| author | Christian Urban <christian.urban@kcl.ac.uk> |
| Sat, 11 Oct 2025 08:33:35 +0100 | |
| changeset 1006 | 674d1258a2fb |
| parent 976 | 4be299d9b41b |
| child 1008 | 7431e391682f |
| permissions | -rw-r--r-- |
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% !TEX program = xelatex |
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\usepackage{../slides}
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\usepackage{../graphicss}
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\usepackage{../langs}
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\pgfplotsset{compat=1.17}
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% beamer stuff |
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\renewcommand{\slidecaption}{CFL 03, King's College London}
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\begin{document}
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\begin{frame}[t]
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\\[-3mm] |
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\LARGE Compilers and \\[-1mm] |
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\LARGE Formal Languages\\[-5mm] |
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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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Office Hour: & Friday 11:30 -- 12:30\\ |
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Location: & N7.07 (North Wing, Bush House)\\ |
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Slides \& Progs: & KEATS\\ |
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Pollev: & \texttt{\alert{https://pollev.com/cfltutoratki576}}\\
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\end{tabular}
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\end{center}
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\begin{tikzpicture}
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\node[drop shadow,fill=white,inner sep=0pt] |
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{\footnotesize\rowcolors{1}{capri!10}{white}
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\begin{tabular}{|p{4.8cm}|p{4.8cm}|}\hline
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1 Introduction, Languages & 6 While-Language \\ |
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2 Regular Expressions, Derivatives & 7 Compilation, JVM \\ |
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\cellcolor{blue!50}
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3 Automata, Regular Languages & 8 Compiling Functional Languages \\ |
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4 Lexing, Tokenising & 9 Optimisations \\ |
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5 Grammars, Parsing & 10 LLVM \\ \hline |
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\end{tabular}%
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}; |
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\begin{textblock}{5}(12,3)
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\includegraphics[scale=0.35]{qr01}\\
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\small{}Wifi: \\
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\small{}Pwd:
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\end{textblock}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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{
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\setbeamercolor{background canvas}{bg=cream}
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\begin{frame}[c]
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\frametitle{For Installation Problems}
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\begin{itemize}
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\item Harry Dilnot (harry.dilnot@kcl.ac.uk) \\ |
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\;\;Windows expert |
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\item Oliver Iliffe (oliver.iliffe@kcl.ac.uk) |
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\end{itemize}
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} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[c]
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\begin{mybox3}{From Pollev last week}\it
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Is the equivalence of two regexes belong in the P or NP class of problems? |
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\end{mybox3}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[c]
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\begin{mybox3}{From Pollev last week}\it
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If state machines are not efficient, then how/why do many lexer |
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packages like the logos crate in rust compile down a lexer |
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definition down to a jump table driven state machine? |
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\textcolor{gray}{Could we
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achieve quicker lexing with things like SIMD instructions?} |
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\end{mybox3}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[c]
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%%\frametitle{\boldmath$(abcdef)^{\{n\}}$ in Rust and Scala}
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\begin{textblock}{14}(1,3)
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\begin{tikzpicture}
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\begin{axis}[
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xlabel={\bl{n} copies of \bl{\texttt{"}abcdef\texttt{"}}},
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x label style={at={(0.45,-0.16)}},
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ylabel={time in secs},
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enlargelimits=false, |
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ytick={0,10,...,60},
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ymax=65, |
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xmax=50000, |
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xtick={0,10000,...,40000},
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scaled ticks=false, |
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axis lines=left, |
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width=10cm, |
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height=6cm, |
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legend entries={Rust, Scala V3},
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legend style={font=\small,at={(1.0,0.48)},anchor=north},
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legend cell align=left] |
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\addplot[blue,mark=*, mark options={fill=white}] table {re-rust2.data};
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\only<2>{\addplot[red,mark=*, mark options={fill=white}] table {re-scala2.data};}
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\end{axis}
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\end{tikzpicture}
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\end{textblock}
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\begin{textblock}{9}(1.8,1.8)
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{Regular expression: \bl{\texttt{(abcdef)$^{\{\textrm{n}\}}$}}}
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\end{textblock}
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\begin{textblock}{6}(10.5,11.4)
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{\bl{$n$}}
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%%\frametitle{\boldmath$(abcdef)^{\{n\}}$ in Rust and Scala}
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\begin{textblock}{14}(1,3)
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\begin{tikzpicture}
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\begin{axis}[
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xlabel={strings of the form \bl{$\underbrace{\texttt{a}...\texttt{a}}_n$}},
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x label style={at={(0.45,-0.16)}},
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ylabel={time in secs},
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enlargelimits=false, |
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xtick={0,5,10,...,30},
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xmax=35, |
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ymax=35, |
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ytick={0,10,...,30},
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axis lines=left, |
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width=10cm, |
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height=6cm, |
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legend entries={Python,JavaScript,Swift,Dart},
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legend style={font=\small,at={(1.15,0.78)},anchor=north},
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legend cell align=left] |
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\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};
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\addplot[red,mark=*, mark options={fill=white}] table {re-js.data};
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\addplot[magenta,mark=*, mark options={fill=white}] table {re-swift.data};
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\addplot[brown,mark=*, mark options={fill=white}] table {re-dart.data};
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\end{tikzpicture}
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\end{textblock}
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\begin{textblock}{6}(1.8,1.8)
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{Regular expression: \bl{\texttt{(a*)*$\,$b}}}
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\begin{textblock}{6}(10.5,11.4)
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{\bl{$n$}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% @@@@@@@@@@@@@@@@@ |
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%\end{document}
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% @@@@@@@@@@@@@@@@@@ |
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\begin{frame}[c]
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\begin{mybox3}{From Pollev last week}\it
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For a regular expression $r = r_1 \cdot r_2$, to prove that |
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$der\;c\;r = (der\;c\;r) \cdot r^{\{n-1\}}$, is there a
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way to prove it in the general case instead |
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of how you do the calculations for each $n$ in the videos? |
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\end{mybox3}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%\begin{frame}[c]
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%\frametitle{Scala Book, Exams}
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% |
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%\begin{itemize}
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%\item \small https://nms.kcl.ac.uk/christian.urban/ProgInScala2ed.pdf\normalsize |
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%\item homework (written exam 80\%) |
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%\item coursework (20\%)\bigskip |
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%\item short survey at KEATS; to be answered until Sunday |
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%\end{itemize}
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%\end{frame}
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%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[t]
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\frametitle{(Basic) Regular Expressions}
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\begin{center}
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\begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
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\bl{$r$} & \bl{$::=$} & \bl{$\ZERO$} & nothing\\
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& \bl{$\mid$} & \bl{$\ONE$} & empty string / "" / $[]$\\
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& \bl{$\mid$} & \bl{$c$} & character\\
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& \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\
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& \bl{$\mid$} & \bl{$r_1 + r_2$} & alternative / choice\\
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& \bl{$\mid$} & \bl{$r^*$} & star (zero or more)\\
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\end{tabular}
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\end{center}
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How about ranges \bl{$[a\mbox{-}z]$}, \bl{$r^+$} and \bl{$\sim{}r$}? Do they increase the
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set of languages we can recognise? |
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[t]
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\frametitle{Negation}
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Assume you have an alphabet consisting of the letters \bl{$a$}, \bl{$b$} and \bl{$c$} only.
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Find a (basic!) regular expression that matches all strings \emph{\alert{except}}
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\bl{$ab$} and \bl{$ac$}!
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[t]
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\frametitle{Automata}
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A \alert{\bf deterministic finite automaton}, DFA, consists of:
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\begin{itemize}
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\item an alphabet \bl{$\varSigma$}
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\item a set of states \bl{$\mbox{Qs}$}
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\item one of these states is the start state \bl{$\mbox{Q}_0$}
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\item some states are accepting states \bl{$F$}, and
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\item there is transition function \bl{$\delta$}\bigskip
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\small |
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which takes a state as argument and a character and produces a |
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new state; this function might not be everywhere defined $\Rightarrow$ partial function |
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\end{itemize}
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\begin{center}
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\bl{$A(\varSigma, \mbox{Qs}, \mbox{Q}_0, F, \delta)$}
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\end{center}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{frame}[c]
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\begin{center}
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diff
changeset
|
298 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
| 573 | 299 |
every state/.style={minimum size=0pt,inner sep=2pt,
|
300 |
draw=blue!50,very thick,fill=blue!20},] |
|
| 515 | 301 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
302 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
303 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
304 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
305 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
306 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
307 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
308 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
309 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
310 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
311 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
312 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
313 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
314 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
315 |
\end{tikzpicture}
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
316 |
\end{center}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
317 |
|
| 573 | 318 |
\mbox{}\\[-14mm]
|
319 |
\only<1>{
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
320 |
\begin{itemize}
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
321 |
\item the start state can be an accepting state |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
322 |
\item it is possible that there is no accepting state |
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
323 |
\item all states might be accepting (but this does not necessarily mean all strings are accepted) |
| 573 | 324 |
\end{itemize}}
|
325 |
\only<2>{
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
326 |
for this automaton \bl{$\delta$} is the function\\
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
327 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
328 |
\begin{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
329 |
\begin{tabular}{lll}
|
| 515 | 330 |
\bl{$(\mbox{Q}_0, a) \rightarrow \mbox{Q}_1$} & \bl{$(\mbox{Q}_1, a) \rightarrow \mbox{Q}_4$}
|
331 |
& \bl{$(\mbox{Q}_4, a) \rightarrow \mbox{Q}_4$}\\
|
|
332 |
\bl{$(\mbox{Q}_0, b) \rightarrow \mbox{Q}_2$} & \bl{$(\mbox{Q}_1, b) \rightarrow \mbox{Q}_2$} &
|
|
333 |
\bl{$(\mbox{Q}_4, b) \rightarrow \mbox{Q}_4$}\\
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
334 |
\end{tabular}\ldots
|
| 573 | 335 |
\end{center}
|
336 |
} |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
337 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
338 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
339 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
340 |
|
| 573 | 341 |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
342 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
343 |
\begin{frame}[t]
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
344 |
\frametitle{Accepting a String}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
345 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
346 |
Given |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
347 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
348 |
\begin{center}
|
| 574 | 349 |
\bl{$A(\varSigma, \mbox{Qs}, \mbox{Q}_0, F, \delta)$}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
350 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
351 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
352 |
you can define |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
353 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
354 |
\begin{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
355 |
\begin{tabular}{l}
|
| 778 | 356 |
\bl{$\widehat{\delta}(Q, []) \dn Q$}\\
|
357 |
\bl{$\widehat{\delta}(Q, c::s) \dn \widehat{\delta}(\delta(Q, c), s)$}\\
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
358 |
\end{tabular}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
359 |
\end{center}\pause
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
360 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
361 |
Whether a string \bl{$s$} is accepted by \bl{$A$}?
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
362 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
363 |
\begin{center}
|
| 517 | 364 |
\hspace{5mm}\bl{$\widehat{\delta}(\mbox{Q}_0, s) \in F$}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
365 |
\end{center}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
366 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
367 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
368 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
369 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
370 |
\begin{frame}[c]
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
371 |
\frametitle{Regular Languages}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
372 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
373 |
A \alert{\bf language} is a set of strings.\bigskip
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
374 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
375 |
A \alert{\bf regular expression} specifies a language.\bigskip
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
376 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
377 |
A language is \alert{\bf regular} iff there exists
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
378 |
a regular expression that recognises all its strings.\bigskip\bigskip\pause |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
379 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
380 |
not all languages are regular, e.g.~\bl{$a^nb^n$} is not
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
381 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
382 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
383 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
384 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
385 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
386 |
\frametitle{Regular Languages (2)}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
387 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
388 |
A language is \alert{\bf regular} iff there exists a regular
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
389 |
expression that recognises all its strings.\bigskip\medskip |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
390 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
391 |
or {\bf equivalently}\bigskip\medskip
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
392 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
393 |
A language is \alert{\bf regular} iff there exists a
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
394 |
deterministic finite automaton that recognises all its |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
395 |
strings. |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
396 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
397 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
398 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
399 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
400 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
401 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
402 |
\begin{frame}[c]
|
|
348
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
403 |
\frametitle{\begin{tabular}{c}
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
404 |
Non-Deterministic\\[-1mm] |
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
405 |
Finite Automata\end{tabular}}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
406 |
|
| 779 | 407 |
\bl{$N(\varSigma, \mbox{Qs}, \mbox{Qs}_0, F, \rho)$}\\
|
408 |
A non-deterministic finite automaton (NFA) consists of: |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
409 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
410 |
\begin{itemize}
|
| 779 | 411 |
\item a finite set of states, \bl{$Qs$}
|
412 |
\item \underline{some} these states are the start states, \bl{$Qs_0$}
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
413 |
\item some states are accepting states, and |
| 779 | 414 |
\item there is transition \alert{relation}, \bl{$\rho$}\medskip
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
415 |
\end{itemize}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
416 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
417 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
418 |
\begin{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
419 |
\begin{tabular}{c}
|
| 515 | 420 |
\bl{$(\mbox{Q}_1, a) \rightarrow \mbox{Q}_2$}\\
|
421 |
\bl{$(\mbox{Q}_1, a) \rightarrow \mbox{Q}_3$}\\
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
422 |
\end{tabular}
|
| 515 | 423 |
\bl{\ldots}
|
424 |
\hspace{10mm}\pause
|
|
425 |
\bl{$(\mbox{Q}_1, a) \rightarrow \{\mbox{Q}_2, \mbox{Q}_3\}$}
|
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
426 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
427 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
428 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
429 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
430 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
431 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
432 |
\begin{frame}[c]
|
| 515 | 433 |
\frametitle{An NFA Example}
|
434 |
||
435 |
% A NFA for (ab* + b)*a |
|
436 |
\begin{center}
|
|
437 |
\begin{tikzpicture}[>=stealth',very thick, auto,
|
|
438 |
every state/.style={minimum size=0pt,inner sep=3pt,
|
|
439 |
draw=blue!50,very thick,fill=blue!20},scale=2] |
|
440 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
441 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
442 |
\node[state, accepting] (Q_2) [right=of Q_1] {$\mbox{Q}_2$};
|
|
443 |
\path[->] (Q_0) edge [loop above] node {\alert{$b$}} ();
|
|
444 |
\path[<-] (Q_0) edge node [below] {\alert{$b$}} (Q_1);
|
|
445 |
\path[->] (Q_0) edge [bend left] node [above] {\alert{$a$}} (Q_1);
|
|
446 |
\path[->] (Q_0) edge [bend right] node [below] {\alert{$a$}} (Q_2);
|
|
447 |
\path[->] (Q_1) edge [loop above] node {\alert{$a,b$}} ();
|
|
448 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_2);
|
|
449 |
\end{tikzpicture}
|
|
450 |
\end{center}
|
|
451 |
||
452 |
\end{frame}
|
|
453 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
454 |
||
455 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
456 |
\begin{frame}[c]
|
|
| 574 | 457 |
\frametitle{Another Example}
|
458 |
||
459 |
For the regular expression \bl{$(.^*)a\,(.^{\{n\}})bc$}\bigskip\bigskip
|
|
460 |
||
461 |
% \begin{center}
|
|
| 650 | 462 |
\mbox{}\hspace{-11.5mm}
|
| 574 | 463 |
\begin{tikzpicture}[>=stealth',very thick, auto,
|
464 |
every state/.style={minimum size=5pt,inner sep=1pt,
|
|
465 |
draw=blue!50,very thick,fill=blue!20},scale=1, node distance=5mm, |
|
466 |
decoration={brace,mirror,amplitude=7}]
|
|
467 |
\node[state,initial] (Q_0) {$\;0\;$};
|
|
468 |
\node[state] (Q_1) [right=of Q_0] {$\;1\;$};
|
|
469 |
\node[state] (Q_2) [right=of Q_1] {$\;2\;$};
|
|
470 |
\node (Q_3) [right=of Q_2] {$\hspace{-1mm}\ldots\hspace{-3mm}$};
|
|
471 |
\node[state] (Q_4) [right=of Q_3] {$\;\phantom{1}\;$};
|
|
472 |
\node (Q_5) [right=of Q_4] {$\hspace{-1mm}\ldots\hspace{-3mm}$};
|
|
473 |
\node[state] (Q_6) [right=of Q_5] {\tiny$n+1$};
|
|
474 |
\node[state] (Q_7) [right=of Q_6] {\tiny$n+2$};
|
|
475 |
\node[state,accepting] (Q_8) [right=of Q_7] {\tiny$n+3$};
|
|
476 |
||
477 |
\path[->] (Q_0) edge [loop above] node {$*$} ();
|
|
478 |
\path[->] (Q_0) edge node [above] {$a$} (Q_1);
|
|
479 |
\path[->] (Q_1) edge node [above] {$*$} (Q_2);
|
|
480 |
\path[->] (Q_2) edge node [above] {$*$} (Q_3);
|
|
481 |
\path[->] (Q_4) edge node [above] {$*$} (Q_5);
|
|
482 |
\path[->] (Q_5) edge node [above] {$*$} (Q_6);
|
|
483 |
\path[->] (Q_6) edge node [above] {$b$} (Q_7);
|
|
484 |
\path[->] (Q_7) edge node [above] {$c$} (Q_8);
|
|
485 |
||
486 |
\draw [decorate] ([yshift=-5mm]Q_1.east) --node[below=3mm]{$n$} ([yshift=-5mm]Q_6.west);
|
|
487 |
\end{tikzpicture}\bigskip
|
|
488 |
||
489 |
||
490 |
{\small Note the star-transitions: accept any character.}
|
|
491 |
||
492 |
\end{frame}
|
|
493 |
||
494 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
495 |
\begin{frame}[c]
|
|
| 515 | 496 |
\frametitle{Two Epsilon NFA Examples}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
497 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
498 |
\small |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
499 |
\begin{center}
|
|
143
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
500 |
\begin{tabular}[t]{c@{\hspace{9mm}}c}
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
501 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
502 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
|
215
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
503 |
\node[state,initial] (q_0) {$q_0$};
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
504 |
\node[state] (q_1) [above=of q_0] {$q_1$};
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
505 |
\node[state, accepting] (q_2) [below=of q_0] {$q_2$};
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
506 |
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_1);
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
507 |
\path[->] (q_0) edge node [left] {\alert{$\epsilon$}} (q_2);
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
508 |
\path[->] (q_0) edge [loop right] node {\alert{$a$}} ();
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
509 |
\path[->] (q_1) edge [loop above] node {\alert{$a$}} ();
|
|
828303e8e4af
updated slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
143
diff
changeset
|
510 |
\path[->] (q_2) edge [loop below] node {\alert{$b$}} ();
|
|
143
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
511 |
\end{tikzpicture} &
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
512 |
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
513 |
\raisebox{20mm}{
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
514 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
515 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
516 |
\node[state,initial] (r_1) {$r_1$};
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
517 |
\node[state] (r_2) [above=of r_1] {$r_2$};
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
518 |
\node[state, accepting] (r_3) [right=of r_1] {$r_3$};
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
519 |
\path[->] (r_1) edge node [below] {\alert{$b$}} (r_3);
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
520 |
\path[->] (r_2) edge [bend left] node [above] {\alert{$a$}} (r_3);
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
521 |
\path[->] (r_1) edge [bend left] node [left] {\alert{$\epsilon$}} (r_2);
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
522 |
\path[->] (r_2) edge [bend left] node [right] {\alert{$a$}} (r_1);
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
523 |
\end{tikzpicture}}
|
|
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
524 |
\end{tabular}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
525 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
526 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
527 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
528 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
529 |
|
| 515 | 530 |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
531 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
532 |
\begin{frame}[c]
|
| 779 | 533 |
\frametitle{Thompson: Rexp to $\epsilon$NFA}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
534 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
535 |
\begin{center}
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
536 |
\begin{tabular}[t]{l@{\hspace{10mm}}l}
|
| 445 | 537 |
\raisebox{1mm}{\bl{$\ZERO$}} &
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
538 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 539 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
540 |
\end{tikzpicture}\\\\
|
| 445 | 541 |
\raisebox{1mm}{\bl{$\ONE$}} &
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
542 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 543 |
\node[state, initial, accepting] (Q_0) {$\mbox{}$};
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
544 |
\end{tikzpicture}\\\\
|
| 651 | 545 |
\raisebox{5mm}{\bl{$c$}} &
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
546 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 547 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
548 |
\node[state, accepting] (Q_1) [right=of Q_0] {$\mbox{}$};
|
|
549 |
\path[->] (Q_0) edge node [below] {\alert{$c$}} (Q_1);
|
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
550 |
\end{tikzpicture}\\\\
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
551 |
\end{tabular}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
552 |
\end{center}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
553 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
554 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
555 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
556 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
557 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
558 |
\begin{frame}[t]
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
559 |
\frametitle{Case $r_1\cdot r_2$}
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
560 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
561 |
\mbox{}\bigskip
|
| 515 | 562 |
\onslide<1->{By recursion we are given two automata:\bigskip}
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
563 |
|
| 515 | 564 |
{\centering%
|
565 |
\only<1>{%
|
|
566 |
\begin{tikzpicture}[node distance=3mm,
|
|
567 |
>=stealth',very thick, |
|
568 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20}]
|
|
569 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
|
570 |
\node[state, initial] (Q_01) [below=1mm of Q_0] {$\mbox{}$};
|
|
571 |
\node[state, initial] (Q_02) [above=1mm of Q_0] {$\mbox{}$};
|
|
572 |
\node (R_1) [right=of Q_0] {$\ldots$};
|
|
573 |
\node[state, accepting] (T_1) [right=of R_1] {$\mbox{}$};
|
|
574 |
\node[state, accepting] (T_2) [above=of T_1] {$\mbox{}$};
|
|
575 |
\node[state, accepting] (T_3) [below=of T_1] {$\mbox{}$};
|
|
576 |
||
577 |
\node (A_0) [right=2.5cm of T_1] {$\mbox{}$};
|
|
578 |
\node[state, initial] (A_01) [above=1mm of A_0] {$\mbox{}$};
|
|
579 |
\node[state, initial] (A_02) [below=1mm of A_0] {$\mbox{}$};
|
|
580 |
||
581 |
\node (b_1) [right=of A_0] {$\ldots$};
|
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
582 |
\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
583 |
\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
584 |
\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
585 |
\begin{pgfonlayer}{background}
|
| 515 | 586 |
\node (1) [rounded corners, inner sep=1mm, thick, |
587 |
draw=black!60, fill=black!20, fit= (Q_0) (R_1) (T_1) (T_2) (T_3)] {};
|
|
588 |
\node (2) [rounded corners, inner sep=1mm, thick, |
|
589 |
draw=black!60, fill=black!20, fit= (A_0) (b_1) (c_1) (c_2) (c_3)] {};
|
|
590 |
\node [yshift=2mm] at (1.north) {$r_1$};
|
|
591 |
\node [yshift=2mm] at (2.north) {$r_2$};
|
|
|
143
e3fd4c5995ef
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
139
diff
changeset
|
592 |
\end{pgfonlayer}
|
| 515 | 593 |
\end{tikzpicture}}
|
594 |
\only<2>{%
|
|
595 |
\begin{tikzpicture}[node distance=3mm,
|
|
596 |
>=stealth',very thick, |
|
597 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]
|
|
598 |
\node[state, initial] (Q_0) {$\mbox{}$};
|
|
599 |
\node[state, initial] (Q_01) [below=1mm of Q_0] {$\mbox{}$};
|
|
600 |
\node[state, initial] (Q_02) [above=1mm of Q_0] {$\mbox{}$};
|
|
601 |
\node (r_1) [right=of Q_0] {$\ldots$};
|
|
602 |
\node[state] (t_1) [right=of r_1] {$\mbox{}$};
|
|
603 |
\node[state] (t_2) [above=of t_1] {$\mbox{}$};
|
|
604 |
\node[state] (t_3) [below=of t_1] {$\mbox{}$};
|
|
605 |
||
606 |
\node (A_0) [right=2.5cm of t_1] {$\mbox{}$};
|
|
607 |
\node[state] (A_01) [above=1mm of A_0] {$\mbox{}$};
|
|
608 |
\node[state] (A_02) [below=1mm of A_0] {$\mbox{}$};
|
|
609 |
||
610 |
\node (b_1) [right=of A_0] {$\ldots$};
|
|
611 |
\node[state, accepting] (c_1) [right=of b_1] {$\mbox{}$};
|
|
612 |
\node[state, accepting] (c_2) [above=of c_1] {$\mbox{}$};
|
|
613 |
\node[state, accepting] (c_3) [below=of c_1] {$\mbox{}$};
|
|
614 |
\path[->] (t_1) edge (A_01); |
|
615 |
\path[->] (t_2) edge node [above] {$\epsilon$s} (A_01);
|
|
616 |
\path[->] (t_3) edge (A_01); |
|
617 |
\path[->] (t_1) edge (A_02); |
|
618 |
\path[->] (t_2) edge (A_02); |
|
619 |
\path[->] (t_3) edge node [below] {$\epsilon$s} (A_02);
|
|
620 |
||
621 |
\begin{pgfonlayer}{background}
|
|
622 |
\node (3) [rounded corners, inner sep=1mm, thick, |
|
623 |
draw=black!60, fill=black!20, fit= (Q_0) (c_1) (c_2) (c_3)] {};
|
|
624 |
\node [yshift=2mm] at (3.north) {$r_1\cdot r_2$};
|
|
625 |
\end{pgfonlayer}
|
|
626 |
\end{tikzpicture}}
|
|
627 |
% |
|
628 |
}\bigskip\bigskip |
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
629 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
630 |
\small |
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
631 |
We need to (1) change the accepting nodes of the first automaton into non-accepting nodes, and (2) connect them |
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
632 |
via $\epsilon$-transitions to the starting state of the second automaton. |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
633 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
634 |
\end{frame}
|
|
137
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
635 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
636 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
637 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
changeset
|
638 |
\begin{frame}[t]
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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|
639 |
\frametitle{Case $r_1+ r_2$}
|
| 515 | 640 |
\mbox{}\\[-8mm]
|
|
137
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added pictures
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parents:
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diff
changeset
|
641 |
|
| 515 | 642 |
\onslide<1->{By recursion we are given two automata:\smallskip}
|
|
137
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added pictures
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parents:
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diff
changeset
|
643 |
|
| 515 | 644 |
\hspace{2cm}
|
645 |
\only<1>{%
|
|
646 |
\begin{tikzpicture}[node distance=3mm,
|
|
647 |
>=stealth',very thick, |
|
648 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
649 |
baseline=(current bounding box.center)] |
|
650 |
\node at (0,0) (1) {$\mbox{}$};
|
|
651 |
\node (2) [above=14mm of 1] {};
|
|
652 |
\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
|
|
653 |
\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
|
|
654 |
\node[state, initial] (3) [below=10mm of 1] {$\mbox{}$};
|
|
|
135
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parents:
134
diff
changeset
|
655 |
|
| 515 | 656 |
\node (a) [right=of 2] {$\ldots\,$};
|
657 |
\node (a1) [right=of a] {$$};
|
|
|
137
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parents:
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diff
changeset
|
658 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
|
69cec773736b
added pictures
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parents:
136
diff
changeset
|
659 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
|
69cec773736b
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parents:
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diff
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|
660 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
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diff
changeset
|
661 |
\node (b) [right=of 3] {$\ldots$};
|
|
69cec773736b
added pictures
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parents:
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diff
changeset
|
662 |
\node[state, accepting] (b1) [right=of b] {$\mbox{}$};
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
136
diff
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|
663 |
\node[state, accepting] (b2) [above=of b1] {$\mbox{}$};
|
|
69cec773736b
added pictures
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parents:
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diff
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|
664 |
\node[state, accepting] (b3) [below=of b1] {$\mbox{}$};
|
|
69cec773736b
added pictures
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parents:
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diff
changeset
|
665 |
\begin{pgfonlayer}{background}
|
| 515 | 666 |
\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};
|
667 |
\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (3) (b1) (b2) (b3)] {};
|
|
668 |
\node [yshift=3mm] at (1.north) {$r_1$};
|
|
669 |
\node [yshift=3mm] at (2.north) {$r_2$};
|
|
|
137
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parents:
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|
670 |
\end{pgfonlayer}
|
| 515 | 671 |
\end{tikzpicture}}
|
672 |
\only<2>{%
|
|
673 |
\begin{tikzpicture}[node distance=3mm,
|
|
674 |
>=stealth',very thick, |
|
675 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
676 |
baseline=(current bounding box.center)] |
|
677 |
\node at (0,0) (1) {$\mbox{}$};
|
|
678 |
\node (2) [above=14mm of 1] {$$};
|
|
679 |
\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
|
|
680 |
\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
|
|
681 |
\node[state, initial] (3) [below=10mm of 1] {$\mbox{}$};
|
|
682 |
||
683 |
\node (a) [right=of 2] {$\ldots\,$};
|
|
684 |
\node (a1) [right=of a] {$$};
|
|
685 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
|
686 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
|
687 |
||
688 |
\node (b) [right=of 3] {$\ldots$};
|
|
689 |
\node[state, accepting] (b1) [right=of b] {$\mbox{}$};
|
|
690 |
\node[state, accepting] (b2) [above=of b1] {$\mbox{}$};
|
|
691 |
\node[state, accepting] (b3) [below=of b1] {$\mbox{}$};
|
|
692 |
||
693 |
%\path[->] (1) edge node [above] {$\epsilon$} (2);
|
|
694 |
%\path[->] (1) edge node [below] {$\epsilon$} (3);
|
|
695 |
||
696 |
\begin{pgfonlayer}{background}
|
|
697 |
\node (3) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3) (b2) (b3)] {};
|
|
698 |
\node [yshift=3mm] at (3.north) {$r_1+ r_2$};
|
|
699 |
\end{pgfonlayer}
|
|
700 |
\end{tikzpicture}}
|
|
701 |
% |
|
|
137
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parents:
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diff
changeset
|
702 |
|
|
69cec773736b
added pictures
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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diff
changeset
|
703 |
\small |
| 515 | 704 |
\mbox{}\\\medskip
|
| 517 | 705 |
We can just put both automata together. |
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
706 |
\end{frame}
|
|
135
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parents:
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diff
changeset
|
707 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
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parents:
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diff
changeset
|
708 |
|
|
72b686e1c974
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Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
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diff
changeset
|
709 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
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parents:
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diff
changeset
|
710 |
\begin{frame}[c]
|
|
138
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parents:
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diff
changeset
|
711 |
\frametitle{Case $r^*$}
|
|
b98a0a49c432
added slides
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parents:
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diff
changeset
|
712 |
|
| 515 | 713 |
\onslide<1->{By recursion we are given an automaton for $r$:\smallskip}
|
|
135
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parents:
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diff
changeset
|
714 |
|
| 515 | 715 |
\hspace{2cm}
|
716 |
\only<1>{\begin{tikzpicture}[node distance=3mm,
|
|
717 |
>=stealth',very thick, |
|
718 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
719 |
baseline=(current bounding box.north)] |
|
720 |
\node (2) {$\mbox{}$};
|
|
721 |
\node[state, initial] (4) [above=1mm of 2] {$\mbox{}$};
|
|
722 |
\node[state, initial] (5) [below=1mm of 2] {$\mbox{}$};
|
|
|
138
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parents:
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diff
changeset
|
723 |
\node (a) [right=of 2] {$\ldots$};
|
|
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
724 |
\node[state, accepting] (a1) [right=of a] {$\mbox{}$};
|
|
b98a0a49c432
added slides
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parents:
137
diff
changeset
|
725 |
\node[state, accepting] (a2) [above=of a1] {$\mbox{}$};
|
| 515 | 726 |
\node[state, accepting] (a3) [below=of a1] {$\mbox{}$};
|
727 |
\begin{pgfonlayer}{background}
|
|
728 |
\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};
|
|
729 |
\node [yshift=3mm] at (1.north) {$r$};
|
|
730 |
\end{pgfonlayer}
|
|
731 |
\end{tikzpicture}}
|
|
732 |
\only<2->{\begin{tikzpicture}[node distance=3mm,
|
|
733 |
>=stealth',very thick, |
|
734 |
every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},
|
|
735 |
baseline=(current bounding box.north)] |
|
736 |
\node at (0,0) [state, initial,accepting] (1) {$\mbox{}$};
|
|
737 |
\node (2) [right=16mm of 1] {$\mbox{}$};
|
|
738 |
\node[state] (4) [above=1mm of 2] {$\mbox{}$};
|
|
739 |
\node[state] (5) [below=1mm of 2] {$\mbox{}$};
|
|
740 |
\node (a) [right=of 2] {$\ldots$};
|
|
|
138
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
741 |
\node[state] (a1) [right=of a] {$\mbox{}$};
|
|
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
742 |
\node[state] (a2) [above=of a1] {$\mbox{}$};
|
| 515 | 743 |
\node[state] (a3) [below=of a1] {$\mbox{}$};
|
744 |
\path[->] (1) edge node [below] {$\epsilon$} (4);
|
|
745 |
\path[->] (1) edge node [below] {$\epsilon$} (5);
|
|
746 |
\path[->] (a1) edge [bend left=45] node [below] {$\epsilon$} (1);
|
|
747 |
\path[->] (a2) edge [bend right] node [below] {$\epsilon$} (1);
|
|
748 |
\path[->] (a3) edge [bend left=45] node [below] {$\epsilon$} (1);
|
|
|
138
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
749 |
\begin{pgfonlayer}{background}
|
| 515 | 750 |
\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3)] {};
|
751 |
\node [yshift=3mm] at (2.north) {$r^*$};
|
|
|
138
b98a0a49c432
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
752 |
\end{pgfonlayer}
|
| 515 | 753 |
\end{tikzpicture}}
|
754 |
\bigskip |
|
|
138
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added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
137
diff
changeset
|
755 |
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
756 |
\onslide<3->{
|
|
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
757 |
Why can't we just have an epsilon transition from the accepting states to the starting state?} |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
758 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
759 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
760 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
761 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
762 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
763 |
\begin{frame}[c]
|
| 779 | 764 |
\frametitle{\mbox{NFA Breadth-First: \boldmath{}$a^{?\{n\}}\!\cdot\! a^{\{n\}}$}}
|
765 |
||
766 |
\begin{tikzpicture}
|
|
767 |
\begin{axis}[xlabel={\pcode{a}s},
|
|
768 |
ylabel={time in secs},
|
|
769 |
enlargelimits=false, |
|
770 |
xtick={0,5,...,30},
|
|
771 |
xmax=30, |
|
772 |
ymax=35, |
|
773 |
ytick={0,5,...,30},
|
|
774 |
scaled ticks=false, |
|
775 |
axis lines=left, |
|
776 |
width=9cm, |
|
777 |
height=6cm, |
|
778 |
legend entries={Python,Ruby,my NFA},
|
|
779 |
legend pos=outer north east, |
|
780 |
legend cell align=left] |
|
781 |
\addplot[blue,mark=*, mark options={fill=white}] table {re-python.data};
|
|
782 |
\addplot[brown,mark=pentagon*, mark options={fill=white}] table {re-ruby.data};
|
|
783 |
\addplot[red,mark=triangle*, mark options={fill=red}] table {nfabreadth.data};
|
|
784 |
\end{axis}
|
|
785 |
\end{tikzpicture}
|
|
786 |
||
787 |
\end{frame}
|
|
788 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
789 |
||
790 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
791 |
\begin{frame}[c]
|
|
792 |
\frametitle{NFA Breadth-First:\boldmath{}$(a^*)^* \cdot b$}
|
|
793 |
||
794 |
\bigskip |
|
795 |
\begin{center}
|
|
796 |
\begin{tikzpicture}
|
|
797 |
\begin{axis}[
|
|
798 |
xlabel={\pcode{a}s},
|
|
799 |
%%x label style={at={(1.05,0.0)}},
|
|
800 |
ylabel={time in secs},
|
|
801 |
enlargelimits=false, |
|
802 |
xtick={0,10,...,100},
|
|
803 |
xmax=103, |
|
804 |
ymax=35, |
|
805 |
ytick={0,5,...,30},
|
|
806 |
scaled ticks=false, |
|
807 |
axis lines=left, |
|
808 |
width=9cm, |
|
809 |
height=6cm, |
|
810 |
legend entries={Java 8,Python,JavaScript,Swift,Dart,my NFA},
|
|
811 |
legend pos=outer north east, |
|
812 |
legend cell align=left] |
|
813 |
\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};
|
|
814 |
\addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data};
|
|
815 |
\addplot[red,mark=*, mark options={fill=white}] table {re-js.data};
|
|
816 |
\addplot[magenta,mark=*, mark options={fill=white}] table {re-swift.data};
|
|
817 |
\addplot[brown,mark=*, mark options={fill=white}] table {re-dart.data};
|
|
818 |
||
819 |
\addplot[red,mark=triangle*, mark options={fill=red}] table {nfabreadth2.data};
|
|
820 |
\end{axis}
|
|
821 |
\end{tikzpicture}
|
|
822 |
\end{center}
|
|
823 |
||
824 |
||
825 |
\end{frame}
|
|
826 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
827 |
||
828 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
829 |
\begin{frame}[c]
|
|
830 |
\frametitle{NFA Depth-First: \boldmath$a^{?\{n\}} \cdot a^{\{n\}}$}
|
|
831 |
||
832 |
\begin{tikzpicture}
|
|
833 |
\begin{axis}[xlabel={\pcode{a}s},
|
|
834 |
ylabel={time in secs},
|
|
835 |
enlargelimits=false, |
|
836 |
xtick={0,5,...,30},
|
|
837 |
xmax=30, |
|
838 |
ymax=35, |
|
839 |
ytick={0,5,...,30},
|
|
840 |
scaled ticks=false, |
|
841 |
axis lines=left, |
|
842 |
width=9cm, |
|
843 |
height=6cm, |
|
844 |
legend entries={Python,Ruby,my NFA},
|
|
845 |
legend pos=outer north east, |
|
846 |
legend cell align=left] |
|
847 |
\addplot[blue,mark=*, mark options={fill=white}] table {re-python.data};
|
|
848 |
\addplot[brown,mark=pentagon*, mark options={fill=white}] table {re-ruby.data};
|
|
849 |
\addplot[red,mark=triangle*, mark options={fill=red}] table {nfadepth.data};
|
|
850 |
\end{axis}
|
|
851 |
\end{tikzpicture}
|
|
852 |
||
853 |
\end{frame}
|
|
854 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
855 |
||
856 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
857 |
\begin{frame}[c]
|
|
858 |
\frametitle{NFA Depth-First: \boldmath$(a^*)^* \cdot b$}
|
|
859 |
||
860 |
\bigskip |
|
861 |
\begin{center}
|
|
862 |
\begin{tikzpicture}
|
|
863 |
\begin{axis}[
|
|
864 |
xlabel={\pcode{a}s},
|
|
865 |
%%x label style={at={(1.05,0.0)}},
|
|
866 |
ylabel={time in secs},
|
|
867 |
enlargelimits=false, |
|
868 |
xtick={0,15,...,30},
|
|
869 |
xmax=33, |
|
870 |
ymax=35, |
|
871 |
ytick={0,5,...,30},
|
|
872 |
scaled ticks=false, |
|
873 |
axis lines=left, |
|
874 |
width=9cm, |
|
875 |
height=6cm, |
|
876 |
legend entries={Java 8,Python,JavaScript,Swift,Dart,my NFA},
|
|
877 |
legend pos=outer north east, |
|
878 |
legend cell align=left] |
|
879 |
\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};
|
|
880 |
\addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data};
|
|
881 |
\addplot[red,mark=*, mark options={fill=white}] table {re-js.data};
|
|
882 |
\addplot[magenta,mark=*, mark options={fill=white}] table {re-swift.data};
|
|
883 |
\addplot[brown,mark=*, mark options={fill=white}] table {re-dart.data};
|
|
884 |
||
885 |
\addplot[red,mark=triangle*, mark options={fill=red}] table {nfadepth2.data};
|
|
886 |
\end{axis}
|
|
887 |
\end{tikzpicture}
|
|
888 |
\end{center}
|
|
889 |
||
890 |
The punchline is that many existing libraries do depth-first search |
|
891 |
in NFAs (with backtracking). |
|
892 |
||
893 |
\end{frame}
|
|
894 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
895 |
||
| 782 | 896 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
897 |
\begin{frame}[c]
|
|
898 |
\frametitle{Subset Construction}
|
|
899 |
||
900 |
||
901 |
\begin{textblock}{5}(1,3.5)
|
|
902 |
\begin{center}
|
|
903 |
\begin{tikzpicture}[node distance=6mm,scale=0.7,>=stealth',very thick,
|
|
904 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
|
905 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
906 |
\node[state] (Q_1) [below=of Q_0] {$\mbox{Q}_1$};
|
|
907 |
\node[state, accepting] (Q_2) [below=of Q_1] {$\mbox{Q}_2$};
|
|
908 |
||
909 |
\path[->] (Q_0) edge node [left] {\alert{$1$}} (Q_1);
|
|
910 |
\path[->] (Q_1) edge node [left] {\alert{$0,1$}} (Q_2);
|
|
911 |
\path[->] (Q_0) edge [loop right] node {\alert{$0,1$}} ();
|
|
912 |
%\path[->] (Q_1) edge [loop above] node {\alert{$a$}} ();
|
|
913 |
%\path[->] (Q_2) edge [loop below] node {\alert{$b$}} ();
|
|
914 |
\end{tikzpicture}
|
|
915 |
\end{center}
|
|
916 |
\end{textblock}
|
|
917 |
||
918 |
\begin{textblock}{11}(6.5,4)
|
|
919 |
\begin{tabular}{r|cl}
|
|
920 |
nodes \textcolor{white}{*} & $0$ & $1$\\
|
|
921 |
\hline |
|
922 |
$\{\}$ \textcolor{white}{*} & \onslide<2->{$\{\}$} & \onslide<2->{$\{\}$}\\
|
|
923 |
\onslide<5->{s:} $\{0\}$ \textcolor{white}{*} & \onslide<3->{$\{0\}$} & \onslide<3->{$\{0,1\}$}\\
|
|
924 |
$\{1\}$ \textcolor{white}{*} & \onslide<3->{$\{2\}$} & \onslide<3->{$\{2\}$}\\
|
|
925 |
$\{2\}$ \onslide<5->{*} & \onslide<3->{$\{\}$} & \onslide<3->{$\{\}$}\\
|
|
926 |
$\{0,1\}$ \textcolor{white}{*} &\onslide<4->{$\{0,2\}$} & \onslide<4->{$\{0,1,2\}$}\\
|
|
927 |
$\{0,2\}$ \onslide<5->{*} &\onslide<4->{$\{0\}$} & \onslide<4->{$\{0,1\}$}\\
|
|
928 |
$\{1,2\}$ \onslide<5->{*} & \onslide<4->{$\{2\}$} & \onslide<4->{$\{2\}$}\\
|
|
929 |
$\{0,1,2\}$ \onslide<5->{*}& \onslide<4->{$\{0,2\}$} & \onslide<4->{$\{0,1,2\}$}\\
|
|
930 |
\end{tabular}
|
|
931 |
\end{textblock}
|
|
932 |
||
933 |
\end{frame}
|
|
934 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 779 | 935 |
|
| 782 | 936 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
937 |
\begin{frame}[c]
|
|
938 |
\frametitle{The Result}
|
|
939 |
||
940 |
\footnotesize |
|
941 |
\begin{center}
|
|
942 |
\begin{tikzpicture}[scale=0.5,>=stealth',very thick,
|
|
943 |
every state/.style={minimum size=0pt,
|
|
944 |
draw=blue!50,very thick,fill=blue!20}, |
|
945 |
baseline=0mm] |
|
946 |
\node[state,initial] (q0) {$\{0\}$};
|
|
947 |
\node[state] (q01) [right=of q0] {$\{0,1\}$};
|
|
948 |
\node[state,accepting] (q12) [above=of q01] {$\{1,2\}$};
|
|
949 |
\node[state,accepting] (q02) [below=of q01] {$\{0,2\}$};
|
|
950 |
\node[state] (q1) [right=2cm of q12] {$\{1\}$};
|
|
951 |
\node[state,accepting] (q2) [right=2.5cm of q01] {$\{2\}$};
|
|
952 |
\node[state,accepting] (q012) [right=1.5cm of q02] {$\{0,1,2\}$};
|
|
953 |
\node[state] (Qn) [right=of q2] {$\{\}$};
|
|
954 |
||
955 |
\path[->] (q0) edge [loop above] node {\alert{$0$}} ();
|
|
956 |
\path[->] (q0) edge node [above] {\alert{$1$}} (q01);
|
|
957 |
\path[->] (q02) edge [bend left] node [left] {\alert{$1$}} (q01);
|
|
958 |
\path[->] (q02) edge node [below] {\alert{$0$}} (q0);
|
|
959 |
\path[->] (q01) edge node [above] {\alert{$1$}} (q012);
|
|
960 |
\path[->] (q01) edge [bend left] node [right] {\alert{$0$}} (q02);
|
|
961 |
\path[->] (q12) edge node [below] {\alert{$0,1$}} (q2);
|
|
962 |
\path[->] (q1) edge node [right] {\alert{$0,1$}} (q2);
|
|
963 |
\path[->] (q2) edge node [above] {\alert{$0,1$}} (Qn);
|
|
964 |
\path[->] (q012) edge [loop right] node {\alert{$1$}} ();
|
|
965 |
\path[->] (q012) edge node [below] {\alert{$0$}} (q02);
|
|
966 |
\path[->] (Qn) edge [loop above] node {$\alert{0},\alert{1}$} ();
|
|
967 |
\end{tikzpicture}
|
|
968 |
\end{center}
|
|
969 |
||
970 |
||
971 |
\end{frame}
|
|
972 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
973 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
974 |
\begin{frame}[c]
|
|
975 |
\frametitle{Removing Dead States}
|
|
976 |
||
977 |
\footnotesize |
|
978 |
\begin{center}
|
|
979 |
\begin{tabular}{ll}
|
|
980 |
\large DFA: & \large (original) NFA:\\ |
|
981 |
\raisebox{0mm}{%
|
|
982 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
983 |
every state/.style={minimum size=0pt,
|
|
984 |
draw=blue!50,very thick,fill=blue!20}] |
|
985 |
\node[state,initial] (q0) {$\{0\}$};
|
|
986 |
\node[state] (q01) [right=of q0] {$\{0,1\}$};
|
|
987 |
\node[state,accepting] (q02) [below=of q01] {$\{0,2\}$};
|
|
988 |
\node[state,accepting] (q012) [right=1.5cm of q02] {$\{0,1,2\}$};
|
|
989 |
||
990 |
\path[->] (q0) edge [loop above] node {\alert{$0$}} ();
|
|
991 |
\path[->] (q0) edge node [above] {\alert{$1$}} (q01);
|
|
992 |
\path[->] (q02) edge [bend left] node [left] {\alert{$1$}} (q01);
|
|
993 |
\path[->] (q02) edge node [below] {\alert{$0$}} (q0);
|
|
994 |
\path[->] (q01) edge node [above] {\alert{$1$}} (q012);
|
|
995 |
\path[->] (q01) edge [bend left] node [right] {\alert{$0$}} (q02);
|
|
996 |
\path[->] (q012) edge [loop right] node {\alert{$1$}} ();
|
|
997 |
\path[->] (q012) edge node [below] {\alert{$0$}} (q02);
|
|
998 |
\end{tikzpicture}}
|
|
999 |
& |
|
1000 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
1001 |
every state/.style={minimum size=0pt,
|
|
1002 |
draw=blue!50,very thick,fill=blue!20},] |
|
1003 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
|
1004 |
\node[state] (Q_1) [below=of Q_0] {$\mbox{Q}_1$};
|
|
1005 |
\node[state, accepting] (Q_2) [below=of Q_1] {$\mbox{Q}_2$};
|
|
1006 |
||
1007 |
\path[->] (Q_0) edge node [left] {\alert{$1$}} (Q_1);
|
|
1008 |
\path[->] (Q_1) edge node [left] {\alert{$0,1$}} (Q_2);
|
|
1009 |
\path[->] (Q_0) edge [loop right] node {\alert{$0,1$}} ();
|
|
1010 |
\end{tikzpicture}
|
|
1011 |
\end{tabular}
|
|
1012 |
\end{center}
|
|
1013 |
||
1014 |
\end{frame}
|
|
1015 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
| 779 | 1016 |
|
1017 |
||
1018 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1019 |
\begin{frame}[c]
|
|
| 782 | 1020 |
\frametitle{Subset Construction ($\epsilon$NFA)}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1021 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1022 |
|
| 662 | 1023 |
\begin{textblock}{5}(1,1.5)
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
1024 |
\begin{center}
|
| 515 | 1025 |
\begin{tikzpicture}[node distance=6mm,scale=0.7,>=stealth',very thick,
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
1026 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1027 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
1028 |
\node[state] (Q_1) [above=of Q_0] {$\mbox{Q}_1$};
|
|
1029 |
\node[state, accepting] (Q_2) [below=of Q_0] {$\mbox{Q}_2$};
|
|
1030 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_1);
|
|
1031 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_2);
|
|
1032 |
\path[->] (Q_0) edge [loop right] node {\alert{$a$}} ();
|
|
1033 |
\path[->] (Q_1) edge [loop above] node {\alert{$a$}} ();
|
|
1034 |
\path[->] (Q_2) edge [loop below] node {\alert{$b$}} ();
|
|
|
136
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
1035 |
\end{tikzpicture}
|
|
cc19e6cbcb68
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
135
diff
changeset
|
1036 |
\end{center}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1037 |
\end{textblock}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1038 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1039 |
\begin{textblock}{11}(6.5,4.5)
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1040 |
\begin{tabular}{r|cl}
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
1041 |
nodes \textcolor{white}{*} & $a$ & $b$\\
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1042 |
\hline |
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1043 |
$\{\}$ \textcolor{white}{*} & \onslide<2->{$\{\}$} & \onslide<2->{$\{\}$}\\
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
1044 |
$\{0\}$ \textcolor{white}{*} & \onslide<3->{$\{0,1,2\}$} & \onslide<3->{$\{2\}$}\\
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1045 |
$\{1\}$ \textcolor{white}{*} & \onslide<3->{$\{1\}$} & \onslide<3->{$\{\}$}\\
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1046 |
$\{2\}$ \onslide<5->{*} & \onslide<3->{$\{\}$} & \onslide<3->{$\{2\}$}\\
|
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
1047 |
$\{0,1\}$ \textcolor{white}{*} &\onslide<4->{$\{0,1,2\}$} & \onslide<4->{$\{2\}$}\\
|
|
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
1048 |
$\{0,2\}$ \onslide<5->{*} &\onslide<4->{$\{0,1,2\}$} & \onslide<4->{$\{2\}$}\\
|
|
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
1049 |
$\{1,2\}$ \onslide<5->{*} & \onslide<4->{$\{1\}$} & \onslide<4->{$\{2\}$}\\
|
|
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
1050 |
\onslide<5->{s:} $\{0,1,2\}$ \onslide<5->{*}& \onslide<4->{$\{0,1,2\}$} & \onslide<4->{$\{2\}$}\\
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1051 |
\end{tabular}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1052 |
\end{textblock}
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1053 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1054 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1055 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1056 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1057 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1058 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1059 |
\frametitle{The Result}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1060 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1061 |
\footnotesize |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1062 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1063 |
\begin{tikzpicture}[scale=0.5,>=stealth',very thick,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1064 |
every state/.style={minimum size=0pt,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1065 |
draw=blue!50,very thick,fill=blue!20}, |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1066 |
baseline=0mm] |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1067 |
\node[state,initial,accepting] (q012) {$\{0,1,2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1068 |
\node[state,accepting] (q02) [right=of q012] {$\{0,2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1069 |
\node[state] (q01) [above=of q02] {$\{0,1\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1070 |
\node[state,accepting] (q12) [below=of q02] {$\{1,2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1071 |
\node[state] (q0) [right=2cm of q01] {$\{0\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1072 |
\node[state] (q1) [right=2.5cm of q02] {$\{1\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1073 |
\node[state,accepting] (q2) [right=1.5cm of q12] {$\{2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1074 |
\node[state] (qn) [right=of q1] {$\{\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1075 |
|
|
348
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1076 |
\path[->] (q012) edge [loop below] node {\alert{$a$}} ();
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1077 |
\path[->] (q012) edge node [above] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1078 |
\path[->] (q12) edge [bend left] node [below,pos=0.4] {\alert{$a$}} (q1);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1079 |
\path[->] (q12) edge node [below] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1080 |
\path[->] (q02) edge node [above] {\alert{$a$}} (q012);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1081 |
\path[->] (q02) edge [bend left] node [above, pos=0.8] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1082 |
\path[->] (q01) edge node [below] {\alert{$a$}} (q012);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1083 |
\path[->] (q01) edge [bend left] node [above] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1084 |
\path[->] (q0) edge node [below] {\alert{$a$}} (q012);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1085 |
\path[->] (q0) edge node [right, pos=0.2] {\alert{$b$}} (q2);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1086 |
\path[->] (q1) edge [loop above] node {\alert{$a$}} ();
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1087 |
\path[->] (q1) edge node [above] {\alert{$b$}} (qn);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1088 |
\path[->] (q2) edge [loop right] node {\alert{$b$}} ();
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1089 |
\path[->] (q2) edge node [below] {\alert{$a$}} (qn);
|
|
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1090 |
\path[->] (qn) edge [loop above] node {$\alert{a},\alert{b}$} ();
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1091 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1092 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1093 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1094 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1095 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1096 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1097 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1098 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1099 |
\begin{frame}[c]
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1100 |
\frametitle{Removing Dead States}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1101 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1102 |
\footnotesize |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1103 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1104 |
\begin{tabular}{ll}
|
| 445 | 1105 |
DFA: & (original) NFA:\\ |
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1106 |
\raisebox{10mm}{%
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1107 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1108 |
every state/.style={minimum size=0pt,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1109 |
draw=blue!50,very thick,fill=blue!20}] |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1110 |
\node[state,initial,accepting] (q012) {$\{0,1,2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1111 |
\node[state,accepting] (q2) [right=of q012] {$\{2\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1112 |
\node[state] (qn) [right=of q2] {$\{\}$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1113 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1114 |
\path[->] (q012) edge [loop below] node {\alert{$a$}} ();
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1115 |
\path[->] (q012) edge node [above] {\alert{$b$}} (q2);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1116 |
\path[->] (q2) edge [loop below] node {\alert{$b$}} ();
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1117 |
\path[->] (q2) edge node [below] {\alert{$a$}} (qn);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1118 |
\path[->] (qn) edge [loop above] node |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1119 |
{$\alert{a},\alert{b}$} ();
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1120 |
\end{tikzpicture}}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1121 |
& |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1122 |
\begin{tikzpicture}[scale=0.7,>=stealth',very thick,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1123 |
every state/.style={minimum size=0pt,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1124 |
draw=blue!50,very thick,fill=blue!20},] |
| 515 | 1125 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
1126 |
\node[state] (Q_1) [above=of Q_0] {$\mbox{Q}_1$};
|
|
1127 |
\node[state, accepting] (Q_2) [below=of Q_0] {$\mbox{Q}_2$};
|
|
1128 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_1);
|
|
1129 |
\path[->] (Q_0) edge node [left] {\alert{$\epsilon$}} (Q_2);
|
|
1130 |
\path[->] (Q_0) edge [loop right] node {\alert{$a$}} ();
|
|
1131 |
\path[->] (Q_1) edge [loop above] node {\alert{$a$}} ();
|
|
1132 |
\path[->] (Q_2) edge [loop below] node {\alert{$b$}} ();
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1133 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1134 |
\end{tabular}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1135 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1136 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1137 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1138 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1139 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1140 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1141 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1142 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1143 |
\begin{frame}[c]
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1144 |
\frametitle{Regexps and Automata}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1145 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1146 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1147 |
\begin{tikzpicture}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1148 |
\node (rexp) {\bl{\bf Regexps}};
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1149 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1150 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1151 |
\onslide<2->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1152 |
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1153 |
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1154 |
\onslide<2->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1155 |
\end{tikzpicture}\\
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1156 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1157 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1158 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1159 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1160 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1161 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1162 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1163 |
\frametitle{DFA Minimisation}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1164 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1165 |
\begin{enumerate}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1166 |
\item Take all pairs \bl{$(q, p)$} with \bl{$q \not= p$}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1167 |
\item Mark all pairs that accepting and non-accepting states |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1168 |
\item For all unmarked pairs \bl{$(q, p)$} and all characters \bl{$c$} test whether
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1169 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1170 |
\bl{$(\delta(q, c), \delta(p,c))$}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1171 |
\end{center}
|
|
348
31e89128ccd2
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
346
diff
changeset
|
1172 |
are marked. If yes in at least one case, then also mark \bl{$(q, p)$}.
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1173 |
\item Repeat last step until no change. |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1174 |
\item All unmarked pairs can be merged. |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1175 |
\end{enumerate}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1176 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1177 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1178 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1179 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1180 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1181 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1182 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1183 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1184 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1185 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1186 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
1187 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
1188 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
1189 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
1190 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
1191 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
1192 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
1193 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
1194 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
1195 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
1196 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
1197 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
1198 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
1199 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1200 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1201 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1202 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1203 |
\mbox{}\\[-20mm]\mbox{}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1204 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1205 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1206 |
\begin{tikzpicture}[scale=0.8,line width=0.8mm]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1207 |
\draw (0,0) -- (4,0); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1208 |
\draw (0,1) -- (4,1); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1209 |
\draw (0,2) -- (3,2); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1210 |
\draw (0,3) -- (2,3); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1211 |
\draw (0,4) -- (1,4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1212 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1213 |
\draw (0,0) -- (0, 4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1214 |
\draw (1,0) -- (1, 4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1215 |
\draw (2,0) -- (2, 3); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1216 |
\draw (3,0) -- (3, 2); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1217 |
\draw (4,0) -- (4, 1); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1218 |
|
| 515 | 1219 |
\draw (0.5,-0.5) node {$\mbox{Q}_0$};
|
1220 |
\draw (1.5,-0.5) node {$\mbox{Q}_1$};
|
|
1221 |
\draw (2.5,-0.5) node {$\mbox{Q}_2$};
|
|
1222 |
\draw (3.5,-0.5) node {$\mbox{Q}_3$};
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1223 |
|
| 515 | 1224 |
\draw (-0.5, 3.5) node {$\mbox{Q}_1$};
|
1225 |
\draw (-0.5, 2.5) node {$\mbox{Q}_2$};
|
|
1226 |
\draw (-0.5, 1.5) node {$\mbox{Q}_3$};
|
|
1227 |
\draw (-0.5, 0.5) node {$\mbox{Q}_4$};
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1228 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1229 |
\draw (0.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1230 |
\draw (1.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1231 |
\draw (2.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1232 |
\draw (3.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1233 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1234 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1235 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1236 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1237 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1238 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1239 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1240 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1241 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1242 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1243 |
\begin{tabular}{@{\hspace{-8mm}}cc@{}}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1244 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1245 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1246 |
\node[state,initial] (Q_0) {$\mbox{Q}_0$};
|
1247 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
1248 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
1249 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
1250 |
\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};
|
|
1251 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
|
1252 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
1253 |
\path[->] (Q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
1254 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
1255 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
1256 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
1257 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
1258 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
1259 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1260 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1261 |
& |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1262 |
\raisebox{9mm}{\begin{tikzpicture}[scale=0.6,line width=0.8mm]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1263 |
\draw (0,0) -- (4,0); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1264 |
\draw (0,1) -- (4,1); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1265 |
\draw (0,2) -- (3,2); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1266 |
\draw (0,3) -- (2,3); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1267 |
\draw (0,4) -- (1,4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1268 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1269 |
\draw (0,0) -- (0, 4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1270 |
\draw (1,0) -- (1, 4); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1271 |
\draw (2,0) -- (2, 3); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1272 |
\draw (3,0) -- (3, 2); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1273 |
\draw (4,0) -- (4, 1); |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1274 |
|
| 515 | 1275 |
\draw (0.5,-0.5) node {$\mbox{Q}_0$};
|
1276 |
\draw (1.5,-0.5) node {$\mbox{Q}_1$};
|
|
1277 |
\draw (2.5,-0.5) node {$\mbox{Q}_2$};
|
|
1278 |
\draw (3.5,-0.5) node {$\mbox{Q}_3$};
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1279 |
|
| 515 | 1280 |
\draw (-0.5, 3.5) node {$\mbox{Q}_1$};
|
1281 |
\draw (-0.5, 2.5) node {$\mbox{Q}_2$};
|
|
1282 |
\draw (-0.5, 1.5) node {$\mbox{Q}_3$};
|
|
1283 |
\draw (-0.5, 0.5) node {$\mbox{Q}_4$};
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1284 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1285 |
\draw (0.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1286 |
\draw (1.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1287 |
\draw (2.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1288 |
\draw (3.5,0.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1289 |
\draw (0.5,1.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1290 |
\draw (2.5,1.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1291 |
\draw (0.5,3.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1292 |
\draw (1.5,2.5) node {\large$\star$};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1293 |
\end{tikzpicture}}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1294 |
\end{tabular}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1295 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1296 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1297 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1298 |
\mbox{}\\[-20mm]\mbox{}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1299 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1300 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1301 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1302 |
every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1303 |
\node[state,initial] (Q_02) {$\mbox{Q}_{0, 2}$};
|
1304 |
\node[state] (Q_13) [right=of Q_02] {$\mbox{Q}_{1, 3}$};
|
|
1305 |
\node[state, accepting] (Q_4) [right=of Q_13] {$\mbox{Q}_{4\phantom{,0}}$};
|
|
1306 |
\path[->] (Q_02) edge [bend left] node [above] {\alert{$a$}} (Q_13);
|
|
1307 |
\path[->] (Q_13) edge [bend left] node [below] {\alert{$b$}} (Q_02);
|
|
1308 |
\path[->] (Q_02) edge [loop below] node {\alert{$b$}} ();
|
|
1309 |
\path[->] (Q_13) edge node [above] {\alert{$a$}} (Q_4);
|
|
1310 |
\path[->] (Q_4) edge [loop above] node {\alert{$a, b$}} ();
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1311 |
\end{tikzpicture}\\
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1312 |
minimal automaton |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1313 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1314 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1315 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1316 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1317 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1318 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1319 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1320 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1321 |
\frametitle{Alternatives}
|
| 573 | 1322 |
\mbox{}\\[-14mm]\mbox{}
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1323 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1324 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1325 |
\begin{tikzpicture}[>=stealth',very thick,auto,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1326 |
every state/.style={minimum size=0pt,
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1327 |
inner sep=2pt,draw=blue!50,very thick,fill=blue!20}] |
| 515 | 1328 |
\only<1>{\node[state,initial] (Q_0) {$\mbox{Q}_0$};}
|
1329 |
\only<2->{\node[state,accepting] (Q_0) {$\mbox{Q}_0$};}
|
|
1330 |
\node[state] (Q_1) [right=of Q_0] {$\mbox{Q}_1$};
|
|
1331 |
\node[state] (Q_2) [below right=of Q_0] {$\mbox{Q}_2$};
|
|
1332 |
\node[state] (Q_3) [right=of Q_2] {$\mbox{Q}_3$};
|
|
1333 |
\only<1>{\node[state, accepting] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};}
|
|
1334 |
\only<2->{\node[state, initial right] (Q_4) [right=of Q_1] {$\mbox{Q}_4$};}
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1335 |
\only<1-2>{
|
| 515 | 1336 |
\path[->] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
1337 |
\path[->] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
1338 |
\path[->] (Q_4) edge [loop above] node {\alert{$a, b$}} ();
|
|
1339 |
\path[->] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
1340 |
\path[->] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
1341 |
\path[->] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
1342 |
\path[->] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
1343 |
\path[->] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
1344 |
\path[->] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);}
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1345 |
\only<3->{
|
| 515 | 1346 |
\path[<-] (Q_0) edge node [above] {\alert{$a$}} (Q_1);
|
1347 |
\path[<-] (Q_1) edge node [above] {\alert{$a$}} (Q_4);
|
|
1348 |
\path[<-] (Q_4) edge [loop above] node {\alert{$a, b$}} ();
|
|
1349 |
\path[<-] (Q_3) edge node [right] {\alert{$a$}} (Q_4);
|
|
1350 |
\path[<-] (Q_2) edge node [above] {\alert{$a$}} (Q_3);
|
|
1351 |
\path[<-] (Q_1) edge node [right] {\alert{$b$}} (Q_2);
|
|
1352 |
\path[<-] (Q_0) edge node [above] {\alert{$b$}} (Q_2);
|
|
1353 |
\path[<-] (Q_2) edge [loop left] node {\alert{$b$}} ();
|
|
1354 |
\path[<-] (Q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (Q_0);}
|
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1355 |
\end{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1356 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1357 |
\mbox{}\\[-18mm]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1358 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1359 |
\small |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1360 |
\begin{itemize}
|
| 573 | 1361 |
\item<1-> exchange initial / accepting states\\[-2mm] |
1362 |
\item<2-> reverse all edges\\[-2mm] |
|
1363 |
\item<3-> subset construction $\Rightarrow$ DFA\\[-2mm] |
|
1364 |
\item<4-> remove dead states\\[-2mm] |
|
1365 |
\item<5-> repeat once more |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1366 |
\onslide<6->{$\Rightarrow$ minimal DFA}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1367 |
\end{itemize}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1368 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1369 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1370 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1371 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1372 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1373 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1374 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1375 |
\frametitle{Regexps and Automata}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1376 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1377 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1378 |
\begin{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1379 |
\node (rexp) {\bl{\bf Regexps}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1380 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1381 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1382 |
\onslide<1->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1383 |
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1384 |
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1385 |
\onslide<1->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1386 |
\onslide<2->{\path[->, red, line width=2mm] (dfa) edge [bend left=45] (rexp);}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1387 |
\end{tikzpicture}\\
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1388 |
\end{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1389 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1390 |
\end{frame}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1391 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1392 |
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1393 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1394 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1395 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1396 |
\begin{frame}<2>[c]
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1397 |
\frametitle{DFA to Rexp}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1398 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1399 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1400 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1401 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1402 |
\only<1>{\node[state, initial] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
1403 |
\only<2->{\node[state, initial,accepting] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
|
1404 |
\only<1>{\node[state] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
1405 |
\only<2->{\node[state,accepting] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
1406 |
\only<1>{\node[state, accepting] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
|
1407 |
\only<2->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1408 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1409 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1410 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1411 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1412 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1413 |
(q0) edge [loop below] node {\alert{$b$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1414 |
; |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1415 |
\end{tikzpicture}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1416 |
\end{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1417 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1418 |
\onslide<3>{How to get from a DFA to a regular expression?}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1419 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1420 |
\end{frame}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1421 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1422 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1423 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1424 |
\begin{frame}[c]
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1425 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1426 |
\begin{center}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1427 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1428 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
| 783 | 1429 |
\only<1->{\node[state, initial,accepting] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
1430 |
\only<1->{\node[state,accepting] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
| 515 | 1431 |
\only<1->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1432 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1433 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1434 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1435 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1436 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1437 |
(q0) edge [loop below] node {\alert{$b$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1438 |
; |
|
139
6e7c3db9023d
added
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
138
diff
changeset
|
1439 |
\end{tikzpicture}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1440 |
\end{center}\pause\bigskip
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1441 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1442 |
\onslide<2->{
|
| 445 | 1443 |
You know how to solve since school days, no? |
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1444 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1445 |
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
|
| 515 | 1446 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$2\, \mbox{Q}_0 + 3 \,\mbox{Q}_1 + 4\, \mbox{Q}_2$}\\
|
1447 |
\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$2 \,\mbox{Q}_0 + 3\, \mbox{Q}_1 + 1\, \mbox{Q}_2$}\\
|
|
1448 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$1\, \mbox{Q}_0 + 5\, \mbox{Q}_1 + 2\, \mbox{Q}_2$}\\
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1449 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1450 |
\end{tabular}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1451 |
\end{center}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1452 |
} |
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1453 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1454 |
\end{frame}
|
|
135
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1455 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1456 |
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1457 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1458 |
\begin{frame}[c]
|
|
72b686e1c974
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
134
diff
changeset
|
1459 |
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1460 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1461 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1462 |
every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
|
| 515 | 1463 |
\only<1->{\node[state, initial] (q0) at ( 0,1) {$\mbox{Q}_0$};}
|
1464 |
\only<1->{\node[state] (q1) at ( 1,1) {$\mbox{Q}_1$};}
|
|
1465 |
\only<1->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1466 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1467 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1468 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1469 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1470 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1471 |
(q0) edge [loop below] node {\alert{$b$}} ()
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1472 |
; |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1473 |
\end{tikzpicture}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1474 |
\end{center}\bigskip
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1475 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1476 |
\onslide<2->{
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1477 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1478 |
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
|
| 650 | 1479 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,b + \mbox{Q}_1\,b + \mbox{Q}_2\,b + \ONE$}\\
|
| 515 | 1480 |
\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a$}\\
|
1481 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_1\,a + \mbox{Q}_2\,a$}\\
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1482 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1483 |
\end{tabular}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1484 |
\end{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1485 |
} |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1486 |
|
| 783 | 1487 |
|
1488 |
\only<3-9>{\small
|
|
1489 |
\begin{textblock}{6}(1,0.8)
|
|
1490 |
\begin{bubble}[6.7cm]
|
|
1491 |
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
1492 |
\multicolumn{3}{@{}l}{substitute \bl{$\mbox{Q}_1$} into \bl{$\mbox{Q}_0$} \& \bl{$\mbox{Q}_2$}:}\\
|
|
1493 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,b + \mbox{Q}_0\,a\,b + \mbox{Q}_2\,b + \ONE$}\\
|
|
1494 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a + \mbox{Q}_2\,a$}
|
|
1495 |
\end{tabular}
|
|
1496 |
\end{bubble}
|
|
1497 |
\end{textblock}}
|
|
1498 |
||
1499 |
\only<4-9>{\small
|
|
1500 |
\begin{textblock}{6}(2,4.15)
|
|
1501 |
\begin{bubble}[6.7cm]
|
|
1502 |
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
1503 |
\multicolumn{3}{@{}l}{simplifying \bl{$\mbox{Q}_0$}:}\\
|
|
1504 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,(b + a\,b) + \mbox{Q}_2\,b + \ONE$}\\
|
|
1505 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a + \mbox{Q}_2\,a$}
|
|
1506 |
\end{tabular}
|
|
1507 |
\end{bubble}
|
|
1508 |
\end{textblock}}
|
|
1509 |
||
1510 |
\only<6-9>{\small
|
|
1511 |
\begin{textblock}{6}(3,7.55)
|
|
1512 |
\begin{bubble}[6.7cm]
|
|
1513 |
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
1514 |
\multicolumn{3}{@{}l}{Arden for \bl{$\mbox{Q}_2$}:}\\
|
|
1515 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,(b + a\,b) + \mbox{Q}_2\,b + \ONE$}\\
|
|
1516 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a\,(a^*)$}
|
|
1517 |
\end{tabular}
|
|
1518 |
\end{bubble}
|
|
1519 |
\end{textblock}}
|
|
1520 |
||
1521 |
\only<7-9>{\small
|
|
1522 |
\begin{textblock}{6}(4,10.9)
|
|
1523 |
\begin{bubble}[7.5cm]
|
|
1524 |
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
1525 |
\multicolumn{3}{@{}l}{Substitute \bl{$\mbox{Q}_2$} and simplify:}\\
|
|
1526 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,(b + a\,b + a\,a\,(a^*)\,b) + \ONE$}\\
|
|
1527 |
\end{tabular}
|
|
1528 |
\end{bubble}
|
|
1529 |
\end{textblock}}
|
|
1530 |
||
1531 |
\only<8-9>{\small
|
|
1532 |
\begin{textblock}{6}(5,13.4)
|
|
1533 |
\begin{bubble}[7.5cm]
|
|
1534 |
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
1535 |
\multicolumn{3}{@{}l}{Arden again for \bl{$\mbox{Q}_0$}:}\\
|
|
1536 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*$}\\
|
|
1537 |
\end{tabular}
|
|
1538 |
\end{bubble}
|
|
1539 |
\end{textblock}}
|
|
1540 |
||
1541 |
||
1542 |
\only<9-10>{\small
|
|
1543 |
\begin{textblock}{6}(6,11.5)
|
|
1544 |
\begin{bubble}[6.7cm]
|
|
1545 |
\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
1546 |
\multicolumn{3}{@{}l}{Finally:}\\
|
|
1547 |
\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*$}\\
|
|
1548 |
\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*\,a$}\\
|
|
1549 |
\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*\,a\,a\,(a^*)$}\\
|
|
1550 |
\end{tabular}
|
|
1551 |
\end{bubble}
|
|
1552 |
\end{textblock}}
|
|
1553 |
||
1554 |
||
1555 |
||
1556 |
||
1557 |
||
1558 |
\only<5-6>{
|
|
1559 |
\begin{textblock}{6}(0.7,11.9)
|
|
1560 |
\begin{bubble}[6.7cm]
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1561 |
Arden's Lemma: |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1562 |
\begin{center}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1563 |
If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1564 |
\end{center}
|
| 783 | 1565 |
\end{bubble}
|
1566 |
\end{textblock}}
|
|
1567 |
||
1568 |
\only<8>{
|
|
1569 |
\begin{textblock}{6}(1.1,7.8)
|
|
1570 |
\begin{bubble}[6.7cm]
|
|
1571 |
Arden's Lemma: |
|
1572 |
\begin{center}
|
|
1573 |
If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
|
|
1574 |
\end{center}
|
|
1575 |
\end{bubble}
|
|
1576 |
\end{textblock}}
|
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1577 |
|
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1578 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1579 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1580 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1581 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1582 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1583 |
\frametitle{Regexps and Automata}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1584 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1585 |
\begin{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1586 |
\begin{tikzpicture}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1587 |
\node (rexp) {\bl{\bf Regexps}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1588 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1589 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1590 |
\onslide<1->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1591 |
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1592 |
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1593 |
\onslide<1->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1594 |
\onslide<1->{\path[->, red, line width=2mm] (dfa) edge [bend left=45] (rexp);}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1595 |
\end{tikzpicture}\\
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1596 |
\end{center}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1597 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1598 |
\end{frame}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1599 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1600 |
|
| 940 | 1601 |
{
|
1602 |
\setbeamercolor{background canvas}{bg=cream}
|
|
1603 |
\begin{frame}<1-10>[c]
|
|
1604 |
\end{frame}
|
|
1605 |
} |
|
1606 |
||
1607 |
||
|
346
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1608 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1609 |
\begin{frame}[c]
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1610 |
\frametitle{Regular Languages (3)}
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1611 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1612 |
A language is \alert{\bf regular} iff there exists a regular
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1613 |
expression that recognises all its strings.\bigskip\medskip |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1614 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1615 |
or {\bf equivalently}\bigskip\medskip
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1616 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1617 |
A language is \alert{\bf regular} iff there exists a
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1618 |
deterministic finite automaton that recognises all its |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1619 |
strings.\bigskip\medskip\pause |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1620 |
|
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1621 |
Why is every finite set of strings a regular language? |
|
a98794b11ac4
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
265
diff
changeset
|
1622 |
\end{frame}
|
|
265
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1623 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1624 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1625 |
|
|
332fbe9c91ab
added slides
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
215
diff
changeset
|
1626 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 783 | 1627 |
%\begin{frame}[c]
|
1628 |
% |
|
1629 |
%Given the function |
|
1630 |
% |
|
1631 |
%\begin{center}
|
|
1632 |
%\bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
|
|
1633 |
%$rev(\ZERO)$ & $\dn$ & $\ZERO$\\ |
|
1634 |
%$rev(\ONE)$ & $\dn$ & $\ONE$\\ |
|
1635 |
%$rev(c)$ & $\dn$ & $c$\\ |
|
1636 |
%$rev(r_1 + r_2)$ & $\dn$ & $rev(r_1) + rev(r_2)$\\ |
|
1637 |
%$rev(r_1 \cdot r_2)$ & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\ |
|
1638 |
%$rev(r^*)$ & $\dn$ & $rev(r)^*$\\ |
|
1639 |
%\end{tabular}}
|
|
1640 |
%\end{center}
|
|
1641 |
% |
|
1642 |
% |
|
1643 |
%and the set |
|
1644 |
% |
|
1645 |
%\begin{center}
|
|
1646 |
%\bl{$Rev\,A \dn \{s^{-1} \;|\; s \in A\}$}
|
|
1647 |
%\end{center}
|
|
1648 |
% |
|
1649 |
%prove whether |
|
1650 |
% |
|
1651 |
%\begin{center}
|
|
1652 |
%\bl{$L(rev(r)) = Rev (L(r))$}
|
|
1653 |
%\end{center}
|
|
1654 |
% |
|
1655 |
%\end{frame}
|
|
|
133
09efdf5cf07c
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
93
diff
changeset
|
1656 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 25 | 1657 |
|
1658 |
||
| 743 | 1659 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1660 |
\begin{frame}[c]
|
|
1661 |
\frametitle{Regexps and Automata}
|
|
1662 |
||
1663 |
\begin{center}
|
|
1664 |
\begin{tikzpicture}
|
|
1665 |
\node (rexp) {\bl{\bf Regexps}};
|
|
1666 |
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
|
|
1667 |
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
|
|
1668 |
\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};
|
|
1669 |
\path[->,red, line width=2mm] (rexp) edge node [above=4mm, black] |
|
1670 |
{\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
|
|
1671 |
\path[->,red, line width=2mm] (nfa) edge node [above=4mm, black] |
|
1672 |
{\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
|
|
1673 |
\path[->, red, line width=2mm] (dfa) edge node [below=5mm, black] {minimisation} (mdfa);
|
|
1674 |
%%\path[->, red, line width=2mm] (dfa) edge [bend left=45] (rexp); |
|
1675 |
\path[->, red, line width=2mm] (dfa) edge [bend left=45] node [below, black] {\begin{tabular}{l}Brzozowski's\\ method\end{tabular}} (rexp);
|
|
1676 |
||
1677 |
\end{tikzpicture}\\
|
|
1678 |
\end{center}
|
|
1679 |
||
1680 |
\end{frame}
|
|
1681 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1682 |
||
1683 |
||
1684 |
||
1685 |
||
1686 |
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1687 |
% \begin{frame}[c]
|
|
1688 |
% \frametitle{DFA to Rexp}
|
|
1689 |
||
1690 |
% \begin{center}
|
|
1691 |
% \begin{tikzpicture}[scale=2,>=stealth',very thick,
|
|
1692 |
% every state/.style={minimum size=0pt,
|
|
1693 |
% draw=blue!50,very thick,fill=blue!20},] |
|
1694 |
% \node[state, initial] (q0) at ( 0,1) {$q_0$};
|
|
1695 |
% \node[state] (q1) at ( 1,1) {$q_1$};
|
|
1696 |
% \node[state, accepting] (q2) at ( 2,1) {$q_2$};
|
|
1697 |
% \path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
|
1698 |
% (q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
|
1699 |
% (q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
|
1700 |
% (q1) edge node[above] {\alert{$a$}} (q2)
|
|
1701 |
% (q2) edge [loop right] node {\alert{$a$}} ()
|
|
1702 |
% (q0) edge [loop below] node {\alert{$b$}} ();
|
|
1703 |
% \end{tikzpicture}
|
|
1704 |
% \end{center}\bigskip
|
|
1705 |
||
1706 |
% \begin{center}
|
|
1707 |
% \begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l@{\hspace{7mm}}l}
|
|
1708 |
% \bl{$q_0$} & \bl{$=$} & \bl{$\ONE + q_0\,b + q_1\,b + q_2\,b$} & (start state)\\
|
|
1709 |
% \bl{$q_1$} & \bl{$=$} & \bl{$q_0\,a$}\\
|
|
1710 |
% \bl{$q_2$} & \bl{$=$} & \bl{$q_1\,a + q_2\,a$}\\
|
|
1711 |
||
1712 |
% \end{tabular}
|
|
1713 |
% \end{center}
|
|
1714 |
||
1715 |
% Arden's Lemma: |
|
1716 |
% \begin{center}
|
|
1717 |
% If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
|
|
1718 |
% \end{center}
|
|
1719 |
||
1720 |
||
1721 |
% \end{frame}
|
|
1722 |
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1723 |
||
1724 |
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1725 |
% \begin{frame}[c]
|
|
1726 |
% \frametitle{DFA Minimisation}
|
|
1727 |
||
1728 |
% \begin{enumerate}
|
|
1729 |
% \item Take all pairs \bl{$(q, p)$} with \bl{$q \not= p$}
|
|
1730 |
% \item Mark all pairs that accepting and non-accepting states |
|
1731 |
% \item For all unmarked pairs \bl{$(q, p)$} and all characters \bl{$c$} test whether
|
|
1732 |
% \begin{center}
|
|
1733 |
% \bl{$(\delta(q, c), \delta(p,c))$}
|
|
1734 |
% \end{center}
|
|
1735 |
% are marked. If yes, then also mark \bl{$(q, p)$}.
|
|
1736 |
% \item Repeat last step until no change. |
|
1737 |
% \item All unmarked pairs can be merged. |
|
1738 |
% \end{enumerate}
|
|
1739 |
||
1740 |
% \end{frame}
|
|
1741 |
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1742 |
||
1743 |
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1744 |
% \begin{frame}[c]
|
|
1745 |
||
1746 |
% \begin{center}
|
|
1747 |
% \begin{tabular}{@{\hspace{-8mm}}cc@{}}
|
|
1748 |
% \begin{tikzpicture}[>=stealth',very thick,auto,
|
|
1749 |
% every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
|
1750 |
% \node[state,initial] (q_0) {$q_0$};
|
|
1751 |
% \node[state] (q_1) [right=of q_0] {$q_1$};
|
|
1752 |
% \node[state] (q_2) [below right=of q_0] {$q_2$};
|
|
1753 |
% \node[state] (q_3) [right=of q_2] {$q_3$};
|
|
1754 |
% \node[state, accepting] (q_4) [right=of q_1] {$q_4$};
|
|
1755 |
% \path[->] (q_0) edge node [above] {\alert{$a$}} (q_1);
|
|
1756 |
% \path[->] (q_1) edge node [above] {\alert{$a$}} (q_4);
|
|
1757 |
% \path[->] (q_4) edge [loop right] node {\alert{$a, b$}} ();
|
|
1758 |
% \path[->] (q_3) edge node [right] {\alert{$a$}} (q_4);
|
|
1759 |
% \path[->] (q_2) edge node [above] {\alert{$a$}} (q_3);
|
|
1760 |
% \path[->] (q_1) edge node [right] {\alert{$b$}} (q_2);
|
|
1761 |
% \path[->] (q_0) edge node [above] {\alert{$b$}} (q_2);
|
|
1762 |
% \path[->] (q_2) edge [loop left] node {\alert{$b$}} ();
|
|
1763 |
% \path[->] (q_3) edge [bend left=95, looseness=1.3] node [below] {\alert{$b$}} (q_0);
|
|
1764 |
% \end{tikzpicture}
|
|
1765 |
% & |
|
1766 |
% \raisebox{9mm}{\begin{tikzpicture}[scale=0.6,line width=0.8mm]
|
|
1767 |
% \draw (0,0) -- (4,0); |
|
1768 |
% \draw (0,1) -- (4,1); |
|
1769 |
% \draw (0,2) -- (3,2); |
|
1770 |
% \draw (0,3) -- (2,3); |
|
1771 |
% \draw (0,4) -- (1,4); |
|
1772 |
||
1773 |
% \draw (0,0) -- (0, 4); |
|
1774 |
% \draw (1,0) -- (1, 4); |
|
1775 |
% \draw (2,0) -- (2, 3); |
|
1776 |
% \draw (3,0) -- (3, 2); |
|
1777 |
% \draw (4,0) -- (4, 1); |
|
1778 |
||
1779 |
% \draw (0.5,-0.5) node {$q_0$};
|
|
1780 |
% \draw (1.5,-0.5) node {$q_1$};
|
|
1781 |
% \draw (2.5,-0.5) node {$q_2$};
|
|
1782 |
% \draw (3.5,-0.5) node {$q_3$};
|
|
1783 |
||
1784 |
% \draw (-0.5, 3.5) node {$q_1$};
|
|
1785 |
% \draw (-0.5, 2.5) node {$q_2$};
|
|
1786 |
% \draw (-0.5, 1.5) node {$q_3$};
|
|
1787 |
% \draw (-0.5, 0.5) node {$q_4$};
|
|
1788 |
||
1789 |
% \draw (0.5,0.5) node {\large$\star$};
|
|
1790 |
% \draw (1.5,0.5) node {\large$\star$};
|
|
1791 |
% \draw (2.5,0.5) node {\large$\star$};
|
|
1792 |
% \draw (3.5,0.5) node {\large$\star$};
|
|
1793 |
% \draw (0.5,1.5) node {\large$\star$};
|
|
1794 |
% \draw (2.5,1.5) node {\large$\star$};
|
|
1795 |
% \draw (0.5,3.5) node {\large$\star$};
|
|
1796 |
% \draw (1.5,2.5) node {\large$\star$};
|
|
1797 |
% \end{tikzpicture}}
|
|
1798 |
% \end{tabular}
|
|
1799 |
% \end{center}
|
|
1800 |
||
1801 |
||
1802 |
% \mbox{}\\[-20mm]\mbox{}
|
|
1803 |
||
1804 |
% \begin{center}
|
|
1805 |
% \begin{tikzpicture}[>=stealth',very thick,auto,
|
|
1806 |
% every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
|
|
1807 |
% \node[state,initial] (q_02) {$q_{0, 2}$};
|
|
1808 |
% \node[state] (q_13) [right=of q_02] {$q_{1, 3}$};
|
|
1809 |
% \node[state, accepting] (q_4) [right=of q_13] {$q_{4\phantom{,0}}$};
|
|
1810 |
% \path[->] (q_02) edge [bend left] node [above] {\alert{$a$}} (q_13);
|
|
1811 |
% \path[->] (q_13) edge [bend left] node [below] {\alert{$b$}} (q_02);
|
|
1812 |
% \path[->] (q_02) edge [loop below] node {\alert{$b$}} ();
|
|
1813 |
% \path[->] (q_13) edge node [above] {\alert{$a$}} (q_4);
|
|
1814 |
% \path[->] (q_4) edge [loop above] node {\alert{$a, b$}} ();
|
|
1815 |
% \end{tikzpicture}\\
|
|
1816 |
% minimal automaton |
|
1817 |
% \end{center}
|
|
1818 |
||
1819 |
% \end{frame}
|
|
| 966 | 1820 |
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1821 |
||
1822 |
||
| 743 | 1823 |
|
1824 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1825 |
\begin{frame}[c]
|
|
1826 |
\frametitle{Regular Languages}
|
|
1827 |
||
1828 |
Two equivalent definitions:\bigskip |
|
1829 |
||
1830 |
\begin{quote}\rm A language is \alert{regular} iff there exists a
|
|
1831 |
regular expression that recognises all its strings. |
|
1832 |
\end{quote}
|
|
1833 |
||
1834 |
\begin{quote}\rm A language is \alert{regular} iff there exists an
|
|
1835 |
automaton that recognises all its strings. |
|
1836 |
\end{quote}\bigskip\bigskip
|
|
1837 |
||
1838 |
||
1839 |
\small |
|
1840 |
for example \bl{$a^nb^n$} is not regular
|
|
1841 |
\end{frame}
|
|
1842 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1843 |
||
1844 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1845 |
\begin{frame}[c]
|
|
1846 |
\frametitle{Negation}
|
|
1847 |
||
1848 |
Regular languages are closed under negation:\bigskip |
|
1849 |
||
1850 |
\begin{center}
|
|
1851 |
\begin{tikzpicture}[scale=2,>=stealth',very thick,
|
|
1852 |
every state/.style={minimum size=0pt,
|
|
1853 |
draw=blue!50,very thick,fill=blue!20}] |
|
1854 |
\only<1>{\node[state,initial] (q0) at ( 0,1) {$q_0$};}
|
|
1855 |
\only<2>{\node[state,initial,accepting] (q0) at ( 0,1) {$q_0$};}
|
|
1856 |
\only<1>{\node[state] (q1) at ( 1,1) {$q_1$};}
|
|
1857 |
\only<2>{\node[state,accepting] (q1) at ( 1,1) {$q_1$};}
|
|
1858 |
\only<1>{\node[state, accepting] (q2) at ( 2,1) {$q_2$};}
|
|
1859 |
\only<2>{\node[state] (q2) at ( 2,1) {$q_2$};}
|
|
1860 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
|
1861 |
(q1) edge[bend left] node[above] {\alert{$b$}} (q0)
|
|
1862 |
(q2) edge[bend left=50] node[below] {\alert{$b$}} (q0)
|
|
1863 |
(q1) edge node[above] {\alert{$a$}} (q2)
|
|
1864 |
(q2) edge [loop right] node {\alert{$a$}} ()
|
|
1865 |
(q0) edge [loop below] node {\alert{$b$}} ();
|
|
1866 |
\end{tikzpicture}
|
|
1867 |
\end{center}\bigskip\bigskip
|
|
1868 |
||
1869 |
But requires that the automaton is \alert{completed}!
|
|
1870 |
||
1871 |
\end{frame}
|
|
1872 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1873 |
||
| 891 | 1874 |
{
|
1875 |
\setbeamercolor{background canvas}{bg=cream}
|
|
1876 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1877 |
\begin{frame}[c]
|
|
1878 |
\frametitle{\boldmath{}$(a^*)^* \cdot b$}
|
|
1879 |
||
1880 |
\begin{center}
|
|
1881 |
\begin{tikzpicture}
|
|
1882 |
\begin{axis}[
|
|
1883 |
xlabel={\pcode{a}s},
|
|
1884 |
%%x label style={at={(1.05,0.0)}},
|
|
1885 |
ylabel={time in secs},
|
|
1886 |
enlargelimits=false, |
|
1887 |
xtick={0,15,...,30},
|
|
1888 |
xmax=33, |
|
1889 |
ymax=35, |
|
1890 |
ytick={0,5,...,30},
|
|
1891 |
scaled ticks=false, |
|
1892 |
axis lines=left, |
|
1893 |
width=9cm, |
|
1894 |
height=6cm, |
|
1895 |
legend entries={Java 8,Python,JavaScript,Swift,Dart},
|
|
1896 |
legend pos=outer north east, |
|
1897 |
legend cell align=left] |
|
1898 |
\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};
|
|
1899 |
\addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data};
|
|
1900 |
\addplot[red,mark=*, mark options={fill=white}] table {re-js.data};
|
|
1901 |
\addplot[magenta,mark=*, mark options={fill=white}] table {re-swift.data};
|
|
1902 |
\addplot[brown,mark=*, mark options={fill=white}] table {re-dart.data};
|
|
1903 |
||
1904 |
%\addplot[red,mark=triangle*, mark options={fill=red}] table {nfadepth2.data};
|
|
1905 |
\end{axis}
|
|
1906 |
\end{tikzpicture}
|
|
1907 |
\end{center}
|
|
1908 |
||
1909 |
\end{frame}}
|
|
1910 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1911 |
||
| 966 | 1912 |
|
1913 |
||
| 891 | 1914 |
\begin{frame}<1-3>[c]
|
1915 |
\end{frame}
|
|
1916 |
||
| 889 | 1917 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1918 |
{
|
|
1919 |
\setbeamercolor{background canvas}{bg=cream}
|
|
1920 |
\begin{frame}[c]
|
|
1921 |
||
1922 |
\begin{center}
|
|
1923 |
\begin{tikzpicture}[scale=1.5,>=stealth',very thick,
|
|
1924 |
every state/.style={minimum size=0pt,
|
|
1925 |
draw=blue!50,very thick,fill=blue!20}] |
|
1926 |
\node[state,initial] (q0) at (0,2) {$q_0$};
|
|
1927 |
\node[state,accepting] (q1) at (2,2) {$q_1$};
|
|
1928 |
\node[state] (q2) at (0,0) {$q_2$};
|
|
1929 |
\node[state] (q3) at (2,0) {$q_3$};
|
|
1930 |
||
1931 |
\path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
|
|
1932 |
(q1) edge[bend left] node[above] {\alert{$a$}} (q0)
|
|
1933 |
(q2) edge[bend left] node[above] {\alert{$a$}} (q3)
|
|
1934 |
(q3) edge[bend left] node[above] {\alert{$a$}} (q2)
|
|
1935 |
(q0) edge[bend left] node[right] {\alert{$b$}} (q2)
|
|
1936 |
(q2) edge[bend left] node[left] {\alert{$b$}} (q0)
|
|
1937 |
(q1) edge[bend left] node[right] {\alert{$b$}} (q3)
|
|
1938 |
(q3) edge[bend left] node[left] {\alert{$b$}} (q1);
|
|
1939 |
\end{tikzpicture}
|
|
1940 |
\end{center}
|
|
1941 |
||
1942 |
\hfill{}Which language?
|
|
1943 |
||
1944 |
\end{frame}
|
|
1945 |
} |
|
1946 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1947 |
||
1948 |
||
| 966 | 1949 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
1950 |
\begin{frame}[c]
|
|
1951 |
\frametitle{\mbox{CW1: Regexes and \boldmath$L$-function}}
|
|
1952 |
||
1953 |
Given |
|
1954 |
\begin{center}
|
|
1955 |
\begin{tabular}{@ {}l@ {\hspace{20mm}}l@ {\hspace{2mm}}l@ {\hspace{2mm}}l}
|
|
1956 |
\bl{$r^+$} & \bl{$L(r^+)$} & \bl{$\dn$} & \bl{$\bigcup_{1\le i}.\;L(r)^i$}\\
|
|
1957 |
\bl{$r^?$} & \bl{$L(r^?)$} & \bl{$\dn$} & \bl{$L(r) \cup \{[]\}$}\\
|
|
1958 |
\bl{$r_1 \,\&\, r_2$} & \bl{$L(r_1 \& r_2)$} & \bl{$\dn$} & \bl{$L(r_1) \cap L(r_2)$} \\
|
|
1959 |
\bl{$r^{\{n\}}$} & \bl{$L(r^{\{n\}})$} & \bl{$\dn$} & \bl{$L(r)^n$}\\
|
|
1960 |
\bl{$r^{\{..m\}}$} & \bl{$L(r^{\{..m\}})$} & \bl{$\dn$} & \bl{$\bigcup_{0\le i \le m}.\;L(r)^i$}\\
|
|
1961 |
\bl{$r^{\{n..\}}$} & \bl{$L(r^{\{n..\}})$} & \bl{$\dn$} & \bl{$\bigcup_{n\le i}.\;L(r)^i$}\\
|
|
1962 |
\bl{$r^{\{n..m\}}$} & \bl{$L(r^{\{n..m\}})$} & \bl{$\dn$} & \bl{$\bigcup_{n\le i \le m}.\;L(r)^i$}\\
|
|
1963 |
\bl{$\sim{}r$} & \bl{$L(\sim{}r)$} & \bl{$\dn$} & \bl{$\Sigma^* - L(r)$}\\
|
|
1964 |
\end{tabular}
|
|
1965 |
\end{center}
|
|
1966 |
||
1967 |
\end{frame}
|
|
1968 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1969 |
||
1970 |
||
1971 |
||
1972 |
||
1973 |
||
1974 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1975 |
\begin{frame}[c]
|
|
1976 |
\frametitle{Nullable}
|
|
1977 |
||
1978 |
\begin{center}
|
|
1979 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
|
|
1980 |
\bl{$nullable(r^+)$} & \bl{$\dn$} & \bl{$nullable(r)$}\\
|
|
1981 |
\bl{$nullable(r^?)$} & \bl{$\dn$} & \bl{\textit{true}}\\
|
|
1982 |
\bl{$nullable(r_1 \,\&\, r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \wedge nullable(r_2)$}\\
|
|
1983 |
\bl{$nullable(r^{\{n\}})$} & \bl{$\dn$} & \bl{\textit{if} $n = 0$ \textit{then} \textit{true} \textit{else} $nullable(r)$} \\
|
|
1984 |
\bl{$nullable(r^{\{..m\}})$} & \bl{$\dn$} & \bl{\textit{true}} \\
|
|
1985 |
\bl{$nullable (r^{\{n..\}})$} & \bl{$\dn$} & \bl{\textit{if} $n = 0$ \textit{then} \textit{true} \textit{else} $nullable(r)$}\\
|
|
1986 |
\bl{$nullable (r^{\{n..m\}})$} & \bl{$\dn$} & \bl{\textit{if} $n = 0$ \textit{then} \textit{true} \textit{else} $nullable(r)$}\\
|
|
1987 |
\bl{$nullable (\sim{}r)$} & \bl{$\dn$} & \bl{$!\,nullable(r)$}\\
|
|
1988 |
\end{tabular}
|
|
1989 |
\end{center}
|
|
1990 |
||
1991 |
\end{frame}
|
|
1992 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1993 |
||
1994 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1995 |
\begin{frame}[c]
|
|
1996 |
\frametitle{Derivative}
|
|
1997 |
||
1998 |
\begin{center}
|
|
1999 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
|
|
2000 |
\bl{$der\,c\,(r^+)$} & \bl{$\dn$} & \bl{$(der\,c\,r)\cdot r^*$}\\
|
|
2001 |
\bl{$der\,c\,(r^?)$} & \bl{$\dn$} & \bl{$der\,c\,r$}\\
|
|
2002 |
\bl{$der\,c\,(r_1 \,\&\, r_2)$} & \bl{$\dn$} & \bl{$(der\,c\,r_1) \;\&\; (der\,c\,r_2)$}\\
|
|
2003 |
\bl{$der\,c\,(r^{\{n\}})$} & \bl{$\dn$} & \bl{\textit{if} $n = 0$ \textit{then} $\ZERO$ \textit{else} $(der\,c\,r)\cdot r^{\{n - 1\}}$} \\
|
|
2004 |
\bl{$der\,c\,(r^{\{..m\}})$} & \bl{$\dn$} & \bl{\textit{if} $m = 0$ \textit{then} $\ZERO$ \textit{else} $(der\,c\,r)\cdot r^{\{..m - 1\}}$}\\
|
|
2005 |
\bl{$der\,c\,(r^{\{n..\}})$} & \bl{$\dn$} & \bl{\textit{if} $n = 0$ \textit{then} $(der\,c\,r)\cdot r^*$ \textit{else} $(der\,c\,r)\cdot r^{\{n - 1..\}}$}\\
|
|
2006 |
\bl{$der\,c\,(r^{\{n..m\}})$} & \bl{$\dn$} & \bl{\textit{if} $n = 0 \wedge m = 0$ \textit{then} $\ZERO$ \textit{else}}\\
|
|
2007 |
& & \bl{\textit{if} $ n = 0$ \textit{then} $(der\,c\,r)\cdot r^{\{..m-1\}}$
|
|
2008 |
\textit{else} $(der\,c\,r)\cdot r^{\{n-1..m-1\}}$}\\
|
|
2009 |
\bl{$der\,c\,(\sim{}r)$} & \bl{$\dn$} & \bl{$\sim\,(der\,c\,r)$}\\
|
|
2010 |
\end{tabular}
|
|
2011 |
\end{center}
|
|
2012 |
||
2013 |
\end{frame}
|
|
2014 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
2015 |
||
2016 |
||
| 781 | 2017 |
|
| 847 | 2018 |
\begin{frame}<1-15>[c]
|
2019 |
||
| 781 | 2020 |
\end{frame}
|
2021 |
||
| 966 | 2022 |
% \begin{frame}[t]
|
2023 |
% \begin{mybox3}{}
|
|
2024 |
% I always thought dfa's needed a transition for each state for each |
|
2025 |
% character, and if not it would be an nfa not a dfa. Is there an |
|
2026 |
% example that disproves this? |
|
2027 |
% \end{mybox3}
|
|
2028 |
% \end{frame}
|
|
| 781 | 2029 |
|
| 966 | 2030 |
% \begin{frame}<1-12>[c]
|
2031 |
% \end{frame}
|
|
| 781 | 2032 |
|
| 966 | 2033 |
% \begin{frame}[t]
|
2034 |
% \begin{mybox3}{}
|
|
2035 |
% Do the regular expression matchers in Python and Java 8 have more |
|
2036 |
% features than the one implemented in this module? Or is there |
|
2037 |
% another reason for their inefficiency? |
|
2038 |
% \end{mybox3}
|
|
2039 |
% \end{frame}
|
|
| 782 | 2040 |
|
| 891 | 2041 |
|
| 966 | 2042 |
% \begin{frame}[c]
|
2043 |
% \begin{itemize}
|
|
2044 |
% \item CW / censored some msgs |
|
2045 |
% \item power law / proof |
|
2046 |
% \item CW feedback |
|
2047 |
% \item too polished CW submissions |
|
2048 |
% \item no open book |
|
2049 |
% \end{itemize}
|
|
2050 |
% \end{frame}
|
|
| 891 | 2051 |
|
2052 |
||
| 25 | 2053 |
\end{document}
|
2054 |
||
2055 |
%%% Local Variables: |
|
2056 |
%%% mode: latex |
|
2057 |
%%% TeX-master: t |
|
2058 |
%%% End: |
|
2059 |