# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1450559483 0
# Node ID 2d0a59127694ee1e3bffa5e4f4f8af714fd0e622
# Parent  f352cb238c32a9f9de689f4d600fbea77c9f003a
updated

diff -r f352cb238c32 -r 2d0a59127694 handouts/ho06.pdf
Binary file handouts/ho06.pdf has changed
diff -r f352cb238c32 -r 2d0a59127694 handouts/ho06.tex
--- a/handouts/ho06.tex	Mon Dec 07 17:50:05 2015 +0000
+++ b/handouts/ho06.tex	Sat Dec 19 21:11:23 2015 +0000
@@ -37,10 +37,10 @@
 to process from the input and the second is the unprocessed
 part of the input. As we shall see shortly, a parser
 combinator might return more than one such pair, the idea
-being that there are potentially several ways how to interpret
-the input. To simplify matters we will first look at the case
-where the input to the parser combinators is just strings. As
-a concrete example, consider the string
+being that there are potentially several ways of how to
+interpret the input. To simplify matters we will first look at
+the case where the input to the parser combinators is just
+strings. As a concrete example, consider the string
 
 \begin{center}
 \tt\Grid{iffoo\VS testbar}
@@ -65,15 +65,14 @@
 ``went wrong''\ldots or more precisely, nothing could be
 parsed.
 
-The main attraction of parser combinators is that we can
-easily build parser combinators out of smaller components
-following very closely the structure of a grammar. In order to
-implement this in an object-oriented programming language,
-like Scala, we need to specify an abstract class for parser
-combinators. This abstract class requires the implementation
-of the function \texttt{parse} taking an argument of type
-\texttt{I} and returns a set of type \mbox{\texttt{Set[(T,
-I)]}}.
+The idea of parser combinators is that we can easily build
+parser combinators out of smaller components following very
+closely the structure of a grammar. In order to implement this
+in an object-oriented programming language, like Scala, we
+need to specify an abstract class for parser combinators. This
+abstract class requires the implementation of the function
+\texttt{parse} taking an argument of type \texttt{I} and
+returns a set of type \mbox{\texttt{Set[(T, I)]}}.
 
 \begin{center}
 \begin{lstlisting}[language=Scala]
@@ -132,7 +131,7 @@
 More interesting are the parser combinators that build larger
 parsers out of smaller component parsers. For example the
 alternative parser combinator is as follows: given two
-parsers, say, $p$ and $q$, then we apply both parsers to the
+parsers, say, $p$ and $q$, we apply both parsers to the
 input (remember parsers are functions) and combine the output
 (remember they are sets)
  
@@ -194,7 +193,7 @@
 
 \noindent We receive in the first two cases a successful
 output (that is a non-empty set). In each case, either
-\pcode{a} or \pcode{b} are in the processed parts, and
+\pcode{a} or \pcode{b} is in the processed part, and
 \pcode{c} in the unprocessed part. Clearly this parser cannot
 parse anything in the string \pcode{cc}, therefore the empty
 set.
diff -r f352cb238c32 -r 2d0a59127694 progs/LOOP.j
--- a/progs/LOOP.j	Mon Dec 07 17:50:05 2015 +0000
+++ b/progs/LOOP.j	Sat Dec 19 21:11:23 2015 +0000
@@ -2,13 +2,13 @@
 .class public LOOP.LOOP
 .super java/lang/Object
 
-.method public <init>()V
+.method public <init> : ()V
    aload_0
    invokenonvirtual java/lang/Object/<init>()V
    return
 .end method
 
-.method public static write(I)V 
+.method public static write : (I)V 
     .limit locals 5 
     .limit stack 5 
     iload 0 
@@ -19,7 +19,7 @@
 .end method
 
 
-.method public static main([Ljava/lang/String;)V
+.method public static main : ([Ljava/lang/String;)V
    .limit locals 200
    .limit stack 200
 
diff -r f352cb238c32 -r 2d0a59127694 progs/token2.scala
--- a/progs/token2.scala	Mon Dec 07 17:50:05 2015 +0000
+++ b/progs/token2.scala	Sat Dec 19 21:11:23 2015 +0000
@@ -225,7 +225,7 @@
 
 // Lexing Rules for a Small While Language
 
-val SYM = CRANGE("abcdefghijklmnopqrstuvwxyz")
+val SYM = CRANGE("abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ")
 val DIGIT = CRANGE("0123456789")
 val ID = SYM ~ (SYM | DIGIT).% 
 val NUM = PLUS(DIGIT)
@@ -252,7 +252,7 @@
 
 // filters out all white spaces
 def tokenise(r: Rexp, s: String) = 
-  env(lexing_simp(r, s)).filterNot { (s) => s._1 == "w"}.mkString("\n")
+  env(lexing_simp(r, s)).filterNot { (s) => s._1 == "w"}
 
 //   Testing
 //============
@@ -293,8 +293,13 @@
 // Big Test
 //==========
 
+def escape(raw: String): String = {
+  import scala.reflect.runtime.universe._
+  Literal(Constant(raw)).toString
+}
+
 val prog2 = """
-write "fib";
+write "Fib";
 read n;
 minus1 := 0;
 minus2 := 1;
@@ -304,16 +309,155 @@
   minus1 := temp;
   n := n - 1
 };
-write "result";
+write "Result";
 write minus2
 """
 
+val prog3 = """
+start := 1000;
+x := start;
+y := start;
+z := start;
+while 0 < x do {
+ while 0 < y do {
+  while 0 < z do {
+    z := z - 1
+  };
+  z := start;
+  y := y - 1
+ };     
+ y := start;
+ x := x - 1
+}
+"""
+
+
 println("Tokens")
-println(tokenise(WHILE_REGS, prog2))
+println(tokenise(WHILE_REGS, prog2).mkString("\n"))
+
+val fib_tokens = tokenise(WHILE_REGS, prog2)
+fib_tokens.map{case (s1, s2) => (escape(s1), escape(s2))}.mkString(",\n")
+
 
+val test_tokens = tokenise(WHILE_REGS, prog3)
+test_tokens.map{case (s1, s2) => (escape(s1), escape(s2))}.mkString(",\n")
+
+
+/*
 for (i <- 1 to 120 by 10) {
   print(i.toString + ":  ")
   time(lexing_simp(WHILE_REGS, prog2 * i))
 }
+*/
 
+val toks_fib = 
+  List(("k","write"),
+       ("str","Fib"),
+       ("s",";"),
+       ("k","read"),
+       ("i","n"),
+       ("s",";"),
+       ("i","minus1"),
+       ("o",":="),
+       ("n","0"),
+       ("s",";"),
+       ("i","minus2"),
+       ("o",":="),
+       ("n","1"),
+       ("s",";"),
+       ("k","while"),
+       ("i","n"),
+       ("o",">"),
+       ("n","0"),
+       ("k","do"),
+       ("b","{"),
+       ("i","temp"),
+       ("o",":="),
+       ("i","minus2"),
+       ("s",";"),
+       ("i","minus2"),
+       ("o",":="),
+       ("i","minus1"),
+       ("o","+"),
+       ("i","minus2"),
+       ("s",";"),
+       ("i","minus1"),
+       ("o",":="),
+       ("i","temp"),
+       ("s",";"),
+       ("i","n"),
+       ("o",":="),
+       ("i","n"),
+       ("o","-"),
+       ("n","1"),
+       ("b","}"),
+       ("s",";"),
+       ("k","write"),
+       ("str","Result"),
+       ("s",";"),
+       ("k","write"),
+       ("i","minus2"))
 
+val toks_test =
+  List(("i","start"),
+       ("o",":="),
+       ("n","1000"),
+       ("s",";"),
+       ("i","x"),
+       ("o",":="),
+       ("i","start"),
+       ("s",";"),
+       ("i","y"),
+       ("o",":="),
+       ("i","start"),
+       ("s",";"),
+       ("i","z"),
+       ("o",":="),
+       ("i","start"),
+       ("s",";"),
+       ("k","while"),
+       ("n","0"),
+       ("o","<"),
+       ("i","x"),
+       ("k","do"),
+       ("b","{"),
+       ("k","while"),
+       ("n","0"),
+       ("o","<"),
+       ("i","y"),
+       ("k","do"),
+       ("b","{"),
+       ("k","while"),
+       ("n","0"),
+       ("o","<"),
+       ("i","z"),
+       ("k","do"),
+       ("b","{"),
+       ("i","z"),
+       ("o",":="),
+       ("i","z"),
+       ("o","-"),
+       ("n","1"),
+       ("b","}"),
+       ("s",";"),
+       ("i","z"),
+       ("o",":="),
+       ("i","start"),
+       ("s",";"),
+       ("i","y"),
+       ("o",":="),
+       ("i","y"),
+       ("o","-"),
+       ("n","1"),
+       ("b","}"),
+       ("s",";"),
+       ("i","y"),
+       ("o",":="),
+       ("i","start"),
+       ("s",";"),
+       ("i","x"),
+       ("o",":="),
+       ("i","x"),
+       ("o","-"),
+       ("n","1"),
+       ("b","}"))
diff -r f352cb238c32 -r 2d0a59127694 slides/slides12.pdf
Binary file slides/slides12.pdf has changed
diff -r f352cb238c32 -r 2d0a59127694 slides/slides12.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/slides/slides12.tex	Sat Dec 19 21:11:23 2015 +0000
@@ -0,0 +1,547 @@
+\documentclass[dvipsnames,14pt,t]{beamer}
+\usepackage{../slides}
+\usepackage{../graphics}
+\usepackage{../langs}
+\usepackage{../data}
+\usepackage{../grammar}
+
+% beamer stuff 
+\renewcommand{\slidecaption}{AFL, King's College London}
+\newcommand{\bl}[1]{\textcolor{blue}{#1}}       
+
+
+\begin{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[t]
+\frametitle{%
+  \begin{tabular}{@ {}c@ {}}
+  \\[-3mm]
+  \LARGE Automata and \\[-2mm] 
+  \LARGE Formal Languages\\[3mm] 
+  \end{tabular}}
+
+  \normalsize
+  \begin{center}
+  \begin{tabular}{ll}
+  Email:  & christian.urban at kcl.ac.uk\\
+  Office: & S1.27 (1st floor Strand Building)\\
+  Slides: & KEATS (also home work is there)\\
+  \end{tabular}
+  \end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+
+\Large\bf There are more problems, than there are
+programs.\bigskip\bigskip\pause\\
+
+There must be a problem for which there is no program.
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{Subsets}
+
+\Large
+If \bl{$A \subseteq B$} then \bl{$A$} has fewer elements
+than \bl{$B$}\bigskip\bigskip
+
+\Large
+\bl{$A \subseteq B$} and \bl{$B \subseteq A$}\bigskip
+
+then \bl{$A = B$}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \begin{frame}[c]
+  
+  \begin{center}
+  \begin{tikzpicture}
+  
+  \draw (-4,2.5) node [scale=2.5] (X) 
+    {\begin{tabular}{l}
+     $\{$  
+     \!\includegraphics[scale=0.02]{../pics/o1.jpg},
+     \includegraphics[scale=0.02]{../pics/o2.jpg},
+     \!\includegraphics[scale=0.02]{../pics/o3.jpg},
+     \includegraphics[scale=0.02]{../pics/o4.jpg},
+     \!\includegraphics[scale=0.027]{../pics/o5.jpg}
+     $\}$
+    \end{tabular}};
+    
+    \draw (-5.6,-2.5) node [scale=2.5] (Y) 
+    {\begin{tabular}{l}
+     $\{$  
+     \!\includegraphics[scale=0.059]{../pics/a1.jpg},
+     \includegraphics[scale=0.048]{../pics/a2.jpg},
+     \includegraphics[scale=0.02]{../pics/a3.jpg}
+     $\}$
+    \end{tabular}};
+    
+     \draw (0,1.5) node (X1) {5 elements};
+     \draw (0,-3.5) node (y1) {3 elements};
+  \end{tikzpicture}
+  \end{center}
+
+  \end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \begin{frame}[c]
+  \frametitle{Newton vs Feynman}
+  
+  \begin{center}
+  \begin{tabular}{cc}
+  \includegraphics[scale=0.2]{../pics/newton.jpg} &
+  \includegraphics[scale=0.2]{../pics/feynman.jpg}\\
+  classical physics & quantum physics
+  \end{tabular}
+  \end{center}
+  \end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \begin{frame}[c]
+  \frametitle{The Goal of the Talk}
+ \large
+  \begin{itemize}
+  \item show you that something very unintuitive happens with very large sets	
+  \bigskip\bigskip
+  
+  \item convince you that there are more {\bf problems} than {\bf programs}
+  \end{itemize}	
+  \end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[t]
+%
+  \begin{center}
+  \begin{tikzpicture}
+ 
+  \draw (-5,2.5) node [scale=2.3] (X) 
+    {\begin{tabular}{@ {\hspace{-3mm}}l}
+     \bl{$B$ $=$ $\{$  
+     \!\includegraphics[scale=0.02]{../pics/o1.jpg},
+     \includegraphics[scale=0.02]{../pics/o2.jpg},
+     \!\includegraphics[scale=0.02]{../pics/o3.jpg},
+     \includegraphics[scale=0.02]{../pics/o4.jpg},
+     \!\includegraphics[scale=0.027]{../pics/o5.jpg}
+     $\}$}
+    \end{tabular}};
+    
+    \draw (-6.6,-0.5) node [scale=2.3] (Y) 
+    {\begin{tabular}{@ {\hspace{-3mm}}l}
+     \bl{$A$ $=$ $\{$  
+     \!\includegraphics[scale=0.059]{../pics/a1.jpg},
+     \includegraphics[scale=0.048]{../pics/a2.jpg},
+     \includegraphics[scale=0.02]{../pics/a3.jpg}
+     $\}$}
+     \end{tabular}};
+  
+     \only<1>{\draw (-5, -3) node[scale=2] 
+       {\bl{$|A|$ $=$ $5$}, \bl{$|B|$ $=$ $3$}};}
+     \only<2>{
+       \draw [->, line width=1mm, red] (-7.4, 0.2) -- (-6.1, 2.1);
+       \draw [->, line width=1mm, red] (-5.8, 0.2) -- (-3.1, 2.1);
+       \draw [->, line width=1mm, red] (-4.5, 0.2) -- (-7.6, 2.1);
+       \draw (-5, -3) node[scale=2] {then \bl{$|A|$ $\le$ $|B|$}};
+       }
+    \only<3>{
+       \draw [<-, line width=1mm, red] (-7.5, 0.2) -- (-6.1, 2.1);
+       \draw [<-, line width=1mm, red] (-7.3, 0.2) -- (-3.1, 2.1);
+       \draw [<-, line width=1mm, red] (-6, 0.2) -- (-7.5, 2.1);
+       \draw [<-, line width=1mm, red] (-4.5, 0.2) -- (-4.5, 2.1);
+       \draw [<-, line width=1mm, red] (-4.3, 0.2) -- (-1.3, 2.1);
+       
+       \draw (-5, -3) node[scale=1.5] {\small{}for \bl{$=$}
+        has to be a {\bf one-to-one} mapping};
+       }
+
+    
+  \end{tikzpicture}
+  \end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{Cardinality}
+
+\Large
+\bl{$|A|$} $\dn$ ``how many elements''\bigskip\\
+
+\bl{$A \subseteq B  \Rightarrow |A| \leq |B|$}\bigskip\\\pause
+
+if there is an injective function \bl{$f: A \rightarrow B$} then \bl{$|A| \leq |B|$}\
+
+\begin{center}
+\bl{\large$\forall x y.\; f(x) = f(y) \Rightarrow x = y$}
+\end{center}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[t]
+
+  \begin{center}
+  \begin{tikzpicture}
+  
+  \draw (-6.6,2.5) node [scale=2.3] (X) 
+    {\begin{tabular}{@ {\hspace{-3mm}}l}
+     $A$ $=$ $\{$  
+     \!\includegraphics[scale=0.02]{../pics/o1.jpg},
+     \includegraphics[scale=0.02]{../pics/o2.jpg},
+     \!\includegraphics[scale=0.02]{../pics/o3.jpg}
+     $\}$
+    \end{tabular}};
+    
+    \draw (-6.6,-0.5) node [scale=2.3] (Y) 
+    {\begin{tabular}{@ {\hspace{-3mm}}l}
+     $B$ $=$ $\{$  
+     \!\includegraphics[scale=0.059]{../pics/a1.jpg},
+     \includegraphics[scale=0.048]{../pics/a2.jpg},
+     \includegraphics[scale=0.02]{../pics/a3.jpg}
+     $\}$
+     \end{tabular}};
+   \onslide<3->{\draw (-7, -3) node[scale=1.5] 
+      {then \bl{$|A|$ \alert{$=$} $|B|$}};}
+     \only<1>{
+       \draw [->, line width=1mm, red] (-7.4, 0.2) -- (-6.1, 2.1);
+       \draw [->, line width=1mm, red] (-5.8, 0.2) -- (-4.3, 2.1);
+       \draw [->, line width=1mm, red] (-4.5, 0.2) -- (-7.6, 2.1);
+     }
+    \only<2->{
+       \draw [<-, line width=1mm, blue] (-7.5, 0.2) -- (-7.5, 2.1);
+       \draw [<-, line width=1mm, blue] (-5.8, 0.2) -- (-4.3, 2.1);
+       \draw [<-, line width=1mm, blue] (-4.5, 0.2) -- (-6.1, 2.1);
+       }
+   
+    
+  \end{tikzpicture}
+  \end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{Natural Numbers}
+
+\Large
+\bl{$\mathbb{N}$} \bl{$\dn$} \bl{$\{0, 1, 2, 3, .......\}$}\bigskip\pause 
+
+\bl{$A$} is \alert{countable} iff \bl{$|A| \leq |\mathbb{N}|$}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{First Question}
+
+\Large
+\bl{$|\mathbb{N} - \{0\}|   \;\;\;\alert{?}\;\;\;  |\mathbb{N}| $}\bigskip\bigskip 
+
+\large
+\bl{$\geq$} or \bl{$\leq$} or \bl{$=$} ?
+\bigskip\bigskip\bigskip\pause
+
+\bl{$x$ $\mapsto$ $x + 1$},\\  \bl{$|\mathbb{N} - \{0\}|$ $=$  
+$|\mathbb{N}|$}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\Large
+\bl{$|\mathbb{N} - \{0, 1\}|   \;\;\;\alert{?}\;\;\;  |\mathbb{N}| $}\bigskip\pause 
+
+\bl{$|\mathbb{N} - \mathbb{O}|   \;\;\;\alert{?}\;\;\;  |\mathbb{N}| $}\bigskip\bigskip
+
+\normalsize
+\bl{$\mathbb{O}$} $\dn$ odd numbers\quad   \bl{$\{1,3,5......\}$}\\ \pause
+\bl{$\mathbb{E}$} $\dn$ even numbers\quad   \bl{$\{0,2,4......\}$}\\
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\Large
+\bl{$|\mathbb{N} \cup \mathbb{-N}|   \;\;\;\alert{?}\;\;\;  |\mathbb{N}| $}\bigskip\bigskip
+
+
+\normalsize
+\bl{$\mathbb{\phantom{-}N}$} $\dn$ positive numbers\quad   \bl{$\{0,1,2,3,......\}$}\\
+\bl{$\mathbb{-N}$} $\dn$ negative numbers\quad   \bl{$\{0,-1,-2,-3,......\}$}\\
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\Large
+\bl{$A$} is \alert{countable} if there exists an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip
+
+\bl{$A$} is \alert{uncountable} if there does not exist an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip\bigskip 
+
+
+countable:  \bl{$|A| \leq |\mathbb{N}|$}\\
+uncountable:  \bl{$|A| > |\mathbb{N}|$}\pause\bigskip
+
+
+Does there exist such an \bl{$A$} ?
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Hilbert's Hotel}
+
+  \begin{center}
+ \includegraphics[scale=0.8]{../pics/hilberts_hotel.jpg}
+  \end{center}
+
+  \begin{itemize}
+  \item \ldots has as many rooms as there are natural numbers
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+  
+  
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[t]
+ \frametitle{\begin{tabular}{c}Real Numbers between\\[-2mm] 0 and 1\end{tabular}}
+
+  \begin{center}
+  \begin{tikzpicture}
+  \draw [fill, color=black!50] (1,4) rectangle (2, 3);
+  \draw [fill, color=black!50] (2,3) rectangle (3, 2);
+  \draw [fill, color=black!50] (3,2) rectangle (4, 1);
+  \draw [fill, color=black!50] (4,1) rectangle (5, 0);
+  \draw (0, 0) grid (8, 5);
+  \draw [line width = 1.mm] (1,0) -- (1, 5);    
+  \draw [line width = 1.mm] (0, 4) -- (8, 4);    
+  \draw (0.5,3.5) node {$1$};
+  \draw (0.5,2.5) node {$2$};
+  \draw (0.5,1.5) node {$3$};
+  \draw (0.5,0.5) node {$4$};
+  
+  \draw (1.5,3.5) node {\only<1>{$3$}\only<2->{$4$}};
+  \draw (2.5,3.5) node {$3$};
+  \draw (3.5,3.5) node {$3$};
+  \draw (4.5,3.5) node {$3$};
+  \draw (5.5,3.5) node {$3$};
+  \draw (6.5,3.5) node {$3$};
+  \draw (7.5,3.5) node {$\ldots$};
+  
+  \draw (1.5,2.5) node {$1$};
+  \draw (2.5,2.5) node {\only<1-2>{$2$}\only<3->{$3$}};
+  \draw (3.5,2.5) node {$3$};
+  \draw (4.5,2.5) node {$4$};
+  \draw (5.5,2.5) node {$5$};
+  \draw (6.5,2.5) node {$6$};
+  \draw (7.5,2.5) node {$7$};
+
+  \draw (1.5,1.5) node {$0$};
+  \draw (2.5,1.5) node {$1$};
+  \draw (3.5,1.5) node {\only<1-3>{$0$}\only<4->{$1$}};
+  \draw (4.5,1.5) node {$1$};
+  \draw (5.5,1.5) node {$0$};
+  \draw (6.5,1.5) node {$\ldots$};
+  
+   \draw (1.5,0.5) node {$7$};
+  \draw (2.5,0.5) node {$8$};
+  \draw (3.5,0.5) node {$5$};
+  \draw (4.5,0.5) node {\only<1-4>{$3$}\only<5->{$4$}};
+  \draw (5.5,0.5) node {$9$};
+  \draw (6.5,0.5) node {$\ldots$};
+  
+   \draw (1.5,-0.5) node {$\ldots$};
+   \draw (8.5,3.5) node {$\ldots$};
+  \end{tikzpicture}
+  \end{center}
+  \mbox{}\\[-20mm]\mbox{}
+  
+  \onslide<6->{
+  \begin{center}
+  \Large\bl{$|\mathbb{N}| < |R|$}
+  \end{center}
+  }
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+ \frametitle{The Set of Problems}
+
+  $\aleph_0$
+
+  \begin{center}
+  \begin{tikzpicture}
+  \draw [fill, color=black!50] (1,4) rectangle (2, 3);
+  \draw [fill, color=black!50] (2,3) rectangle (3, 2);
+  \draw [fill, color=black!50] (3,2) rectangle (4, 1);
+  \draw [fill, color=black!50] (4,1) rectangle (5, 0);
+  \draw (0, 0) grid (8, 5);
+  \draw [line width = 1.mm] (1,0) -- (1, 5);    
+  \draw [line width = 1.mm] (0, 4) -- (8, 4);    
+  \draw (0.5,3.5) node {$1$};
+  \draw (0.5,2.5) node {$2$};
+  \draw (0.5,1.5) node {$3$};
+  \draw (0.5,0.5) node {$4$};
+  
+  \draw (1.5,4.5) node {$0$};
+  \draw (2.5,4.5) node {$1$};
+  \draw (3.5,4.5) node {$2$};
+  \draw (4.5,4.5) node {$3$};
+  \draw (5.5,4.5) node {$4$};
+  \draw (6.5,4.5) node {$5$};
+  \draw (7.5,4.5) node {$\ldots$}; 
+  
+  \draw (1.5,3.5) node {$0$};
+  \draw (2.5,3.5) node {$1$};
+  \draw (3.5,3.5) node {$0$};
+  \draw (4.5,3.5) node {$1$};
+  \draw (5.5,3.5) node {$0$};
+  \draw (6.5,3.5) node {$1$};
+  \draw (7.5,3.5) node {$\ldots$};
+  
+  \draw (1.5,2.5) node {$0$};
+  \draw (2.5,2.5) node {$0$};
+  \draw (3.5,2.5) node {$0$};
+  \draw (4.5,2.5) node {$1$};
+  \draw (5.5,2.5) node {$1$};
+  \draw (6.5,2.5) node {$0$};
+  \draw (7.5,2.5) node {$0$};
+
+  \draw (1.5,1.5) node {$0$};
+  \draw (2.5,1.5) node {$0$};
+  \draw (3.5,1.5) node {$0$};
+  \draw (4.5,1.5) node {$0$};
+  \draw (5.5,1.5) node {$0$};
+  \draw (6.5,1.5) node {$\ldots$};
+  
+   \draw (1.5,0.5) node {$1$};
+  \draw (2.5,0.5) node {$1$};
+  \draw (3.5,0.5) node {$0$};
+  \draw (4.5,0.5) node {$1$};
+  \draw (5.5,0.5) node {$1$};
+  \draw (6.5,0.5) node {$\ldots$};
+  
+  \draw (1.5,-0.5) node {$\ldots$};
+   \draw (8.5,3.5) node {$\ldots$};
+
+  \end{tikzpicture}
+  \end{center}
+  
+  
+  \onslide<2>{
+  \begin{center}
+  \large \bl{|Progs| $=$ $|\mathbb{N}|$ $<$ |Probs|}
+ \end{center}
+  }
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Halting Problem}
+
+\large
+Assume a program \bl{$H$} that decides for all programs \bl{$A$} and all 
+input data \bl{$D$} whether\bigskip
+
+\begin{itemize}
+\item \bl{$H(A, D) \dn 1$} iff \bl{$A(D)$} terminates
+\item \bl{$H(A, D) \dn 0$} otherwise
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Halting Problem (2)}
+
+\large
+Given such a program \bl{$H$} define the following program \bl{$C$}:
+for all programs \bl{$A$}\bigskip
+
+\begin{itemize}
+\item \bl{$C(A) \dn 0$} iff \bl{$H(A, A) = 0$} 
+\item \bl{$C(A) \dn$ loops} otherwise
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Contradiction}
+
+
+\bl{$H(C, C)$} is either \bl{$0$} or \bl{$1$}.
+
+\begin{itemize}
+\item \bl{$H(C, C) = 1$} $\stackrel{\text{def}\,H}{\Rightarrow}$ \bl{$C(C)\downarrow$} $\stackrel{\text{def}\,C}{\Rightarrow}$ \bl{$H(C, C)=0$} 
+\item \bl{$H(C, C) = 0$} $\stackrel{\text{def}\,H}{\Rightarrow}$ \bl{$C(C)$} loops $\stackrel{\text{def}\,C}{\Rightarrow}$\\ 
+\hspace{7cm}\bl{$H(C, C)=1$} 
+\end{itemize}
+
+Contradiction in both cases. So \bl{$H$} cannot exist.
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Take Home Points}
+ \large
+ 
+  \begin{itemize}
+  \item there are sets that are more infinite than others\bigskip
+  \item even  with the most powerful computer we can imagine, there 
+  are problems that cannot be solved by any program\bigskip\bigskip
+  
+  \item in CS we actually hit quite often such problems (halting problem)
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
+  
+
+\end{document}
+
+%%% Local Variables:  
+%%% mode: latex
+%%% TeX-master: t
+%%% End: 
+