\documentclass[dvipsnames,14pt,t]{beamer}

\usepackage{../slides}

\usepackage{../langs}

\usepackage{../graphics}

\usepackage{../grammar}

\makeatletter

\newenvironment<>{btHighlight}[1][]

{\begin{onlyenv}#2\begingroup\tikzset{bt@Highlight@par/.style={#1}}\begin{lrbox}{\@tempboxa}}

{\end{lrbox}\bt@HL@box[bt@Highlight@par]{\@tempboxa}\endgroup\end{onlyenv}}

\newcommand<>\btHL[1][]{%

  \only#2{\begin{btHighlight}[#1]\bgroup\aftergroup\bt@HL@endenv}%

}

\def\bt@HL@endenv{%

  \end{btHighlight}%   

  \egroup

}

\newcommand{\bt@HL@box}[2][]{%

  \tikz[#1]{%

    \pgfpathrectangle{\pgfpoint{1pt}{0pt}}{\pgfpoint{\wd #2}{\ht #2}}%

    \pgfusepath{use as bounding box}%

    \node[anchor=base west, fill=orange!30,outer sep=0pt,inner xsep=1pt, inner ysep=0pt, rounded corners=3pt, minimum height=\ht\strutbox+1pt,#1]{\raisebox{1pt}{\strut}\strut\usebox{#2}};

  }%

}

\makeatother

% beamer stuff 

\renewcommand{\slidecaption}{APP 09, King's College London}

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

\begin{document}

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

\begin{frame}[t]

\frametitle{%

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

  \\

  \LARGE Access Control and \\[-3mm] 

  \LARGE Privacy Policies (9)\\[-6mm] 

  \end{tabular}}\bigskip\bigskip\bigskip

  \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}

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

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

  \begin{frame}[c]

  \begin{center}

  \includegraphics[scale=0.6]{../pics/bridge-limits.png}

  \end{center}

  \end{frame}

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

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

  \begin{frame}[c]

  \frametitle{Old-Fashioned Eng.~vs.~CS}

  \begin{center}

  \begin{tabular}{@{}cc@{}}

  \begin{tabular}{@{}p{5.2cm}} 

  \includegraphics[scale=0.058]{../pics/towerbridge.jpg}\\ 

  {\bf bridges}: \\

  \raggedright\small

  engineers can ``look'' at a bridge and have a pretty good

  intuition about whether it will hold up or not\\ 

  (redundancy; predictive theory)

  \end{tabular} &

  \begin{tabular}{p{5cm}} 

  \includegraphics[scale=0.265]{../pics/code.jpg}\\

  \raggedright

  {\bf code}: \\

  \raggedright\small

  programmers have very little intuition about their code; 

  often it is too expensive to have redundancy;

  not ``continuous'' 

  \end{tabular}

  \end{tabular}

  \end{center}

  \end{frame}

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

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

  \begin{frame}[c]

  \frametitle{Dijkstra on Testing}

  \begin{bubble}[10cm]

  ``Program testing can be a very effective way to show the

  presence of bugs, but it is hopelessly inadequate for showing

  their absence.''

  \end{bubble}\bigskip

  unfortunately attackers exploit bugs (Satan's computer vs 

  Murphy's)

  \vfill

  \small

  Dijkstra: shortest path algorithm, 

  dining philosophers problem,

  semaphores

  \end{frame}

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

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

\begin{frame}[c]

\frametitle{\Large Proving Programs to be Correct}

\begin{bubble}[10cm]

\small

{\bf Theorem:} There are infinitely many prime 

numbers.\medskip\\

{\bf Proof} \ldots\\

\end{bubble}\bigskip

similarly\bigskip

\begin{bubble}[10cm]

\small

{\bf Theorem:} The program is doing what 

it is sup+ed to be doing.\medskip

{\bf Long, long proof} \ldots\\

\end{bubble}\bigskip\medskip

\small This can be a gigantic proof. The only hope is to have

help from the computer. `Program' is here to be understood to be

quite general (protocol, OS,\ldots).

\end{frame}

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

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

  \begin{frame}[c]

  \frametitle{\Large{}Mars Pathfinder Mission 1997}

  \begin{center}

  \includegraphics[scale=0.15]{../pics/marspath1.png}

  \includegraphics[scale=0.16]{../pics/marspath3.png}

  \includegraphics[scale=0.3]{../pics/marsrover.png}

  \end{center}

  \begin{itemize}

  \item despite NASA's famous testing procedures, the lander crashed frequently on Mars

  \item a scheduling algorithm was not used in the OS

  \end{itemize}

  \end{frame}

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

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

  \begin{frame}[c]

  \begin{textblock}{11}(1,3)

  \begin{tabular}{@{\hspace{-10mm}}l}

  \begin{tikzpicture}[scale=1.1]

  \node at (-2.5,-1.5) {\textcolor{white}{a}};

  \node at (8,4) {\textcolor{white}{a}};

  \only<1>{

   \draw[fill, blue!50] (1,0) rectangle (3,0.6);

  }

  \only<2>{

   \draw[fill, blue!50] (1,0) rectangle (2,0.6);

   \draw[fill, blue!50] (3,0) rectangle (5,0.6);

   \draw[fill, blue!100] (2,3) rectangle (3,3.6);

  }

  \only<3>{

   \draw[fill, blue!50] (1,0) rectangle (2,0.6);

   \draw[fill, blue!50] (3,0) rectangle (4.5,0.6);

   \draw[fill, blue!50] (5.5,0) rectangle (6,0.6);

   \draw[fill, blue!100] (2,3) rectangle (3,3.6);

   \draw[fill, blue!100] (4.5,3) rectangle (5.5,3.6);

  }

  \only<4>{

   \node at (2.5,0.9) {\small locked a resource};

   \draw[fill, blue!50] (1,0) rectangle (2,0.6);

   \draw[blue!50, very thick] (2,0) rectangle (4,0.6);

   \draw[blue!100, very thick] (2,3) rectangle (3, 3.6);

   \draw[red, <-, line width = 2mm] (2,-0.1) -- (2, -1);

  }

  \only<5>{

   \node at (2.5,0.9) {\small locked a resource};

   \draw[fill, blue!50] (1,0) rectangle (4,0.6);

   \draw[blue!100, fill] (4,3) rectangle (5, 3.6);

  }

  \only<6>{

   \node at (2.5,0.9) {\small locked a resource};

   \draw[fill, blue!50] (1,0) rectangle (2,0.6);

   \draw[blue!50, very thick] (2,0) rectangle (4,0.6);

   \draw[blue!100, very thick] (2,3) rectangle (3, 3.6);

   \draw[red, <-, line width = 2mm] (2,-0.1) -- (2, -1);

  }

  \only<7>{

   \node at (2.5,0.9) {\small locked a resource};

   \draw[fill, blue!50] (1,0) rectangle (2.5,0.6);

   \draw[blue!50, very thick] (2.5,0) rectangle (4,0.6);

   \draw[blue!100, very thick] (2.5,3) rectangle (3.5, 3.6);

   \draw[red, <-, line width = 2mm] (2.5,-0.1) -- (2.5, -1);

  }

  \only<8>{

   \node at (2.5,0.9) {\small locked a resource}; 

   \draw[fill, blue!50] (1,0) rectangle (2.5,0.6);

   \draw[blue!50, very thick] (2.5,0) rectangle (4,0.6);

   \draw[blue!100, very thick] (2.5,3) rectangle (3.5, 3.6);

   \draw[blue!100, very thick] (2.5,3) rectangle (3.5, 3.6);

   \draw[red, fill] (2.5,1.5) rectangle (3.5,2.1);

   \draw[red, <-, line width = 2mm] (2.5,-0.1) -- (2.5, -1);

  }

  \only<9>{

   \node at (2.5,0.9) {\small locked a resource}; 

   \draw[fill, blue!50] (1,0) rectangle (2.5,0.6);

   \draw[blue!50, very thick] (3.5,0) rectangle (5,0.6);

   \draw[blue!100, very thick] (3.5,3) rectangle (4.5, 3.6);

   \draw[red, fill] (2.5,1.5) rectangle (3.5,2.1);

   \draw[red, <-, line width = 2mm] (3.5,-0.1) -- (3.5, -1);

  }

  \only<10>{

   \node at (2.5,0.9) {\small locked a resource}; 

   \draw[fill, blue!50] (1,0) rectangle (2.5,0.6);

   \draw[blue!50, very thick] (3.5,0) rectangle (5,0.6);

   \draw[blue!100, very thick] (3.5,3) rectangle (4.5, 3.6);

   \draw[red, fill] (2.5,1.5) rectangle (3.5,2.1);

   \draw[red, fill] (3.55,1.5) rectangle (4.5,2.1);

   \draw[red, <-, line width = 2mm] (3.5,-0.1) -- (3.5, -1);

  }

  \only<11>{

   \node at (2.5,0.9) {\small locked a resource};

   \draw[fill, blue!50] (1,0) rectangle (2.5,0.6);

   \draw[blue!50, very thick] (4.5,0) rectangle (6,0.6);

   \draw[blue!100, very thick] (4.5,3) rectangle (5.5, 3.6);

   \draw[red, fill] (2.5,1.5) rectangle (3.5,2.1);

   \draw[red, fill] (3.55,1.5) rectangle (4.5,2.1);

   \draw[red, <-, line width = 2mm] (4.5,-0.1) -- (4.5, -1);

  }

  \only<12>{

   \node at (2.5,0.9) {\small locked a resource}; 

   \draw[fill, blue!50] (1,0) rectangle (2.5,0.6);

   \draw[blue!50, very thick] (5.5,0) rectangle (7,0.6);

   \draw[blue!100, very thick] (5.5,3) rectangle (6.5, 3.6);

   \draw[red, fill] (2.5,1.5) rectangle (3.5,2.1);

   \draw[red, fill] (3.55,1.5) rectangle (4.5,2.1);

   \draw[red, fill] (4.55,1.5) rectangle (5.5,2.1);

   \draw[red, <-, line width = 2mm] (5.5,-0.1) -- (5.5, -1);

   \node [anchor=base] at (6.3, 1.8) {\Large\ldots};

  }

  \only<13>{

   \node at (2.5,0.9) {\small locked a resource}; 

   \draw[fill, blue!50] (1,0) rectangle (2,0.6);

   \draw[blue!50, very thick] (2,0) rectangle (4,0.6);

   \draw[blue!100, very thick] (2,3) rectangle (3, 3.6);

   \draw[red, <-, line width = 2mm] (2,-0.1) -- (2, -1);

  }

  \only<14>{

   \node at (2.5,3.9) {\small locked a resource}; 

   \draw[fill, blue!50] (1,0) rectangle (2,0.6);

   \draw[blue!50, fill] (2,3) rectangle (4,3.6);

   \draw[blue!100, very thick] (4,3) rectangle (5, 3.6);

   \draw[blue!50, ->, line width = 2mm] (2,1) -- (2, 2.5);

   \draw[red, <-, line width = 2mm] (2,-0.1) -- (2, -1);

  }

  \only<15>{

   \node at (2.5,3.9) {\small locked a resource};  

   \draw[fill, blue!50] (1,0) rectangle (2,0.6);

   \draw[blue!50, fill] (2,3) rectangle (4,3.6);

   \draw[blue!100, very thick] (4,3) rectangle (5, 3.6);

   \draw[red, <-, line width = 2mm] (2.5,-0.1) -- (2.5, -1);

   \draw[red, very thick] (2.5,1.5) rectangle (3.5,2.1); 

  }

  \draw[very thick,->](0,0) -- (8,0);

  \node [anchor=base] at (8, -0.3) {\scriptsize time};

  \node [anchor=base] at (0, -0.3) {\scriptsize 0};

  \node [anchor=base] at (-1.2, 0.2) {\small low priority};

  \only<2->{\node [anchor=base] at (-1.2, 3.2) {\small high priority};}

  \only<8->{\node [anchor=base] at (-1.5, 1.7) {\small medium pr.};}

  \end{tikzpicture}

  \end{tabular}

  \end{textblock}

  \begin{textblock}{0}(2.5,13)%

  \small

  \onslide<3->{

  \begin{bubble}[8cm]%

  Scheduling: You want to avoid that a high 

  priority process is staved indefinitely.

  \end{bubble}}

  \end{textblock}

  \end{frame}

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

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

  \begin{frame}[c]

  \frametitle{\Large Priority Inheritance Scheduling}

  \begin{itemize}

  \item Let a low priority process $L$ temporarily inherit 

    the high priority of $H$ until $L$ leaves the critical 

    section unlocking the resource.\bigskip

  \item Once the resource is unlocked $L$ returns to its original 

    priority level.

  \end{itemize}

  \end{frame}

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

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

  \begin{frame}[c]

  \begin{textblock}{11}(1,3)

  \begin{tabular}{@{\hspace{-10mm}}l}

  \begin{tikzpicture}[scale=1.1]

  \node at (-2.5,-1.5) {\textcolor{white}{a}};

  \node at (8,4) {\textcolor{white}{a}};

  \only<1>{

    \draw[fill, blue!50] (1,0) rectangle (6,0.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,0.9) {\small $A_R$};

    \node at (5.7,0.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,0) -- (3.5,0.6);

    \draw[very thick,blue!100] (5.7,0) -- (5.7,0.6);

  }

  \only<2>{

    \draw[fill, blue!50] (1,0) rectangle (3,0.6);

    \draw[very thick, blue!50] (3,0) rectangle (6,0.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,0.9) {\small $A_R$};

    \node at (5.7,0.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,0) -- (3.5,0.6);

    \draw[very thick,blue!100] (5.7,0) -- (5.7,0.6);

  }

  \only<3>{

    \draw[fill, blue!50] (1,0) rectangle (3,0.6);

    \draw[very thick, blue!50] (3,0) rectangle (6,0.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,0.9) {\small $A_R$};

    \node at (5.7,0.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,0) -- (3.5,0.6);

    \draw[very thick,blue!100] (5.7,0) -- (5.7,0.6);

    \draw[very thick,blue!100] (3,3) rectangle (4,3.6);

    \node at (3.5,3.3) {\small $A$};

  }

  \only<4>{

    \draw[fill, blue!50] (1,0) rectangle (3,0.6);

    \draw[very thick, blue!50] (3,0) rectangle (6,0.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,0.9) {\small $A_R$};

    \node at (5.7,0.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,0) -- (3.5,0.6);

    \draw[very thick,blue!100] (5.7,0) -- (5.7,0.6);

    \draw[very thick,blue!100] (3,3) rectangle (4,3.6);

    \node at (3.5,3.3) {\small $A$};

    \draw[very thick,blue!100] (4,3) rectangle (5,3.6);

    \node at (4.5,3.3) {\small $B$};

  }

  \only<5>{

    \draw[fill, blue!50] (1,0) rectangle (3,0.6);

    \draw[very thick, blue!50] (3,3) rectangle (6,3.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,3.9) {\small $A_R$};

    \node at (5.7,3.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);

    \draw[very thick,blue!100] (5.7,3) -- (5.7,3.6);

    \draw[very thick,blue!100] (6,3) rectangle (7,3.6);

    \node at (6.5,3.3) {\small $A$};

    \draw[very thick,blue!100] (7,3) rectangle (8,3.6);

    \node at (7.5,3.3) {\small $B$};

    \draw[blue!50, ->, line width = 2mm] (3,1) -- (3, 2.5);

  }

  \only<6>{

    \draw[fill, blue!50] (1,0) rectangle (3,0.6);

    \draw[fill, blue!50] (3,3) rectangle (3.5,3.6);

    \draw[very thick, blue!50] (3.5,3) rectangle (6,3.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,3.9) {\small $A_R$};

    \node at (5.7,3.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);

    \draw[very thick,blue!100] (5.7,3) -- (5.7,3.6);

    \draw[very thick,blue!100] (6,3) rectangle (7,3.6);

    \node at (6.5,3.3) {\small $A$};

    \draw[very thick,blue!100] (7,3) rectangle (8,3.6);

    \node at (7.5,3.3) {\small $B$}; 

  }

  \only<7>{

   \draw[fill, blue!50] (1,0) rectangle (3,0.6);

    \draw[fill, blue!50] (3,3) rectangle (3.5,3.6);

    \draw[very thick, blue!50] (3.5,0) rectangle (6,0.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,3.9) {\small $A_R$};

    \node at (5.7,0.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);

    \draw[very thick,blue!100] (5.7,0) -- (5.7,0.6);

    \draw[very thick,blue!100] (6,3) rectangle (7,3.6);

    \node at (6.5,3.3) {\small $A$};

    \draw[very thick,blue!100] (7,3) rectangle (8,3.6);

    \node at (7.5,3.3) {\small $B$};  

    \draw[blue!50, <-, line width = 2mm] (3.5,1) -- (3.5, 2.5);

    \draw[blue!50, <-, line width = 2mm] (4,3.3) -- (5.5,3.3);

  }

  \only<8>{

    \draw[fill, blue!50] (1,0) rectangle (3,0.6);

    \draw[fill, blue!50] (3,3) rectangle (3.5,3.6);

    \draw[very thick, blue!50] (4.5,0) rectangle (7,0.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,3.9) {\small $A_R$};

    \node at (6.7,0.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);

    \draw[very thick,blue!100] (6.7,0) -- (6.7,0.6);

    \draw[fill,blue!100] (3.5,3) rectangle (4.5,3.6);

    \node at (4,3.3) {\small $A$};

    \draw[very thick,blue!100] (7,3) rectangle (8,3.6);

    \node at (7.5,3.3) {\small $B$};  

  }

  \only<9>{

    \draw[fill, blue!50] (1,0) rectangle (3,0.6);

    \draw[fill, blue!50] (3,3) rectangle (3.5,3.6);

    \draw[fill, blue!50] (4.5,0) rectangle (5,0.6);

    \draw[very thick, blue!50] (5,0) rectangle (7,0.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,3.9) {\small $A_R$};

    \node at (6.7,0.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);

    \draw[very thick,blue!100] (6.7,0) -- (6.7,0.6);

    \draw[fill,blue!100] (3.5,3) rectangle (4.5,3.6);

    \node at (4,3.3) {\small $A$};

    \draw[very thick,blue!100] (7,3) rectangle (8,3.6);

    \node at (7.5,3.3) {\small $B$};  

  }

  \only<10>{

    \draw[fill, blue!50] (1,0) rectangle (3,0.6);

    \draw[fill, blue!50] (3,3) rectangle (3.5,3.6);

    \draw[fill, blue!50] (4.5,0) rectangle (5,0.6);

    \draw[very thick, blue!50] (5,0) rectangle (7,0.6);

    \node at (1.5,0.9) {\small $A_L$};

    \node at (2.0,0.9) {\small $B_L$};

    \node at (3.5,3.9) {\small $A_R$};

    \node at (6.7,0.9) {\small $B_R$};

    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);

    \draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);

    \draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);

    \draw[very thick,blue!100] (6.7,0) -- (6.7,0.6);

    \draw[fill,blue!100] (3.5,3) rectangle (4.5,3.6);