\documentclass[dvipsnames,14pt,t]{beamer}
\usepackage{beamerthemeplaincu}
\usepackage[absolute,overlay]{textpos}
\usepackage{ifthen}
\usepackage{tikz}
\usepackage{pgf}
\usepackage{calc} 
\usepackage{ulem}
\usepackage{courier}
\usepackage{listings}
\renewcommand{\uline}[1]{#1}
\usetikzlibrary{arrows}
\usetikzlibrary{automata}
\usetikzlibrary{shapes}
\usetikzlibrary{shadows}
\usetikzlibrary{positioning}
\usetikzlibrary{calc}
\usetikzlibrary{plotmarks}
\usepackage{graphicx} 
\usepackage{pgfplots}
\usepackage{soul}
\usepackage{../langs}
\usepackage{../data}


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  \only<#1>{\pgfkeysalso{#2}} % \pgfkeysalso doesn't change the path
}}

\makeatletter
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  }%
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\makeatother


% beamer stuff 
\renewcommand{\slidecaption}{AFL 10, King's College London, 4.~December 2013}
\newcommand{\bl}[1]{\textcolor{blue}{#1}}       
\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[t]
\frametitle{%
  \begin{tabular}{@ {}c@ {}}
  \\[-3mm]
  \LARGE Automata and \\[-2mm] 
  \LARGE Formal Languages (10)\\[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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{Last Week}

\begin{center}
if \bl{$\varnothing$} does not occur in \bl{$r$} \;\;then\;\;\bl{$L(r) \not= \{\}$}
\end{center}

\noindent
holds, or equivalently

\begin{center}
\bl{$L(r) = \{\}$} \;\;implies\;\; \bl{$\varnothing$} occurs in \bl{$r$}.\pause
\end{center}

\begin{center}
\bl{\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
$occurs(\varnothing)$      & $\dn$ & $true$\\
$occurs(\epsilon)$           & $\dn$ &  $f\!alse$\\
$occurs (c)$                    & $\dn$ &  $f\!alse$\\
$occurs (r_1 + r_2)$       & $\dn$ &  $occurs(r_1) \vee occurs(r_2)$\\ 
$occurs (r_1 \cdot r_2)$ & $\dn$ &  $occurs(r_1) \vee occurs(r_2)$\\
$occurs (r^*)$                & $\dn$ & $occurs(r)$ \\
\end{tabular}}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c,fragile]
\frametitle{Functional Programming}

\footnotesize
\begin{textblock}{13}(0.9,3)
\lstset{emph={def,if,then,else},emphstyle=\color{javapurple}}
\begin{lstlisting}[basicstyle=\ttfamily, numbers=none]
def fib(n) = if n == 0 then 0 
             else if n == 1 then 1 
             else fib(n - 1) + fib(n - 2);

def fact(n) = if n == 0 then 1 else n * fact(n - 1);

def ack(m, n) = if m == 0 then n + 1
                else if n == 0 then ack(m - 1, 1)
                else ack(m - 1, ack(m, n - 1));
                
def gcd(a, b) = if b == 0 then a else gcd(b, a % b);                
\end{lstlisting}
\end{textblock}

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

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

\begin{center}
\bl{\begin{tabular}{@{}lcl@{}}
\textit{Exp} & $\rightarrow$ &  \textit{Var} $|$ \textit{Num}\\
              & $|$ & \textit{Exp} \texttt{+} \textit{Exp} $|$ ... $|$ ( \textit{Exp} )\\
              & $|$ & \texttt{if}\; \textit{BExp} \;\texttt{then}\; \textit{Exp} \;\texttt{else}\; \textit{Exp}\\
              & $|$ & \texttt{write}\;\textit{Exp}\\
              & $|$ & \textit{Exp}\;\texttt{;}\;\textit{Exp}\\
              & $|$ & \textit{FunName} \texttt{(}\textit{Exp},...,\textit{Exp}\texttt{)}\medskip\\
\textit{BExp} & $\rightarrow$ & \ldots\medskip\\
\textit{Decl} & $\rightarrow$ &  \textit{Def} \;\texttt{;}\; \textit{Decl}\\
              & $|$ & \textit{Exp}\medskip\\
\textit{Def} & 
$\rightarrow$ &  \texttt{def} \textit{FunName}\texttt{(}\textit{x}$_1$,..., \textit{x}$_n$\texttt{)} \texttt{=} \textit{Exp}\\
\end{tabular}}
\end{center}


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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c, fragile]
\frametitle{Abstract Syntax Tree}

\footnotesize
\begin{textblock}{13}(0.3,2)
\begin{lstlisting}[language=Scala,basicstyle=\ttfamily, numbers=none]
abstract class Exp
abstract class BExp 
abstract class Decl

case class 
  Def(name: String, args: List[String], body: Exp) 
                                          extends Decl
case class Main(e: Exp) extends Decl

case class Call(name: String, args: List[Exp]) extends Exp
case class If(a: BExp, e1: Exp, e2: Exp) extends Exp
case class Write(e: Exp) extends Exp
case class Var(s: String) extends Exp
case class Num(i: Int) extends Exp
case class Aop(o: String, a1: Exp, a2: Exp) extends Exp
case class Sequ(e1: Exp, e2: Exp) extends Exp

case class Bop(o: String, a1: Exp, a2: Exp) extends BExp
\end{lstlisting}
\end{textblock}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c, fragile]
\frametitle{Mathematical Functions}

Compilation of some mathematical functions:

\begin{center}
\begin{tabular}{lcl}
\texttt{Aop("+", a1, a2)} & $\Rightarrow$ & \texttt{...iadd}\\
\texttt{Aop("-", a1, a2)} & $\Rightarrow$ & \texttt{...isub}\\
\texttt{Aop("*", a1, a2)} & $\Rightarrow$ & \texttt{...imul}\\
\texttt{Aop("/", a1, a2)} & $\Rightarrow$ & \texttt{...idiv}\\
\texttt{Aop("\%", a1, a2)} & $\Rightarrow$ & \texttt{...irem}\\
\end{tabular}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c, fragile]
\frametitle{Boolean Expressions}

Compilation of boolean expressions:

\begin{center}
\begin{tikzpicture}[node distance=2mm and 4mm,
 block/.style={rectangle, minimum size=1cm, draw=black, line width=1mm},
 point/.style={rectangle, inner sep=0mm, minimum size=0mm, fill=red},
 skip loop/.style={red, line width=1mm, to path={-- ++(0,-10mm) -| (\tikztotarget)}}]
\node (A1) [point] {};
\node (b) [block, right=of A1] {code of \bl{$b$}};
\node (A2) [point, right=of b] {};
\node (cs1) [block, right=of A2] {code of \bl{$cs_1$}};
\node (A3) [point, right=of cs1] {};
\node (cs2) [block, right=of A3] {code of \bl{$cs_2$}};
\node (A4) [point, right=of cs2] {};

\only<1>{
\draw (A1) edge [->, red, line width=1mm] (b);
\draw (b) edge [->, red, line width=1mm] (A2);
\draw (A2) edge [skip loop] (A3);
\draw (A3) edge [->, red, line width=1mm] (cs2);
\draw (cs2) edge [->,red, line width=1mm] (A4);
\node [below=of cs1] {\raisebox{-5mm}{\small{}conditional jump}};}
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tabular}{lcl}
\texttt{Bop("==", a1, a2)} & $\Rightarrow$ & \texttt{...if\_icmpne...}\\
\texttt{Bop("!=", a1, a2)} & $\Rightarrow$ & \texttt{...if\_icmpeq...}\\
\texttt{Bop("<", a1, a2)} & $\Rightarrow$ & \texttt{...if\_icmpge...}\\
\texttt{Bop("<=", a1, a2)} & $\Rightarrow$ & \texttt{...if\_icmpgt...}\\
\end{tabular}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t, fragile]
\frametitle{Sequences}

Compiling \texttt{arg1 ; arg2}:


\begin{textblock}{7}(2,5)\footnotesize
\begin{minipage}{6cm}
\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
...arg1...
pop
...arg1...
\end{lstlisting}
\end{minipage}
\end{textblock}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t, fragile]
\frametitle{Write}

Compiling \texttt{write(arg)}:


\begin{textblock}{7}(2,4)\footnotesize
\begin{minipage}{6cm}
\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
...arg...
dup
invokestatic XXX/XXX/write(I)V
\end{lstlisting}
\end{minipage}
\end{textblock}



\begin{textblock}{7}(2,8)\footnotesize
\begin{minipage}{6cm}
\begin{lstlisting}[language=Scala,basicstyle=\ttfamily, numbers=none]
case Write(a1) => {
    compile_exp(a1, env) ++
    List("dup\n",
         "invokestatic XXX/XXX/write(I)V\n")
  }
\end{lstlisting}
\end{minipage}
\end{textblock}
  
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c, fragile]
\frametitle{Functions}

\begin{textblock}{7}(1,2)\footnotesize
\begin{minipage}{6cm}
\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
.method public static write(I)V 
   .limit locals 5 
   .limit stack 5 
   iload 0 
   getstatic java/lang/System/out Ljava/io/PrintStream; 
   swap 
   invokevirtual java/io/PrintStream/println(I)V 
   return 
.end method
\end{lstlisting}
\end{minipage}
\end{textblock}


\begin{textblock}{10}(1,9.8)
\small We will need for definitions\\[-8mm]\mbox{}\footnotesize
\begin{center}
\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
.method public static f (I...I)I
  .limit locals ??
  .limit stack ?? 
  ??
.end method
\end{lstlisting}
\end{center}
\end{textblock}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c, fragile]
\frametitle{Stack Estimation}
\footnotesize
\begin{center}
\begin{lstlisting}[language=Scala,basicstyle=\ttfamily, numbers=none]
def max_stack_exp(e: Exp): Int = e match {
  case Call(_, args) => args.map(max_stack_exp).sum
  case If(a, e1, e2) => max_stack_bexp(a) + 
  	(List(max_stack_exp(e1), max_stack_exp(e1)).max)
  case Write(e) => max_stack_exp(e) + 1
  case Var(_) => 1
  case Num(_) => 1
  case Aop(_, a1, a2) => 
  	max_stack_exp(a1) + max_stack_exp(a2)
  case Sequ(e1, e2) => 
  	List(max_stack_exp(e1), max_stack_exp(e2)).max
}

def max_stack_bexp(e: BExp): Int = e match {
  case Bop(_, a1, a2) => 
  	max_stack_exp(a1) + max_stack_exp(a2)
}
\end{lstlisting}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
\frametitle{Successor}

\begin{textblock}{7}(1,2.5)\footnotesize
\begin{minipage}{6cm}
\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
.method public static suc(I)I 
.limit locals 1
.limit stack
  iload 0
  ldc 1
  iadd
  ireturn
.end method 
\end{lstlisting}
\end{minipage}
\end{textblock}

\begin{textblock}{7}(6,8)
\begin{tikzpicture}\small
\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
{\begin{minipage}{5cm}
\begin{lstlisting}[language=Lisp,basicstyle=\ttfamily, numbers=none]
def suc(x) = x + 1;
\end{lstlisting}
\end{minipage}};
\end{tikzpicture}
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
\frametitle{Addition}

\begin{textblock}{7}(1,1.5)\footnotesize
\begin{minipage}{6cm}
\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
.method public static add(II)I 
.limit locals 2
.limit stack 4
  iload 0
  ldc 0
  if_icmpne If_else_2
  iload 1
  goto If_end_3
If_else_2:
  iload 0
  ldc 1
  isub
  iload 1
  invokestatic defs/defs/add(II)I
  invokestatic defs/defs/suc(I)I
If_end_3:
  ireturn
.end method
\end{lstlisting}
\end{minipage}
\end{textblock}

\begin{textblock}{7}(6,6.2)
\begin{tikzpicture}\small
\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
{\begin{minipage}{7cm}
\begin{lstlisting}[language=Lisp,basicstyle=\ttfamily, numbers=none]
def add(x, y) = 
    if x == 0 then y 
    else suc(add(x - 1, y));
\end{lstlisting}
\end{minipage}};
\end{tikzpicture}
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
\frametitle{Factorial}

\begin{textblock}{7}(1,1.5)\footnotesize
\begin{minipage}{6cm}
\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none]
.method public static facT(II)I 
.limit locals 2
.limit stack 4
  iload 0
  ldc 0	
  if_icmpne If_else_2
  iload 1
  goto If_end_3
If_else_2:
  iload 0
  ldc 1
  isub
  iload 0
  iload 1
  imul
  invokestatic fact/fact/facT(II)I
If_end_3:
  ireturn
.end method 
\end{lstlisting}
\end{minipage}
\end{textblock}

\begin{textblock}{7}(6,7)
\begin{tikzpicture}\small
\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
{\begin{minipage}{7cm}
\begin{lstlisting}[language=Lisp,basicstyle=\ttfamily, numbers=none]
def facT(n, acc) = 
  if n == 0 then acc 
  else facT(n - 1, n * acc);
\end{lstlisting}
\end{minipage}};
\end{tikzpicture}
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
%\frametitle{Factorial}

\begin{textblock}{7}(1,-0.2)\footnotesize
\begin{minipage}{6cm}
\begin{lstlisting}[language=JVMIS,basicstyle=\ttfamily, numbers=none, escapeinside={(*@}{@*)}]
.method public static facT(II)I 
.limit locals 2
.limit stack 4
(*@\hl{facT\_Start:} @*)
  iload 0
  ldc 0
  if_icmpne If_else_2
  iload 1
  goto If_end_3
If_else_2:
  iload 0
  ldc 1
  isub
  iload 0
  iload 1
  imul
  (*@\hl{istore 1} @*)
  (*@\hl{istore 0} @*)
  (*@\hl{goto facT\_Start} @*)
If_end_3:
  ireturn
.end method 
\end{lstlisting}
\end{minipage}
\end{textblock}

\begin{textblock}{7}(6,7)
\begin{tikzpicture}\small
\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
{\begin{minipage}{7cm}
\begin{lstlisting}[language=Lisp,basicstyle=\ttfamily, numbers=none]
def facT(n, acc) = 
  if n == 0 then acc 
  else facT(n - 1, n * acc);
\end{lstlisting}
\end{minipage}};
\end{tikzpicture}
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c, fragile]
\frametitle{Tail Recursion}

A call to \texttt{f(args)} is usually compiled as\medskip

{\small\begin{lstlisting}[basicstyle=\ttfamily, numbers=none]
  args onto stack
  invokestatic  .../f 
\end{lstlisting}}\pause


A call is in tail position provided:\medskip

{\small\begin{itemize}
\item \texttt{if Bexp then \hl{Exp} else \hl{Exp}}
\item \texttt{Exp ; \hl{Exp}}
\item \texttt{Exp  op Exp}
\end{itemize}}\medskip

then a call \texttt{f(args)} can be compiled as\medskip\small

\begin{lstlisting}[basicstyle=\ttfamily, numbers=none]
  prepare environment
  jump to start of function
\end{lstlisting}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c, fragile]
\frametitle{Tail Recursive Call}
\footnotesize
\begin{textblock}{13}(0.3,2)
\begin{lstlisting}[language=Scala,basicstyle=\ttfamily, numbers=none]
def compile_expT(a: Exp, env: Mem, name: String): Instrs = 
  ...
  case Call(n, args) => if (name == n) 
  { 
    val stores = args.zipWithIndex.map 
       { case (x, y) => "istore " + y.toString + "\n" } 
    args.flatMap(a => compile_expT(a, env, "")) ++
    stores.reverse ++ 
    List ("goto " + n + "_Start\n") 
  } 
  else 
  {
    val is = "I" * args.length
    args.flatMap(a => compile_expT(a, env, "")) ++
    List ("invokestatic XXX/XXX/" + n + "(" + is + ")I\n")
  }
\end{lstlisting}
\end{textblock}

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

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

\Large\bf
There are more problems, than there are programs.\bigskip\bigskip\\
There must be a problem for which there is no program.
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

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

\Large
\bl{$A \subseteq B$} and \bl{$B \subseteq A$}\bigskip

then \bl{$A = B$}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

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

\Large
\bl{f} is an injective function iff \bigskip

\bl{$\forall x y.\; f(x) = f(y) \Rightarrow x = y$}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\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{$|A| \leq |B|$}\

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

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

\Large
\bl{$|\mathbb{N} - \{0\}|   \;\;\;\alert{?}\;\;\;  |\mathbb{N}| $}\bigskip\bigskip 

\normalsize
\bl{$\geq$} or \bl{$\leq$} or \bl{$=$} 
\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{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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

\end{document}

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