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

\usepackage{../slides}

\usepackage{../graphics}

\usepackage{../langs}

\usepackage{../data}

% beamer stuff 

\renewcommand{\slidecaption}{AFL 06, King's College London}

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

%\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions

\newcommand{\qq}{\mbox{\texttt{"}}}

\begin{document}

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\begin{frame}[t]

\frametitle{%

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

  \\[-3mm]

  \LARGE Automata and \\[-2mm] 

  \LARGE Formal Languages (6)\\[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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\begin{frame}[c]

\mbox{}\\[-18mm]\mbox{}

{\lstset{language=Scala}\fontsize{10}{12}\selectfont

\texttt{\lstinputlisting[xleftmargin=0mm]{../progs/pow.scala}}}

\end{frame}

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\begin{frame}[c]

\small

\mbox{}\\[5mm]

%\begin{textblock}{10}(3,5)

\begin{tikzpicture}[scale=1.5,

                    node distance=1cm,

                    every node/.style={minimum size=7mm}]

\node (r1)  {\bl{$r_1$}};

\node (r2) [right=of r1] {\bl{$r_2$}};

\draw[->,line width=1mm]  (r1) -- (r2) node[above,midway] {\bl{$der\,a$}};

\node (r3) [right=of r2] {\bl{$r_3$}};

\draw[->,line width=1mm]  (r2) -- (r3) node[above,midway] {\bl{$der\,b$}};

\node (r4) [right=of r3] {\bl{$r_4$}};

\draw[->,line width=1mm]  (r3) -- (r4) node[above,midway] {\bl{$der\,c$}};

\draw (r4) node[anchor=west] {\;\raisebox{3mm}{\bl{$nullable$}}};

\node (v4) [below=of r4] {\bl{$v_4$}};

\draw[->,line width=1mm]  (r4) -- (v4);

\node (v3) [left=of v4] {\bl{$v_3$}};

\draw[->,line width=1mm]  (v4) -- (v3) node[below,midway] {\bl{$inj\,c$}};

\node (v2) [left=of v3] {\bl{$v_2$}};

\draw[->,line width=1mm]  (v3) -- (v2) node[below,midway] {\bl{$inj\,b$}};

\node (v1) [left=of v2] {\bl{$v_1$}};

\draw[->,line width=1mm]  (v2) -- (v1) node[below,midway] {\bl{$inj\,a$}};

\draw[->,line width=0.5mm]  (r3) -- (v3);

\draw[->,line width=0.5mm]  (r2) -- (v2);

\draw[->,line width=0.5mm]  (r1) -- (v1);

\draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{\bl{$mkeps$}}};

\end{tikzpicture}

%\end{textblock}

\begin{center}

\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}

  \\[-10mm]

  \bl{$inj\,(c)\,c\,Empty$} & \bl{$\dn$}  & \bl{$Char\,c$}\\

  \bl{$inj\,(r_1 + r_2)\,c\,Left(v)$} & \bl{$\dn$}  & \bl{$Left(inj\,r_1\,c\,v)$}\\

  \bl{$inj\,(r_1 + r_2)\,c\,Right(v)$} & \bl{$\dn$}  & \bl{$Right(inj\,r_2\,c\,v)$}\\

  \bl{$inj\,(r_1 \cdot r_2)\,c\,Seq(v_1,v_2)$} & \bl{$\dn$}  & \bl{$Seq(inj\,r_1\,c\,v_1,v_2)$}\\

  \bl{$inj\,(r_1 \cdot r_2)\,c\,Left(Seq(v_1,v_2))$} & \bl{$\dn$}  & \bl{$Seq(inj\,r_1\,c\,v_1,v_2)$}\\

  \bl{$inj\,(r_1 \cdot r_2)\,c\,Right(v)$} & \bl{$\dn$}  & \bl{$Seq(mkeps(r_1),inj\,r_2\,c\,v)$}\\

  \bl{$inj\,(r^*)\,c\,Seq(v,vs)$} & \bl{$\dn$}  & \bl{$inj\,r\,c\,v\,::\,vs$}\\

\end{tabular}

\end{center}

\end{frame}

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\begin{frame}[c]

\frametitle{\begin{tabular}{c}Hierarchy of Languages\end{tabular}}

Recall that languages are sets of strings.

\begin{center}

\begin{tikzpicture}

[rect/.style={draw=black!50, top color=white,bottom color=black!20, rectangle, very thick, rounded corners}]

\draw (0,0) node [rect, text depth=39mm, text width=68mm] {all languages};

\draw (0,-0.4) node [rect, text depth=28.5mm, text width=64mm] {decidable languages};

\draw (0,-0.85) node [rect, text depth=17mm] {context sensitive languages};

\draw (0,-1.14) node [rect, text depth=9mm, text width=50mm] {context-free languages};

\draw (0,-1.4) node [rect] {regular languages};

\end{tikzpicture}

\end{center}

\end{frame}

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\begin{frame}[c]

\frametitle{Two Grammars}

Which languages are recognised by the following two grammars?

\begin{center}

\bl{\begin{tabular}{lcl}

$S$ & $\rightarrow$ &  $1 \cdot S \cdot S$\\

        & $|$ & $\epsilon$

\end{tabular}}

\end{center}\bigskip

\begin{center}

\bl{\begin{tabular}{lcl}

$U$ & $\rightarrow$ &  $1 \cdot U$\\

        & $|$ & $\epsilon$

\end{tabular}}

\end{center}

\end{frame}

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\begin{frame}[c]

Atomic parsers, for example

\begin{center}

\bl{$1::rest \;\Rightarrow\; \{(1, rest)\}$} 

\end{center}\bigskip

\begin{itemize}

\item you consume one or more token from the\\ 

  input (stream)

\item also works for characters and strings

\end{itemize}

\end{frame}

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\begin{frame}[c]

Alternative parser (code \bl{$p\;||\;q$})\bigskip

\begin{itemize}

\item apply \bl{$p$} and also \bl{$q$}; then combine 

  the outputs

\end{itemize}

\begin{center}

\large \bl{$p(\text{input}) \cup q(\text{input})$}

\end{center}

\end{frame}

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\begin{frame}[c]

Sequence parser (code \bl{$p\sim q$})\bigskip

\begin{itemize}

\item apply first \bl{$p$} producing a set of pairs

\item then apply \bl{$q$} to the unparsed parts

\item then combine the results:\medskip 

\begin{center}

((output$_1$, output$_2$), unparsed part)

\end{center}

\end{itemize}

\begin{center}

\begin{tabular}{l}

\large \bl{$\{((o_1, o_2), u_2) \;|\;$}\\[2mm] 

\large\mbox{}\hspace{15mm} \bl{$(o_1, u_1) \in p(\text{input}) \wedge$}\\[2mm]

\large\mbox{}\hspace{15mm} \bl{$(o_2, u_2) \in q(u_1)\}$}

\end{tabular}

\end{center}

\end{frame}

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\begin{frame}[c]

Function parser (code \bl{$p \Rightarrow f\;$})\bigskip

\begin{itemize}

\item apply \bl{$p$} producing a set of pairs

\item then apply the function \bl{$f$} to each first component

\end{itemize}

\begin{center}

\begin{tabular}{l}

\large \bl{$\{(f(o_1), u_1) \;|\; (o_1, u_1) \in p(\text{input})\}$}

\end{tabular}

\end{center}

\end{frame}

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\begin{frame}[t]

\frametitle{Ambiguous Grammars}

\begin{center}

\begin{tikzpicture}

\begin{axis}[xlabel={\pcode{1}s},ylabel={time in secs},

    enlargelimits=false,

    xtick={0,100,...,1000},

    xmax=1050,

    ymax=33,

    ytick={0,5,...,30},

    scaled ticks=false,

    axis lines=left,

    width=11cm,

    height=7cm, 

    legend entries={unambiguous,ambiguous},  

    legend pos=north east,

    legend cell align=left,

    x tick label style={font=\small,/pgf/number format/1000 sep={}}]

\addplot[blue,mark=*, mark options={fill=white}] 

  table {s-grammar1.data};

\only<2>{

  \addplot[red,mark=triangle*, mark options={fill=white}] 

  table {s-grammar2.data};}  

\end{axis}

\end{tikzpicture}

\end{center}

\end{frame}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}Arithmetic Expressions\end{tabular}}

A grammar for arithmetic expressions and numbers:

\begin{center}

\bl{\begin{tabular}{lcl}

$E$ & $\rightarrow$ &  $E \cdot + \cdot E \;|\;E \cdot * \cdot E \;|\;( \cdot E \cdot ) \;|\;N$ \\

$N$ & $\rightarrow$ & $N \cdot N \;|\; 0 \;|\; 1 \;|\: \ldots \;|\; 9$ 

\end{tabular}}

\end{center}

Unfortunately it is left-recursive (and ambiguous).\medskip\\

A problem for \alert{recursive descent parsers} (e.g.~parser combinators).

\bigskip\pause

\end{frame}}

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\mode<presentation>{

\begin{frame}[t]

\frametitle{\begin{tabular}{c}Numbers\end{tabular}}

\begin{center}

\bl{\begin{tabular}{lcl}

$N$ & $\rightarrow$ &  $N \cdot N \;|\; 0 \;|\; 1 \;|\; \ldots \;|\; 9$\\

\end{tabular}}

\end{center}

A non-left-recursive, non-ambiguous grammar for numbers:

\begin{center}

\bl{\begin{tabular}{lcl}

$N$ & $\rightarrow$ &  $0 \cdot N \;|\;1 \cdot N \;|\;\ldots\;|\; 0 \;|\; 1 \;|\; \ldots \;|\; 9$\\

\end{tabular}}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}Removing Left-Recursion\end{tabular}}

The rule for numbers is directly left-recursive:

\begin{center}

\bl{\begin{tabular}{lcl}

$N$ & $\rightarrow$ & $N \cdot N \;|\; 0 \;|\; 1\;\;\;\;(\ldots)$ 

\end{tabular}}

\end{center}

Translate

\begin{center}

\begin{tabular}{ccc}

\bl{\begin{tabular}{lcl}

$N$ & $\rightarrow$ & $N \cdot \alpha$\\

 &  $\;|\;$ & $\beta$\\

 \\ 

\end{tabular}} 

& {\Large$\Rightarrow$} &

\bl{\begin{tabular}{lcl}

$N$ & $\rightarrow$ & $\beta \cdot N'$\\

$N'$ & $\rightarrow$ & $\alpha \cdot N'$\\

 &  $\;|\;$ & $\epsilon$ 

\end{tabular}}

\end{tabular}

\end{center}\pause

Which means

\begin{center}

\bl{\begin{tabular}{lcl}

$N$ & $\rightarrow$ & $0 \cdot N' \;|\; 1 \cdot N'$\\

$N'$ & $\rightarrow$ & $N \cdot N' \;|\; \epsilon$\\

\end{tabular}}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}Operator Precedences\end{tabular}}

To disambiguate

\begin{center}

\bl{\begin{tabular}{lcl}

$E$ & $\rightarrow$ &  $E \cdot + \cdot E \;|\;E \cdot * \cdot E \;|\;( \cdot E \cdot ) \;|\;N$ \\

\end{tabular}}

\end{center}

Decide on how many precedence levels, say\medskip\\

\hspace{5mm}highest for \bl{$()$}, medium for \bl{*}, lowest for \bl{+}

\begin{center}

\bl{\begin{tabular}{lcl}

$E_{low}$ & $\rightarrow$ & $E_{med} \cdot + \cdot E_{low} \;|\; E_{med}$ \\

$E_{med}$ & $\rightarrow$ & $E_{hi} \cdot * \cdot E_{med} \;|\; E_{hi}$\\

$E_{hi}$ & $\rightarrow$ &  $( \cdot E_{low} \cdot ) \;|\;N$ \\

\end{tabular}}

\end{center}\pause

\small What happens with \bl{$1 + 3  + 4$}?

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}Chomsky Normal Form\end{tabular}}

All rules must be of the form

\begin{center}

\bl{$A \rightarrow a$}

\end{center}

or

\begin{center}

\bl{$A \rightarrow B\cdot C$}

\end{center}

No rule can contain \bl{$\epsilon$}.

\end{frame}}

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\begin{frame}[c]