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  Office: & S1.27 (1st floor Strand Building)\\

A \alert{regular expression} specifies a set of strings, or language.

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

Different ways of writing strings:

\begin{center}

\begin{tabular}{ccc}

\bl{\consolas"$hello$"}\quad{} &  \bl{$[h, e, l, l, o]$} & \quad\bl{$h\!::\!e\!::\!l\!::\!l\!::\!o\!::\!N\!il$}\bigskip\\

\bl{\consolas ""}       &       \bl{$[]$}                & \bl{$N$\!$il$}

\end{tabular}

\end{center}\pause

The concatenation operation on strings and sets of strings:

\begin{center}

\begin{tabular}{l}

\bl{{\consolas"$f\!oo$"}$\;@\;${\consolas"$bar$"}$\;=\;${\consolas"$f\!oobar$"}}\medskip\\

\bl{$A \;@\; B \dn \{ s_1 @ s_2 \mid s_1 \in A \wedge s_2 \in B\}$}

\end{tabular}

\end{center}

\end{frame}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\mode<presentation>{

\begin{frame}[t]

\begin{textblock}{6}(2,6.5)

         & \bl{$\mid$} & \bl{$\epsilon$}        & empty string / {\consolas""} / $[]$\\

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\hspace{5mm}\mbox{{\lstset{language=Scala}\fontsize{8}{10}\selectfont

\texttt{\lstinputlisting{../progs/app51.scala}}}}

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 \bl{$L(\varnothing)$}     & \bl{$\dn$} & \bl{$\varnothing$}\\

 \bl{$L(\epsilon)$}          & \bl{$\dn$} & \bl{$\{$""$\}$}\\

 \bl{$L(c)$}                     & \bl{$\dn$} & \bl{$\{"c"\}$}\\

 \bl{$L(r_1 + r_2)$}        & \bl{$\dn$} & \bl{$L(r_1) \cup L(r_2)$}\\

 \bl{$L(r_1 \cdot r_2)$}  & \bl{$\dn$} & \bl{$L(r_1) \,@\, L(r_2)$}\\

 \bl{$L(r^*)$}                   & \bl{$\dn$} & \bl{$\bigcup_{n \ge 0} L(r)^n$}\\

\only<2->{  

\hspace{5mm}\textcolor{blue}{$L(r)^0 \;\dn\; \{""\}$}\\

\textcolor{blue}{$L(r)^{n+1} \;\dn\; L(r) \,@\, L(r)^n$}}  

\only<1->{

\bl{$L$ : Rexp $\Rightarrow$ Set$[$String$]$}

What is \bl{$L(a^*)$}?

&\bl{$(a + b)  + c$} & \bl{$\equiv^?$} & \bl{$a + (b + c)$} & \onslide<2->{\YES}\\

&\bl{$a + a$} & \bl{$\equiv^?$} & \bl{$a$} & \onslide<3->{\YES}\\

&\bl{$(a \cdot b)  \cdot c$} & \bl{$\equiv^?$} & \bl{$a \cdot (b \cdot c)$} & \onslide<4->{\YES}\\

&\bl{$a \cdot a$} & \bl{$\equiv^?$} & \bl{$a$} & \onslide<5->{\NO}\\

\FORALLR &\bl{$r \cdot \epsilon$} & \bl{$\equiv^?$} & \bl{$r$} & \onslide<8->{\YES}\\

\FORALLR &\bl{$r + \epsilon$} & \bl{$\equiv^?$} & \bl{$r$} & \onslide<9->{\NO}\\

\FORALLR &\bl{$r + \varnothing$} & \bl{$\equiv^?$} & \bl{$r$} & \onslide<10->{\YES}\\

\FORALLR &\bl{$r \cdot \varnothing$} & \bl{$\equiv^?$} & \bl{$r$} & \onslide<11->{\NO}\\

&\bl{$c \cdot (a + b)$} & \bl{$\equiv^?$} & \bl{$(c \cdot a) + (c \cdot b)$} & \onslide<12->{\YES}\\

&\bl{$a^*$} & \bl{$\equiv^?$} & \bl{$\epsilon + (a \cdot a^*)$} & \onslide<13->{\YES}

\frametitle{\begin{tabular}{c}The Specification\\[-1mm] of Matching\end{tabular}}

\end{frame}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\mode<presentation>{

\begin{frame}[t]

\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}

\mbox{}\\[-13mm]

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		file {re-python.data};

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         \draw[color=brown] plot[mark=pentagon*, mark options={fill=white} ] 

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\small

\ldots{}whether a regular expression can match the empty string:

\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}

\bl{$nullable(\varnothing)$}      & \bl{$\dn$} & \bl{$f\!\/alse$}\\

\bl{$nullable(\epsilon)$}           & \bl{$\dn$} &  \bl{$true$}\\

\bl{$nullable (c)$}                    & \bl{$\dn$} &  \bl{$f\!alse$}\\

\bl{$nullable (r_1 + r_2)$}       & \bl{$\dn$} &  \bl{$nullable(r_1) \vee nullable(r_2)$} \\ 

\bl{$nullable (r_1 \cdot r_2)$} & \bl{$\dn$} &  \bl{$nullable(r_1) \wedge nullable(r_2)$} \\

\bl{$nullable (r^*)$}                 & \bl{$\dn$} & \bl{$true$} \\

\end{tabular}

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\bl{$der\,c\,r$} gives the answer

  \bl{$der\, c\, (\varnothing)$}      & \bl{$\dn$} & \bl{$\varnothing$} & \\

  \bl{$der\, c\, (\epsilon)$}           & \bl{$\dn$} & \bl{$\varnothing$} & \\

  \bl{$der\, c\, (d)$}                     & \bl{$\dn$} & \bl{if $c = d$ then $\epsilon$ else $\varnothing$} & \\

  \bl{$der\, c\, (r_1 + r_2)$}        & \bl{$\dn$} & \bl{$der\, c\, r_1 + der\, c\, r_2$} & \\

  \bl{$der\, c\, (r_1 \cdot r_2)$}  & \bl{$\dn$}  & \bl{if $nullable (r_1)$}\\

  & & \bl{then $(der\,c\,r_1) \cdot r_2 + der\, c\, r_2$}\\ 

  & & \bl{else $(der\, c\, r_1) \cdot r_2$}\\

  \bl{$der\, c\, (r^*)$}          & \bl{$\dn$} & \bl{$(der\,c\,r) \cdot (r^*)$} &\smallskip\\\pause

  \bl{$der\!s\, [] r$}     & \bl{$\dn$} & \bl{$r$} & \\

  \bl{$der\!s\, (c\!::\!s) r$} & \bl{$\dn$} & \bl{$der\!s\,s\,(der\,c\,r)$} & \\

\only<3->{

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\texttt{\lstinputlisting{../progs/app6.scala}}}}

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\end{textblock}}

\texttt{\lstinputlisting{../progs/app7.scala}}}