  \begin{center}
+  \bl{\mbox{\infer{\mbox{F}}
+     {\mbox{\begin{tabular}{l}
+      Authority controls (Person controls F)\\
+      Ticket says (Person controls F)\\
+      \isa{Ticket\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Authority}\\
+      Person says F
+      \end{tabular}}
+     }}}
+  \end{center}
+  \mbox{}\\[1cm]
+
+ 
+  \small
+  \begin{minipage}{1.1\textwidth}
+  \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}}}
+  }}
+  \bl{\mbox{\hspace{6mm}
+  \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q\ says\ F}}}
+        {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ {\isaliteral{5C3C6D617073746F3E}{\isasymmapsto}}\ Q}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F}}}
+  }}
+  \end{center}
+  \end{minipage}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \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{Bell-LaPadula}
+  \small
+  
+  \begin{itemize}
+  \item \alert{Read Rule}: A principal \bl{$P$} can read an object \bl{$O$} if and only if
+  \bl{$P$}'s security level is at least as high as \bl{$O$}'s.
+  \item \alert{Write Rule}: A principal \bl{$P$} can write an object \bl{$O$} if and only if
+  \bl{$O$}'s security level is at least as high as \bl{$P$}'s.\medskip
+
+  \item Meta-Rule: All principals in a system should have a sufficiently high security level
+  in order to access an object.
+  \end{itemize}\bigskip
+
+  This restricts information flow $\Rightarrow$ military\bigskip\bigskip\pause
+
+  Bell-LaPadula: {\bf `no read up'} - {\bf `no write down'}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\begin{tabular}{c}Principle of\\[-2mm] Least Privilege\end{tabular}}
+  
+  \begin{tikzpicture}
+  \draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
+  {\normalsize\color{darkgray}
+  \begin{minipage}{10cm}\raggedright
+  A principal should have as few privileges as possible to access a resource.
+  \end{minipage}};
+  \end{tikzpicture}\bigskip\bigskip
+  \small
+
+  \begin{itemize}
+  \item Bob ($T\!S$) and Alice ($S$) want to communicate
+  \item[] $\Rightarrow$ Bob should lower his security level
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Biba Policy}
+  \small
+  
+  Data Integrity (rather than data confidentiality)
+
+  \begin{itemize}
+  \item Biba: {\bf `no read down'} - {\bf `no write up'}
+  \item \alert{Read Rule}: A principal \bl{$P$} can read an object \bl{$O$} if and only if
+  \bl{$P$}'s security level is lower or equal than \bl{$O$}'s.
+  \item \alert{Write Rule}: A principal \bl{$P$} can write an object \bl{$O$} if and only if
+  \bl{$O$}'s security level is lower or equal than \bl{$P$}'s.
+  \end{itemize}\bigskip\bigskip\pause
+
+  E.g.~Generals write orders to officers; officers write oders to solidiers\\
+  Firewall: you can read from inside the firewall, but not from outside\\
+  Phishing: you can look at an approved PDF, but not one from a random email\\
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Point to Take Home}
+
+\begin{itemize}
+\item Formal methods can be an excellent way of finding 
+bugs as they force the designer 
+to make everything explicit and thus confront dif$\!$ficult design 
+choices that might otherwise be fudged. 
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+
+\end{document}
+
+%%% Local Variables:  
+%%% mode: latex
+%%% TeX-master: t
+%%% End: 
+




