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\documentclass[dvipsnames,14pt,t]{beamer}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{beamerthemeplaincu}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\usepackage[T1]{fontenc}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\usepackage[latin1]{inputenc}
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\usepackage{mathpartir}
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\usepackage[absolute,overlay]{textpos}
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\usepackage{ifthen}
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\usepackage{tikz}
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\usepackage{pgf}
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\usepackage{calc}
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\usepackage{ulem}
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\usepackage{courier}
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\usepackage{listings}
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\renewcommand{\uline}[1]{#1}
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\usetikzlibrary{arrows}
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\usetikzlibrary{automata}
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\usetikzlibrary{shapes}
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\usetikzlibrary{shadows}
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\usetikzlibrary{positioning}
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\usetikzlibrary{calc}
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\usetikzlibrary{plotmarks}
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\usepackage{graphicx}
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\definecolor{javared}{rgb}{0.6,0,0} % for strings
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\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments
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\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords
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\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc
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\lstset{language=Java,
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basicstyle=\ttfamily,
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keywordstyle=\color{javapurple}\bfseries,
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stringstyle=\color{javagreen},
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commentstyle=\color{javagreen},
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morecomment=[s][\color{javadocblue}]{/**}{*/},
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numbers=left,
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numberstyle=\tiny\color{black},
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stepnumber=1,
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numbersep=10pt,
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tabsize=2,
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showspaces=false,
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showstringspaces=false}
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\lstdefinelanguage{scala}{
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morekeywords={abstract,case,catch,class,def,%
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do,else,extends,false,final,finally,%
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for,if,implicit,import,match,mixin,%
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new,null,object,override,package,%
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private,protected,requires,return,sealed,%
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super,this,throw,trait,true,try,%
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type,val,var,while,with,yield},
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otherkeywords={=>,<-,<\%,<:,>:,\#,@},
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sensitive=true,
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morecomment=[l]{//},
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morecomment=[n]{/*}{*/},
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morestring=[b]",
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morestring=[b]',
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morestring=[b]"""
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}
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\lstset{language=Scala,
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basicstyle=\ttfamily,
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keywordstyle=\color{javapurple}\bfseries,
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stringstyle=\color{javagreen},
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commentstyle=\color{javagreen},
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morecomment=[s][\color{javadocblue}]{/**}{*/},
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numbers=left,
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numberstyle=\tiny\color{black},
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stepnumber=1,
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numbersep=10pt,
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tabsize=2,
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showspaces=false,
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showstringspaces=false}
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% beamer stuff
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\renewcommand{\slidecaption}{AFL 07, King's College London, 13.~November 2013}
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\newcommand{\bl}[1]{\textcolor{blue}{#1}}
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\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[t]
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\\[-3mm]
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\LARGE Automata and \\[-2mm]
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\LARGE Formal Languages (7)\\[3mm]
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\end{tabular}}
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\normalsize
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\begin{center}
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\begin{tabular}{ll}
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Email: & christian.urban at kcl.ac.uk\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Office: & S1.27 (1st floor Strand Building)\\
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Slides: & KEATS (also home work is there)\\
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\end{tabular}
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\qq}{\mbox{\texttt{"}}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\frametitle{\begin{tabular}{c}CFGs\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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A \alert{context-free} grammar (CFG) \bl{$G$} consists of:
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\begin{itemize}
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\item a finite set of nonterminal symbols (upper case)
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\item a finite terminal symbols or tokens (lower case)
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\item a start symbol (which must be a nonterminal)
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\item a set of rules
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{$A \rightarrow \text{rhs}_1 | \text{rhs}_2 | \ldots$}
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\end{center}
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where \bl{rhs} are sequences involving terminals and nonterminals (can also be empty).\medskip\pause
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\frametitle{\begin{tabular}{c}Hierarchie of Languages\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Recall that languages are sets of strings.
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{tikzpicture}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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[rect/.style={draw=black!50, top color=white,bottom color=black!20, rectangle, very thick, rounded corners}]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\draw (0,0) node [rect, text depth=39mm, text width=68mm] {all languages};
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\draw (0,-0.4) node [rect, text depth=28.5mm, text width=64mm] {decidable languages};
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\draw (0,-0.85) node [rect, text depth=17mm] {context sensitive languages};
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\draw (0,-1.14) node [rect, text depth=9mm, text width=50mm] {context-free languages};
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\draw (0,-1.4) node [rect] {regular languages};
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tikzpicture}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\frametitle{\begin{tabular}{c}Arithmetic Expressions\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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A grammar for arithmetic expressions and numbers:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$E$ & $\rightarrow$ & $E \cdot + \cdot E \;|\;E \cdot * \cdot E \;|\;( \cdot E \cdot ) \;|\;N$ \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$N$ & $\rightarrow$ & $N \cdot N \;|\; 0 \;|\; 1 \;|\: \ldots \;|\; 9$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Unfortunately left-recursive (and ambiguous).\medskip\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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A problem for \alert{recursive descent parsers} (e.g.~parser combinators).
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bigskip\pause
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\mode<presentation>{
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[t]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\frametitle{\begin{tabular}{c}Numbers\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$N$ & $\rightarrow$ & $N \cdot N \;|\; 0 \;|\; 1 \;|\; \ldots \;|\; 9$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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A non-left-recursive, non-ambiguous grammar for numbers
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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\bl{\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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$N$ & $\rightarrow$ & $0 \cdot N \;|\;1 \cdot N \;|\;\ldots\;|\; 0 \;|\; 1 \;|\; \ldots \;|\; 9$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{frame}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\mode<presentation>{
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\frametitle{\begin{tabular}{c}Operator Precedences\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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To disambiguate
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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209 |
\bl{\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$E$ & $\rightarrow$ & $E \cdot + \cdot E \;|\;E \cdot * \cdot E \;|\;( \cdot E \cdot ) \;|\;N$ \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Decide how many precedence levels, say\medskip\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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215 |
\hspace{5mm}highest for \bl{$()$}, medium for \bl{*}, lowest for \bl{+}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bl{\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$E_{low}$ & $\rightarrow$ & $E_{med} \cdot + \cdot E_{low} \;|\; E_{med}$ \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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$E_{med}$ & $\rightarrow$ & $E_{hi} \cdot * \cdot E_{med} \;|\; E_{hi}$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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221 |
$E_{hi}$ & $\rightarrow$ & $( \cdot E_{low} \cdot ) \;|\;N$ \\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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222 |
\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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223 |
\end{center}\pause
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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224 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\small What happens with \bl{$1 + 3 + 4$}?
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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226 |
\end{frame}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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227 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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228 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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\mode<presentation>{
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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231 |
\begin{frame}[c]
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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232 |
\frametitle{\begin{tabular}{c}Removing Left-Recursion\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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233 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
234 |
The rule for numbers is directly left-recursive:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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235 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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236 |
\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
237 |
\bl{\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
238 |
$N$ & $\rightarrow$ & $N \cdot N \;|\; 0 \;|\; 1\;\;\;\;(\ldots)$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
239 |
\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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240 |
\end{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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241 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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242 |
Translate
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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243 |
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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244 |
\begin{center}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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245 |
\begin{tabular}{ccc}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
246 |
\bl{\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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247 |
$N$ & $\rightarrow$ & $N \cdot \alpha$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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248 |
& $\;|\;$ & $\beta$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
249 |
\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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250 |
\end{tabular}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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251 |
& {\Large$\Rightarrow$} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
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252 |
\bl{\begin{tabular}{lcl}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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253 |
$N$ & $\rightarrow$ & $\beta \cdot N'$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
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|
254 |
$N'$ & $\rightarrow$ & $\alpha \cdot N'$\\
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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255 |
& $\;|\;$ & $\epsilon$
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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256 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
257 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
258 |
\end{center}\pause
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
259 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
260 |
Which means
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
261 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
262 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
263 |
\bl{\begin{tabular}{lcl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
264 |
$N$ & $\rightarrow$ & $0 \cdot N' \;|\; 1 \cdot N'$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
265 |
$N'$ & $\rightarrow$ & $N \cdot N' \;|\; \epsilon$\\
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
266 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
267 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
268 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
269 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
270 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
271 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
272 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
273 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
274 |
\mode<presentation>{
|
63
|
275 |
\begin{frame}[c]
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
276 |
\frametitle{\begin{tabular}{c}Chomsky Normal Form\end{tabular}}
|
63
|
277 |
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
278 |
All rules must be of the form
|
63
|
279 |
|
|
280 |
\begin{center}
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
281 |
\bl{$A \rightarrow a$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
282 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
283 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
284 |
or
|
63
|
285 |
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
286 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
287 |
\bl{$A \rightarrow B\cdot C$}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
288 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
289 |
|
185
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
290 |
No rule can contain \bl{$\epsilon$}.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
291 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
292 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
293 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
294 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
295 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
296 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
297 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
298 |
\frametitle{\begin{tabular}{c}$\epsilon$-Removal\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
299 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
300 |
\begin{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
301 |
\item If \bl{$A\rightarrow \alpha \cdot B \cdot \beta$} and \bl{$B \rightarrow \epsilon$} are in the grammar,
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
302 |
then add \bl{$A\rightarrow \alpha \cdot \beta$} (iterate if necessary).
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
303 |
\item Throw out all \bl{$B \rightarrow \epsilon$}.
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
304 |
\end{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
305 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
306 |
\small
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
307 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
308 |
\begin{tabular}{ccc}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
309 |
\bl{\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
310 |
$N$ & $\rightarrow$ & $0 \cdot N' \;|\; 1\cdot N'$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
311 |
$N'$ & $\rightarrow$ & $N \cdot N'\;|\;\epsilon$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
312 |
\end{tabular}} &
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
313 |
\bl{\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
314 |
$N$ & $\rightarrow$ & $0 \cdot N' \;|\; 1\cdot N'\;|\;0\;|\;1$\\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
315 |
$N'$ & $\rightarrow$ & $N \cdot N'\;|\;N\;|\;\epsilon$
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
316 |
\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
317 |
\end{tabular}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
318 |
\end{center}
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
319 |
|
63
|
320 |
|
|
321 |
\end{frame}}
|
|
322 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
185
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
323 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
324 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
325 |
|
63
|
326 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
327 |
\mode<presentation>{
|
|
328 |
\begin{frame}[c]
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
329 |
\frametitle{\begin{tabular}{c}CYK Algorithm\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
330 |
|
63
|
331 |
|
|
332 |
\begin{center}
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
333 |
\bl{\begin{tabular}{@ {}lcl}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
334 |
$S$ & $\rightarrow$ & $N\cdot P$ \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
335 |
$P$ & $\rightarrow$ & $V\cdot N$ \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
336 |
$N$ & $\rightarrow$ & $N\cdot N$ \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
337 |
$N$ & $\rightarrow$ & $\texttt{students} \;|\; \texttt{Jeff} \;|\; \texttt{geometry} \;|\; \texttt{trains} $ \\
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
338 |
$V$ & $\rightarrow$ & $\texttt{trains}$
|
63
|
339 |
\end{tabular}}
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
340 |
\end{center}
|
63
|
341 |
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
342 |
\bl{\texttt{Jeff trains geometry students}}
|
63
|
343 |
|
|
344 |
\end{frame}}
|
|
345 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
346 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
347 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
348 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
349 |
\frametitle{\begin{tabular}{c}CYK Algorithm\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
350 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
351 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
352 |
\begin{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
353 |
\item runtime is \bl{$O(n^3)$}\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
354 |
\item grammars need to be transferred into CNF
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
355 |
\end{itemize}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
356 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
357 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
358 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
63
|
359 |
|
|
360 |
|
|
361 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
362 |
\mode<presentation>{
|
|
363 |
\begin{frame}[c]
|
|
364 |
\frametitle{\begin{tabular}{c}Parse Trees\end{tabular}}
|
|
365 |
|
|
366 |
\begin{center}
|
|
367 |
\bl{\begin{tabular}{lcl}
|
|
368 |
$E$ & $\rightarrow$ & $F \;|\; F \cdot * \cdot F$\\
|
|
369 |
$F$ & $\rightarrow$ & $T \;|\; T \cdot + \cdot T \;|\; T \cdot - \cdot T$\\
|
|
370 |
$T$ & $\rightarrow$ & $num\_token \;|\; ( \cdot E \cdot )$\\
|
|
371 |
\end{tabular}}
|
|
372 |
\end{center}
|
|
373 |
|
|
374 |
\begin{center}
|
|
375 |
\begin{tikzpicture}[level distance=8mm, blue]
|
|
376 |
\node {$E$}
|
|
377 |
child {node {$F$}
|
|
378 |
child {node {$T$}
|
|
379 |
child {node {(\,$E$\,)}
|
|
380 |
child {node{$F$ *{} $F$}
|
|
381 |
child {node {$T$} child {node {2}}}
|
|
382 |
child {node {$T$} child {node {3}}}
|
|
383 |
}
|
|
384 |
}
|
|
385 |
}
|
|
386 |
child {node {+}}
|
|
387 |
child {node {$T$}
|
|
388 |
child {node {(\,$E$\,)}
|
|
389 |
child {node {$F$}
|
|
390 |
child {node {$T$ +{} $T$}
|
|
391 |
child {node {3}}
|
|
392 |
child {node {4}}
|
|
393 |
}
|
|
394 |
}}
|
|
395 |
}};
|
|
396 |
\end{tikzpicture}
|
|
397 |
\end{center}
|
|
398 |
|
|
399 |
\begin{textblock}{5}(1, 6.5)
|
|
400 |
\bl{\texttt{(2*3)+(3+4)}}
|
|
401 |
\end{textblock}
|
|
402 |
|
|
403 |
\end{frame}}
|
|
404 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
405 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
406 |
\mode<presentation>{
|
|
407 |
\begin{frame}[c]
|
|
408 |
\frametitle{\begin{tabular}{c}Ambiguous Grammars\end{tabular}}
|
|
409 |
|
|
410 |
A CFG is \alert{ambiguous} if there is a string that has at least parse trees.
|
|
411 |
|
|
412 |
|
|
413 |
\begin{center}
|
|
414 |
\bl{\begin{tabular}{lcl}
|
|
415 |
$E$ & $\rightarrow$ & $num\_token$ \\
|
|
416 |
$E$ & $\rightarrow$ & $E \cdot + \cdot E$ \\
|
|
417 |
$E$ & $\rightarrow$ & $E \cdot - \cdot E$ \\
|
|
418 |
$E$ & $\rightarrow$ & $E \cdot * \cdot E$ \\
|
|
419 |
$E$ & $\rightarrow$ & $( \cdot E \cdot )$
|
|
420 |
\end{tabular}}
|
|
421 |
\end{center}
|
|
422 |
|
|
423 |
\bl{\texttt{1 + 2 * 3 + 4}}
|
|
424 |
|
|
425 |
\end{frame}}
|
|
426 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
427 |
|
|
428 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
429 |
\mode<presentation>{
|
|
430 |
\begin{frame}[c]
|
|
431 |
\frametitle{\begin{tabular}{c}Dangling Else\end{tabular}}
|
|
432 |
|
|
433 |
Another ambiguous grammar:\bigskip
|
|
434 |
|
|
435 |
\begin{center}
|
|
436 |
\bl{\begin{tabular}{lcl}
|
|
437 |
$E$ & $\rightarrow$ & if $E$ then $E$\\
|
|
438 |
& $|$ & if $E$ then $E$ else $E$ \\
|
|
439 |
& $|$ & id
|
|
440 |
\end{tabular}}
|
|
441 |
\end{center}\bigskip
|
|
442 |
|
|
443 |
\bl{\texttt{if a then if x then y else c}}
|
|
444 |
|
|
445 |
\end{frame}}
|
|
446 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
447 |
|
184
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
448 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
449 |
\mode<presentation>{
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
450 |
\begin{frame}[c]
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
451 |
\frametitle{\begin{tabular}{c}A CFG Derivation\end{tabular}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
452 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
453 |
\begin{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
454 |
\item Begin with a string with only the start symbol \bl{$S$}\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
455 |
\item Replace any non-terminal \bl{$X$} in the string by the
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
456 |
right-hand side of some production \bl{$X \rightarrow \text{rhs}$}\bigskip
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
457 |
\item Repeat 2 until there are no non-terminals
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
458 |
\end{enumerate}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
459 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
460 |
\begin{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
461 |
\bl{$S \rightarrow \ldots \rightarrow \ldots \rightarrow \ldots \rightarrow \ldots $}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
462 |
\end{center}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
463 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
464 |
\end{frame}}
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
465 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
466 |
|
Christian Urban <christian dot urban at kcl dot ac dot uk>
diff
changeset
|
467 |
|
63
|
468 |
\end{document}
|
|
469 |
|
|
470 |
%%% Local Variables:
|
|
471 |
%%% mode: latex
|
|
472 |
%%% TeX-master: t
|
|
473 |
%%% End:
|
|
474 |
|