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\documentclass[dvipsnames,14pt,t]{beamer}
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\usepackage{beamerthemeplainculight}
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\usepackage[T1]{fontenc}
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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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\renewcommand{\slidecaption}{AFL 06, King's College London, 31.~October 2012}
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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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% The data files, written on the first run.
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\begin{filecontents}{re-python.data}
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1 0.029
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5 0.029
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10 0.029
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15 0.032
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16 0.042
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17 0.042
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18 0.055
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19 0.084
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20 0.136
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21 0.248
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22 0.464
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23 0.899
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24 1.773
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25 3.505
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26 6.993
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27 14.503
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28 29.307
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#29 58.886
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\end{filecontents}
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\begin{filecontents}{re1.data}
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1 0.00179
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2 0.00011
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3 0.00014
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4 0.00026
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5 0.00050
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6 0.00095
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7 0.00190
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8 0.00287
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9 0.00779
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10 0.01399
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11 0.01894
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12 0.03666
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13 0.07994
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14 0.08944
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15 0.02377
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16 0.07392
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17 0.22798
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18 0.65310
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19 2.11360
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20 6.31606
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21 21.46013
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\end{filecontents}
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\begin{filecontents}{re2.data}
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1 0.00240
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2 0.00013
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3 0.00020
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4 0.00030
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5 0.00049
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6 0.00083
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7 0.00146
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8 0.00228
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9 0.00351
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10 0.00640
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11 0.01217
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12 0.02565
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13 0.01382
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14 0.02423
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15 0.05065
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16 0.06522
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17 0.02140
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18 0.05176
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19 0.18254
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20 0.51898
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21 1.39631
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22 2.69501
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23 8.07952
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\end{filecontents}
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\begin{filecontents}{re-internal.data}
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1 0.00069
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301 0.00700
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601 0.00297
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901 0.00470
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1201 0.01301
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1501 0.01175
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1801 0.01761
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2101 0.01787
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2401 0.02717
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2701 0.03932
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3001 0.03470
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3301 0.04349
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3601 0.05411
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3901 0.06181
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4201 0.07119
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4501 0.08578
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\end{filecontents}
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\begin{filecontents}{re3.data}
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1 0.001605
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501 0.131066
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1001 0.057885
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1501 0.136875
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2001 0.176238
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2501 0.254363
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3001 0.37262
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3501 0.500946
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4001 0.638384
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4501 0.816605
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5001 1.00491
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5501 1.232505
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6001 1.525672
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6501 1.757502
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7001 2.092784
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7501 2.429224
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8001 2.803037
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8501 3.463045
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9001 3.609
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9501 4.081504
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10001 4.54569
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\end{filecontents}
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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 (6)\\[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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Of$\!$fice: & 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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\begin{tabular}{c}ReDoS\end{tabular}}
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\begin{itemize}
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\item \alert{R}egular \alert{e}xpression \alert{D}enial \alert{o}f \alert{S}ervice\bigskip
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\item ``Regular Expressions Will Stab You in the Back''\bigskip
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\item Evil regular expressions\medskip
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\begin{itemize}
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\item \bl{$(a?\{n\})a\{n\}$}
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\item \bl{$(a^+)^+$}
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\item \bl{$([a-zA-Z]^+)^*$}
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\item \bl{$(a + aa)^+$}
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\item \bl{$(a + a?)^+$}
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\end{itemize}
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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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\frametitle{\begin{tabular}{c}Regexp Matching\end{tabular}}
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Given a regular expression
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\begin{enumerate}
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\item you might convert it into a DFA (subset construction)
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\item you might try all possible paths in an NFA via backtracking
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\item you might try all paths in an NFA in parallel
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\item you might try to convert the DFA ``lazily''
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\end{enumerate}\bigskip
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Often No~2 is implemented (sometimes there are even good reasons for doing this).
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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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\frametitle{\begin{tabular}{c}\bl{$(a?\{n\})a\{n\}$} in Python\end{tabular}}
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\begin{tikzpicture}[y=.2cm, x=.3cm]
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%axis
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\draw (0,0) -- coordinate (x axis mid) (30,0);
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\draw (0,0) -- coordinate (y axis mid) (0,30);
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%ticks
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\foreach \x in {0,5,...,30}
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\draw (\x,1pt) -- (\x,-3pt)
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node[anchor=north] {\x};
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\foreach \y in {0,5,...,30}
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\draw (1pt,\y) -- (-3pt,\y)
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node[anchor=east] {\y};
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%labels
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\node[below=0.6cm] at (x axis mid) {\bl{a}s};
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\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
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%plots
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\draw[color=blue] plot[mark=*, mark options={fill=white}]
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file {re-python.data};
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\only<2->{
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\draw[color=red] plot[mark=triangle*, mark options={fill=white} ]
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file {re1.data};}
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\only<3->{
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\draw[color=green] plot[mark=square*, mark options={fill=white} ]
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file {re2.data};}
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%legend
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\begin{scope}[shift={(4,20)}]
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\draw[color=blue] (0,0) --
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plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0)
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node[right]{\small Python};
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\only<2->{\draw[yshift=\baselineskip, color=red] (0,0) --
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plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (0.5,0)
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node[right]{\small Scala V1};}
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\only<3->{
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\draw[yshift=2\baselineskip, color=green] (0,0) --
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plot[mark=square*, mark options={fill=white}] (0.25,0) -- (0.5,0)
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node[right]{\small Scala V2 with simplifications};}
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\end{scope}
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\end{tikzpicture}
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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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\begin{tikzpicture}[y=.7cm, x=.0009cm]
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%axis
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\draw (0,0) -- coordinate (x axis mid) (10000,0);
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\draw (0,0) -- coordinate (y axis mid) (0,6);
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%ticks
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\foreach \x in {0,2000,...,10000}
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\draw (\x,1pt) -- (\x,-3pt)
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node[anchor=north] {\x};
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\foreach \y in {0,1,...,6}
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\draw (1pt,\y) -- (-3pt,\y)
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node[anchor=east] {\y};
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%labels
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\node[below=0.6cm] at (x axis mid) {\bl{a}s};
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\node[rotate=90, left=1.2cm] at (y axis mid) {secs};
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%plots
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\draw[color=blue] plot[mark=*, mark options={fill=white}]
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file {re-internal.data};
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\only<2->{
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\draw[color=red] plot[mark=triangle*, mark options={fill=white} ]
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file {re3.data};}
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%legend
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\begin{scope}[shift={(2000,4)}]
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\draw[color=blue] (0,0) --
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plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0)
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node[right]{\small Scala Internal};
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\only<2->{
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\draw[yshift=\baselineskip, color=red] (0,0) --
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plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (0.5,0)
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node[right]{\small Scala V3 with explicit $\_\{\_\}$};}
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\end{scope}
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\end{tikzpicture}
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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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\frametitle{\begin{tabular}{c}Last Week\end{tabular}}
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Last week I showed you\bigskip
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\begin{itemize}
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359 |
\item an algorithm for automata minimisation
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360 |
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361 |
\item an algorithm for transforming a regular expression into an NFA
|
|
|
362 |
|
|
|
363 |
\item an algorithm for transforming an NFA into a DFA (subset construction)
|
|
|
364 |
|
|
|
365 |
\end{itemize}
|
|
|
366 |
|
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|
367 |
\end{frame}}
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368 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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369 |
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370 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
371 |
\mode<presentation>{
|
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|
372 |
\begin{frame}[c]
|
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|
373 |
\frametitle{\begin{tabular}{c}This Week\end{tabular}}
|
|
|
374 |
|
|
|
375 |
Go over the algorithms again, but with two new things and \ldots\medskip
|
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|
376 |
|
|
|
377 |
\begin{itemize}
|
|
|
378 |
\item with the example: what is the regular expression that accepts every string, except those ending
|
|
|
379 |
in \bl{aa}?\medskip
|
|
|
380 |
|
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|
381 |
\item Go over the proof for \bl{$L(rev(r)) = Rev(L(r))$}.\medskip
|
|
|
382 |
|
|
|
383 |
\item Anything else so far.
|
|
|
384 |
\end{itemize}
|
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|
385 |
|
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|
386 |
\end{frame}}
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|
387 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
388 |
|
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|
389 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
390 |
\mode<presentation>{
|
|
|
391 |
\begin{frame}[c]
|
|
|
392 |
\frametitle{\begin{tabular}{c}Proofs By Induction\end{tabular}}
|
|
|
393 |
|
|
|
394 |
\begin{itemize}
|
|
|
395 |
\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip
|
|
|
396 |
\item \bl{$P$} holds for \bl{r$_1$ + r$_2$} under the assumption that \bl{$P$} already
|
|
|
397 |
holds for \bl{r$_1$} and \bl{r$_2$}.\bigskip
|
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|
398 |
\item \bl{$P$} holds for \bl{r$_1$ $\cdot$ r$_2$} under the assumption that \bl{$P$} already
|
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|
399 |
holds for \bl{r$_1$} and \bl{r$_2$}.
|
|
|
400 |
\item \bl{$P$} holds for \bl{r$^*$} under the assumption that \bl{$P$} already
|
|
|
401 |
holds for \bl{r}.
|
|
|
402 |
\end{itemize}
|
|
|
403 |
|
|
|
404 |
\begin{center}
|
|
|
405 |
\bl{$P(r):\;\;L(rev(r)) = Rev(L(r))$}
|
|
|
406 |
\end{center}
|
|
|
407 |
|
|
|
408 |
\end{frame}}
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409 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
410 |
|
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|
411 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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412 |
\mode<presentation>{
|
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|
413 |
\begin{frame}[t]
|
|
|
414 |
|
|
|
415 |
What is the regular expression that accepts every string, except those ending
|
|
|
416 |
in \bl{aa}?\pause\bigskip
|
|
|
417 |
|
|
|
418 |
\begin{center}
|
|
|
419 |
\begin{tabular}{l}
|
|
|
420 |
\bl{(a + b)$^*$ba}\\
|
|
|
421 |
\bl{(a + b)$^*$ab}\\
|
|
|
422 |
\bl{(a + b)$^*$bb}\\\pause
|
|
|
423 |
\bl{a}\\
|
|
|
424 |
\bl{\texttt{""}}
|
|
|
425 |
\end{tabular}
|
|
|
426 |
\end{center}\pause
|
|
|
427 |
|
|
|
428 |
What are the strings to be avoided?\pause\medskip
|
|
|
429 |
|
|
|
430 |
\begin{center}
|
|
|
431 |
\bl{(a + b)$^*$aa}
|
|
|
432 |
\end{center}
|
|
|
433 |
|
|
|
434 |
\end{frame}}
|
|
|
435 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
436 |
|
|
|
437 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
438 |
\mode<presentation>{
|
|
|
439 |
\begin{frame}[t]
|
|
|
440 |
|
|
|
441 |
An NFA for \bl{(a + b)$^*$aa}
|
|
|
442 |
|
|
|
443 |
\begin{center}
|
|
|
444 |
\begin{tikzpicture}[scale=2, line width=0.5mm]
|
|
|
445 |
\node[state, initial] (q0) at ( 0,1) {$q_0$};
|
|
|
446 |
\node[state] (q1) at ( 1,1) {$q_1$};
|
|
|
447 |
\node[state, accepting] (q2) at ( 2,1) {$q_2$};
|
|
|
448 |
\path[->] (q0) edge node[above] {$a$} (q1)
|
|
|
449 |
(q1) edge node[above] {$a$} (q2)
|
|
|
450 |
(q0) edge [loop below] node {$a$} ()
|
|
|
451 |
(q0) edge [loop above] node {$b$} ()
|
|
|
452 |
;
|
|
|
453 |
\end{tikzpicture}
|
|
|
454 |
\end{center}\pause
|
|
|
455 |
|
|
|
456 |
Minimisation for DFAs\\
|
|
|
457 |
Subset Construction for NFAs
|
|
|
458 |
|
|
|
459 |
\end{frame}}
|
|
|
460 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
461 |
|
|
|
462 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
463 |
\mode<presentation>{
|
|
|
464 |
\begin{frame}[c]
|
|
|
465 |
\frametitle{\begin{tabular}{c}DFA Minimisation\end{tabular}}
|
|
|
466 |
|
|
|
467 |
|
|
|
468 |
\begin{enumerate}
|
|
|
469 |
\item Take all pairs \bl{(q, p)} with \bl{q $\not=$ p}
|
|
|
470 |
\item Mark all pairs that accepting and non-accepting states
|
|
|
471 |
\item For all unmarked pairs \bl{(q, p)} and all characters \bl{c} tests wether
|
|
|
472 |
\begin{center}
|
|
|
473 |
\bl{($\delta$(q,c), $\delta$(p,c))}
|
|
|
474 |
\end{center}
|
|
|
475 |
are marked. If yes, then also mark \bl{(q, p)}.
|
|
|
476 |
\item Repeat last step until nothing changed.
|
|
|
477 |
\item All unmarked pairs can be merged.
|
|
|
478 |
\end{enumerate}
|
|
|
479 |
|
|
|
480 |
\end{frame}}
|
|
|
481 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
482 |
|
|
|
483 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
484 |
\mode<presentation>{
|
|
|
485 |
\begin{frame}[c]
|
|
|
486 |
|
|
|
487 |
Minimal DFA \only<1>{\bl{(a + b)$^*$aa}}\only<2->{\alert{not} \bl{(a + b)$^*$aa}}
|
|
|
488 |
|
|
|
489 |
\begin{center}
|
|
|
490 |
\begin{tikzpicture}[scale=2, line width=0.5mm]
|
|
|
491 |
\only<1>{\node[state, initial] (q0) at ( 0,1) {$q_0$};}
|
|
|
492 |
\only<2->{\node[state, initial,accepting] (q0) at ( 0,1) {$q_0$};}
|
|
|
493 |
\only<1>{\node[state] (q1) at ( 1,1) {$q_1$};}
|
|
|
494 |
\only<2->{\node[state,accepting] (q1) at ( 1,1) {$q_1$};}
|
|
|
495 |
\only<1>{\node[state, accepting] (q2) at ( 2,1) {$q_2$};}
|
|
|
496 |
\only<2->{\node[state] (q2) at ( 2,1) {$q_2$};}
|
|
|
497 |
\path[->] (q0) edge[bend left] node[above] {$a$} (q1)
|
|
|
498 |
(q1) edge[bend left] node[above] {$b$} (q0)
|
|
|
499 |
(q2) edge[bend left=50] node[below] {$b$} (q0)
|
|
|
500 |
(q1) edge node[above] {$a$} (q2)
|
|
|
501 |
(q2) edge [loop right] node {$a$} ()
|
|
|
502 |
(q0) edge [loop below] node {$b$} ()
|
|
|
503 |
;
|
|
|
504 |
\end{tikzpicture}
|
|
|
505 |
\end{center}
|
|
|
506 |
|
|
|
507 |
\onslide<3>{How to get from a DFA to a regular expression?}
|
|
|
508 |
|
|
|
509 |
\end{frame}}
|
|
|
510 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
511 |
|
|
|
512 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
513 |
\mode<presentation>{
|
|
|
514 |
\begin{frame}[c]
|
|
|
515 |
|
|
|
516 |
\begin{center}
|
|
|
517 |
\begin{tikzpicture}[scale=2, line width=0.5mm]
|
|
|
518 |
\only<1->{\node[state, initial] (q0) at ( 0,1) {$q_0$};}
|
|
|
519 |
\only<1->{\node[state] (q1) at ( 1,1) {$q_1$};}
|
|
|
520 |
\only<1->{\node[state] (q2) at ( 2,1) {$q_2$};}
|
|
|
521 |
\path[->] (q0) edge[bend left] node[above] {$a$} (q1)
|
|
|
522 |
(q1) edge[bend left] node[above] {$b$} (q0)
|
|
|
523 |
(q2) edge[bend left=50] node[below] {$b$} (q0)
|
|
|
524 |
(q1) edge node[above] {$a$} (q2)
|
|
|
525 |
(q2) edge [loop right] node {$a$} ()
|
|
|
526 |
(q0) edge [loop below] node {$b$} ()
|
|
|
527 |
;
|
|
|
528 |
\end{tikzpicture}
|
|
|
529 |
\end{center}\pause\bigskip
|
|
|
530 |
|
|
|
531 |
\onslide<2->{
|
|
|
532 |
\begin{center}
|
|
|
533 |
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
|
|
|
534 |
\bl{$q_0$} & \bl{$=$} & \bl{$2\, q_0 + 3 \,q_1 + 4\, q_2$}\\
|
|
|
535 |
\bl{$q_1$} & \bl{$=$} & \bl{$2 \,q_0 + 3\, q_1 + 1\, q_2$}\\
|
|
|
536 |
\bl{$q_2$} & \bl{$=$} & \bl{$1\, q_0 + 5\, q_1 + 2\, q_2$}\\
|
|
|
537 |
|
|
|
538 |
\end{tabular}
|
|
|
539 |
\end{center}
|
|
|
540 |
}
|
|
|
541 |
|
|
|
542 |
\end{frame}}
|
|
|
543 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
544 |
|
|
|
545 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
546 |
\mode<presentation>{
|
|
|
547 |
\begin{frame}[c]
|
|
|
548 |
|
|
|
549 |
\begin{center}
|
|
|
550 |
\begin{tikzpicture}[scale=2, line width=0.5mm]
|
|
|
551 |
\only<1->{\node[state, initial] (q0) at ( 0,1) {$q_0$};}
|
|
|
552 |
\only<1->{\node[state] (q1) at ( 1,1) {$q_1$};}
|
|
|
553 |
\only<1->{\node[state] (q2) at ( 2,1) {$q_2$};}
|
|
|
554 |
\path[->] (q0) edge[bend left] node[above] {$a$} (q1)
|
|
|
555 |
(q1) edge[bend left] node[above] {$b$} (q0)
|
|
|
556 |
(q2) edge[bend left=50] node[below] {$b$} (q0)
|
|
|
557 |
(q1) edge node[above] {$a$} (q2)
|
|
|
558 |
(q2) edge [loop right] node {$a$} ()
|
|
|
559 |
(q0) edge [loop below] node {$b$} ()
|
|
|
560 |
;
|
|
|
561 |
\end{tikzpicture}
|
|
|
562 |
\end{center}\bigskip
|
|
|
563 |
|
|
|
564 |
\onslide<2->{
|
|
|
565 |
\begin{center}
|
|
|
566 |
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
|
|
|
567 |
\bl{$q_0$} & \bl{$=$} & \bl{$\epsilon + q_0\,b + q_1\,b + q_2\,b$}\\
|
|
|
568 |
\bl{$q_1$} & \bl{$=$} & \bl{$q_0\,a$}\\
|
|
|
569 |
\bl{$q_2$} & \bl{$=$} & \bl{$q_1\,a + q_2\,a$}\\
|
|
|
570 |
|
|
|
571 |
\end{tabular}
|
|
|
572 |
\end{center}
|
|
|
573 |
}
|
|
|
574 |
|
|
|
575 |
\onslide<3->{
|
|
|
576 |
Arden's Lemma:
|
|
|
577 |
\begin{center}
|
|
|
578 |
If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
|
|
|
579 |
\end{center}
|
|
|
580 |
}
|
|
|
581 |
|
|
|
582 |
\end{frame}}
|
|
|
583 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
584 |
|
|
|
585 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
586 |
\mode<presentation>{
|
|
|
587 |
\begin{frame}[c]
|
|
|
588 |
\frametitle{\begin{tabular}{c}Algorithms on Automata\end{tabular}}
|
|
|
589 |
|
|
|
590 |
|
|
|
591 |
\begin{itemize}
|
|
|
592 |
\item Reg $\rightarrow$ NFA: Thompson-McNaughton-Yamada method\medskip
|
|
|
593 |
\item NFA $\rightarrow$ DFA: Subset Construction\medskip
|
|
|
594 |
\item DFA $\rightarrow$ Reg: Brzozowski's Algebraic Method\medskip
|
|
|
595 |
\item DFA minimisation: Hopcrofts Algorithm\medskip
|
|
|
596 |
\item complement DFA
|
|
|
597 |
\end{itemize}
|
|
|
598 |
|
|
|
599 |
\end{frame}}
|
|
|
600 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
601 |
\newcommand{\qq}{\mbox{\texttt{"}}}
|
|
|
602 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
603 |
\mode<presentation>{
|
|
|
604 |
\begin{frame}[c]
|
|
|
605 |
\frametitle{\begin{tabular}{c}Grammars\end{tabular}}
|
|
|
606 |
|
|
|
607 |
\begin{center}
|
|
|
608 |
\bl{\begin{tabular}{lcl}
|
|
|
609 |
$E$ & $\rightarrow$ & $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
|
|
|
610 |
$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
|
|
|
611 |
$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
|
|
|
612 |
\end{tabular}}
|
|
|
613 |
\end{center}
|
|
|
614 |
|
|
|
615 |
\bl{$E$}, \bl{$F$} and \bl{$T$} are non-terminals\\
|
|
|
616 |
\bl{$E$} is start symbol\\
|
|
|
617 |
\bl{$num$}, \bl{(}, \bl{)}, \bl{+} \ldots are terminals\bigskip\\
|
|
|
618 |
|
|
|
619 |
|
|
|
620 |
\bl{\texttt{(2*3)+(3+4)}}
|
|
|
621 |
|
|
|
622 |
\end{frame}}
|
|
|
623 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
624 |
|
|
|
625 |
|
|
|
626 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
627 |
\mode<presentation>{
|
|
|
628 |
\begin{frame}[c]
|
|
|
629 |
|
|
|
630 |
\begin{center}
|
|
|
631 |
\bl{\begin{tabular}{lcl}
|
|
|
632 |
$E$ & $\rightarrow$ & $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
|
|
|
633 |
$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
|
|
|
634 |
$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
|
|
|
635 |
\end{tabular}}
|
|
|
636 |
\end{center}
|
|
|
637 |
|
|
|
638 |
\begin{center}
|
|
|
639 |
\begin{tikzpicture}[level distance=8mm, blue]
|
|
|
640 |
\node {E}
|
|
|
641 |
child {node {F}
|
|
|
642 |
child {node {T}
|
|
|
643 |
child {node {\qq(\qq\,E\,\qq)\qq}
|
|
|
644 |
child {node{F \qq*\qq{} F}
|
|
|
645 |
child {node {T} child {node {2}}}
|
|
|
646 |
child {node {T} child {node {3}}}
|
|
|
647 |
}
|
|
|
648 |
}
|
|
|
649 |
}
|
|
|
650 |
child {node {\qq+\qq}}
|
|
|
651 |
child {node {T}
|
|
|
652 |
child {node {\qq(\qq\,E\,\qq)\qq}
|
|
|
653 |
child {node {F}
|
|
|
654 |
child {node {T \qq+\qq{} T}
|
|
|
655 |
child {node {3}}
|
|
|
656 |
child {node {4}}
|
|
|
657 |
}
|
|
|
658 |
}}
|
|
|
659 |
}};
|
|
|
660 |
\end{tikzpicture}
|
|
|
661 |
\end{center}
|
|
|
662 |
|
|
|
663 |
\begin{textblock}{5}(1, 5)
|
|
|
664 |
\bl{\texttt{(2*3)+(3+4)}}
|
|
|
665 |
\end{textblock}
|
|
|
666 |
|
|
|
667 |
\end{frame}}
|
|
|
668 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
669 |
|
|
|
670 |
\end{document}
|
|
|
671 |
|
|
|
672 |
%%% Local Variables:
|
|
|
673 |
%%% mode: latex
|
|
|
674 |
%%% TeX-master: t
|
|
|
675 |
%%% End:
|
|
|
676 |
|