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\usepackage{../langs}

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% 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}

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\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}}

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\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}}

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\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}}

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\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}

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\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}}

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\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}

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\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}

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\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}

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\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}

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

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\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}

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\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}

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\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}

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\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}

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\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}

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

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\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);