% !TEX program = xelatex

\documentclass[dvipsnames,14pt,t,xelatex,aspectratio=169,xcolor={table}]{beamer}

\usepackage{../slides}

\usepackage{../graphics}

\usepackage{../langs}

\usepackage{../data}

\usepackage{../grammar}

% beamer stuff 

\renewcommand{\slidecaption}{CFL 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 Compilers and \\[-2mm] 

  \LARGE Formal Languages\\[3mm] 

  \end{tabular}}

  \normalsize

  \begin{center}

  \begin{tabular}{ll}

    Email:  & christian.urban at kcl.ac.uk\\

    %Office Hours: & Thursdays 12 -- 14\\

    %Location: & N7.07 (North Wing, Bush House)\\

    Slides \& Progs: & KEATS (also homework is there)\\  

  \end{tabular}

  \end{center}

  \begin{center}

    \begin{tikzpicture}

      \node[drop shadow,fill=white,inner sep=0pt] 

      {\footnotesize\rowcolors{1}{capri!10}{white}

        \begin{tabular}{|p{4.8cm}|p{4.8cm}|}\hline

          1 Introduction, Languages          & \cellcolor{blue!50} 6 While-Language \\

          2 Regular Expressions, Derivatives & 7 Compilation, JVM \\

          3 Automata, Regular Languages      & 8 Compiling Functional Languages \\

          4 Lexing, Tokenising               & 9 Optimisations \\

          5 Grammars, Parsing                & 10 LLVM \\ \hline

        \end{tabular}%

      };

    \end{tikzpicture}

  \end{center}

\end{frame}

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\begin{frame}[c]

\end{frame}

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\begin{frame}

\frametitle{While-Language}

\mbox{}\\[-23mm]\mbox{}

\bl{\begin{plstx}[rhs style=,one per line]: \meta{Stmt} ::= skip

         | \meta{Id} := \meta{AExp}

         | if \meta{BExp} then \meta{Block} else \meta{Block}

         | while \meta{BExp} do \meta{Block}\\

: \meta{Stmts} ::= \meta{Stmt} ; \meta{Stmts}

          | \meta{Stmt}\\

: \meta{Block} ::= \{ \meta{Stmts} \}

          | \meta{Stmt}\\

: \meta{AExp} ::= \ldots\\

: \meta{BExp} ::= \ldots\\\end{plstx}}

\end{frame}

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\begin{frame}[c]

\begin{center}

\end{tabular}}

\end{center}

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\mode<presentation>{

\begin{frame}[c]

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

\begin{center}

\bl{\begin{tabular}{@{}lcl@{}}

$\text{eval}(n, E)$ & $\dn$ & $n$\\

$\text{eval}(x, E)$ & $\dn$ & $E(x)$ \;\;\;\textcolor{black}{lookup \bl{$x$} in \bl{$E$}}\\

$\text{eval}(a_1 + a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) + \text{eval}(a_2, E)$\\

$\text{eval}(a_1 - a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) - \text{eval}(a_2, E)$\\

$\text{eval}(a_1 * a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) * \text{eval}(a_2, E)$\bigskip\\

$\text{eval}(a_1 = a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) = \text{eval}(a_2, E)$\\

$\text{eval}(a_1\,!\!= a_2, E)$ & $\dn$ & $\neg(\text{eval}(a_1, E) = \text{eval}(a_2, E))$\\

$\text{eval}(a_1 < a_2, E)$ & $\dn$ & $\text{eval}(a_1, E) < \text{eval}(a_2, E)$\

\end{tabular}}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}Interpreter (2)\end{tabular}}

\begin{center}

\bl{\begin{tabular}{@{}lcl@{}}

$\text{eval}(\text{skip}, E)$ & $\dn$ & $E$\\

$\text{eval}(x:=a, E)$ & $\dn$ & \bl{$E(x \mapsto \text{eval}(a, E))$}\\

\multicolumn{3}{@{}l@{}}{$\text{eval}(\text{if}\;b\;\text{then}\;cs_1\;\text{else}\;cs_2 , E) \dn$}\\

\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\text{if}\;\text{eval}(b,E)\;\text{then}\;

\text{eval}(cs_1,E)$}\\

\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\phantom{\text{if}\;\text{eval}(b,E)\;}\text{else}\;\text{eval}(cs_2,E)$}\\

\multicolumn{3}{@{}l@{}}{$\text{eval}(\text{while}\;b\;\text{do}\;cs, E) \dn$}\\

\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\text{if}\;\text{eval}(b,E)$}\\

\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\text{then}\;

\text{eval}(\text{while}\;b\;\text{do}\;cs, \text{eval}(cs,E))$}\\

\multicolumn{3}{@{}l@{}}{\hspace{2cm}$\text{else}\; E$}\\

$\text{eval}(\text{write}\; x, E)$ & $\dn$ & $\{\;\text{println}(E(x))\; ;\;E\;\}$\\

\end{tabular}}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[t]

\frametitle{\begin{tabular}{c}Interpreted Code\end{tabular}}

\begin{center}

\begin{tikzpicture}

\begin{axis}[axis x line=bottom, axis y line=left, xlabel=n, ylabel=secs, legend style=small]

\addplot+[smooth] file {interpreted.data};

\end{axis}

\end{tikzpicture}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}Java Virtual Machine\end{tabular}}

\begin{itemize}

\item introduced in 1995

\item is a stack-based VM (like Postscript, CLR of .Net)

\item contains a JIT compiler

\item many languages take advantage of JVM's infrastructure (JRE)

\item is garbage collected $\Rightarrow$ no buffer overflows

\item some languages compile to the JVM: Scala, Clojure\ldots

\end{itemize}

\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}[t]

\frametitle{Starting Symbol}

\bl{\begin{plstx}[margin=1cm]

  : \meta{S} ::= \meta{A}\cdot\meta{S}\cdot\meta{B} |

                 \meta{B}\cdot\meta{S}\cdot\meta{A} | \epsilon\\

  : \meta{A} ::= a | \epsilon\\

  : \meta{B} ::= b\\  

\end{plstx}}

TODO: Testcases for math expressions

\url{https://github.com/ArashPartow/math-parser-benchmark-project}

\end{frame}

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\begin{frame}[c]

\frametitle{Hierarchy of Languages}

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{Parser Combinators}

Atomic parsers, for example

\begin{center}

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

\end{center}\bigskip

\begin{itemize}

\item you consume one or more tokens 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}[c]

\frametitle{Types of Parsers}

\begin{itemize}

\item {\bf Sequencing}: if \bl{$p$} returns results of type \bl{$T$}, and \bl{$q$} results of type \bl{$S$},

then \bl{$p \sim q$} returns results of type

\begin{center}

\bl{$T \times S$}

\end{center}\pause

\item {\bf Alternative}: if \bl{$p$} returns results of type \bl{$T$} then  \bl{$q$} \alert{must} also have results of type \bl{$T$},

and \bl{$p \;|\; q$} returns results of type

\begin{center}

\bl{$T$}

\end{center}\pause

\item {\bf Semantic Action}: if \bl{$p$} returns results of type \bl{$T$} and \bl{$f$} is a function from

\bl{$T$} to \bl{$S$}, then

\bl{$p \Rightarrow f$} returns results of type

\begin{center}

\bl{$S$}

\end{center}

\end{itemize}

\end{frame}

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\begin{frame}[c]

\frametitle{Two Grammars}

Which languages are recognised by the following two grammars?

\bl{\begin{plstx}[margin=3cm]

: \meta{S} ::= \liningnums{1}\cdot\meta{S}\cdot \meta{S} | \epsilon\\

\end{plstx}}

\bl{\begin{plstx}[margin=3cm]

: \meta{U} ::= \liningnums{1}\cdot\meta{U} | \epsilon\\

\end{plstx}}

\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,