# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1475413662 -3600
# Node ID c08290ee4f1f24197afa10556ecc49fa6b6cb331
# Parent  55be90b2a642d708f286f0edfc9a51864aee9e7e
updated

diff -r 55be90b2a642 -r c08290ee4f1f data.sty
--- a/data.sty	Sat Oct 01 23:34:37 2016 +0100
+++ b/data.sty	Sun Oct 02 14:07:42 2016 +0100
@@ -96,6 +96,7 @@
 21 21.46013
 \end{filecontents}
 
+
 %% Scala V1, example (a*)* b
 \begin{filecontents}{re1a.data}
 1 0.00013
diff -r 55be90b2a642 -r c08290ee4f1f handouts/ho02.pdf
Binary file handouts/ho02.pdf has changed
diff -r 55be90b2a642 -r c08290ee4f1f handouts/ho02.tex
--- a/handouts/ho02.tex	Sat Oct 01 23:34:37 2016 +0100
+++ b/handouts/ho02.tex	Sun Oct 02 14:07:42 2016 +0100
@@ -64,7 +64,7 @@
     axis lines=left,
     width=6.5cm,
     height=5cm]
-\addplot[green,mark=square*,mark options={fill=white}] table {re2.data};
+\addplot[green,mark=square*,mark options={fill=white}] table {re2b.data};
 \addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
 \end{axis}
 \end{tikzpicture}
@@ -387,9 +387,9 @@
 Input: $r_1$, $abc$\medskip\\
 Step 1: & build derivative of $a$ and $r_1$ & $(r_2 = \textit{der}\,a\,r_1)$\smallskip\\
 Step 2: & build derivative of $b$ and $r_2$ & $(r_3 = \textit{der}\,b\,r_2)$\smallskip\\
-Step 3: & build derivative of $c$ and $r_3$ & $(r_4 = \textit{der}\,b\,r_3)$\smallskip\\
-Step 4: & the string is exhausted; test & ($\textit{nullable}(r_4)$)\\
-        & whether $r_4$ can recognise the\\
+Step 3: & build derivative of $c$ and $r_3$ & $(r_4 = \textit{der}\,c\,r_3)$\smallskip\\
+Step 4: & the string is exhausted: & $(\textit{nullable}(r_4))$\\
+        & test whether $r_4$ can recognise the\\
         & empty string\smallskip\\
 Output: & result of this test $\Rightarrow \textit{true} \,\text{or}\, \textit{false}$\\        
 \end{tabular}
diff -r 55be90b2a642 -r c08290ee4f1f progs/re1.scala
--- a/progs/re1.scala	Sat Oct 01 23:34:37 2016 +0100
+++ b/progs/re1.scala	Sun Oct 02 14:07:42 2016 +0100
@@ -68,6 +68,10 @@
   (end - start)/(i * 1.0e9)
 }
 
+for (i <- 1 to 20) {
+  println(i + ": " + "%.5f".format(time_needed(2, matches(EVIL1(i), "a" * i))))
+}
+
 
 for (i <- 1 to 20) {
   println(i + ": " + "%.5f".format(time_needed(2, matches(EVIL1(i), "a" * i))))
diff -r 55be90b2a642 -r c08290ee4f1f slides.sty
--- a/slides.sty	Sat Oct 01 23:34:37 2016 +0100
+++ b/slides.sty	Sun Oct 02 14:07:42 2016 +0100
@@ -18,6 +18,10 @@
 
 \newcommand{\ZERO}{\mbox{\bf 0}}
 \newcommand{\ONE}{\mbox{\bf 1}}
+\newcommand{\Der}{\textit{Der}}
+\newcommand{\der}{\textit{der}}
+\newcommand{\Ders}{\textit{Ders}}
+\newcommand{\ders}{\textit{ders}}
 \newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}%
 \newcommand{\slidecaption}{}
 
diff -r 55be90b2a642 -r c08290ee4f1f slides/compiled.data
--- a/slides/compiled.data	Sat Oct 01 23:34:37 2016 +0100
+++ b/slides/compiled.data	Sun Oct 02 14:07:42 2016 +0100
@@ -1,6 +1,6 @@
 %% LaTeX2e file `compiled.data'
 %% generated by the `filecontents' environment
-%% from source `annonce' on 2016/02/23.
+%% from source `slides02' on 2016/10/02.
 %%
 %1 0.234146
 %5000 0.227539
diff -r 55be90b2a642 -r c08290ee4f1f slides/compiled2.data
--- a/slides/compiled2.data	Sat Oct 01 23:34:37 2016 +0100
+++ b/slides/compiled2.data	Sun Oct 02 14:07:42 2016 +0100
@@ -1,6 +1,6 @@
 %% LaTeX2e file `compiled2.data'
 %% generated by the `filecontents' environment
-%% from source `annonce' on 2016/02/23.
+%% from source `slides02' on 2016/10/02.
 %%
 0 0
 200 0.222058
diff -r 55be90b2a642 -r c08290ee4f1f slides/interpreted.data
--- a/slides/interpreted.data	Sat Oct 01 23:34:37 2016 +0100
+++ b/slides/interpreted.data	Sun Oct 02 14:07:42 2016 +0100
@@ -1,6 +1,6 @@
 %% LaTeX2e file `interpreted.data'
 %% generated by the `filecontents' environment
-%% from source `annonce' on 2016/02/23.
+%% from source `slides02' on 2016/10/02.
 %%
 200 1.005863
 400 7.8296765
diff -r 55be90b2a642 -r c08290ee4f1f slides/interpreted2.data
--- a/slides/interpreted2.data	Sat Oct 01 23:34:37 2016 +0100
+++ b/slides/interpreted2.data	Sun Oct 02 14:07:42 2016 +0100
@@ -1,6 +1,6 @@
 %% LaTeX2e file `interpreted2.data'
 %% generated by the `filecontents' environment
-%% from source `annonce' on 2016/02/23.
+%% from source `slides02' on 2016/10/02.
 %%
 0 0
 200 1.005863
diff -r 55be90b2a642 -r c08290ee4f1f slides/slides02.pdf
Binary file slides/slides02.pdf has changed
diff -r 55be90b2a642 -r c08290ee4f1f slides/slides02.tex
--- a/slides/slides02.tex	Sat Oct 01 23:34:37 2016 +0100
+++ b/slides/slides02.tex	Sun Oct 02 14:07:42 2016 +0100
@@ -51,55 +51,58 @@
 
 \footnotesize
 \begin{center}
+  {\normalsize Graphs: \bl{$a^{?\{n\}} \cdot a^{\{n\}}$} and strings \bl{$\underbrace{\,a\ldots a\,}_{n}$}}\\
 \begin{tabular}{@{}cc@{}}
 \begin{tikzpicture}
 \begin{axis}[
-    xlabel={\pcode{a}s},
+    xlabel={$n$},
+    x label style={at={(1.05,0.0)}},
     ylabel={time in secs},
     enlargelimits=false,
     xtick={0,5,...,30},
-    xmax=30,
+    xmax=33,
     ymax=35,
     ytick={0,5,...,30},
     scaled ticks=false,
     axis lines=left,
-    width=4.5cm,
+    width=5cm,
     height=4.5cm, 
     legend entries={Python,Ruby},  
     legend pos=north west,
     legend cell align=left  
 ]
-\addplot[blue,mark=*, mark options={fill=white}] 
-  table {re-python.data};
-\addplot[brown,mark=pentagon*, mark options={fill=white}] 
-  table {re-ruby.data};  
+\addplot[blue,mark=*, mark options={fill=white}] table {re-python.data};
+\addplot[brown,mark=pentagon*, mark options={fill=white}] table {re-ruby.data};  
 \end{axis}
 \end{tikzpicture}
 &
 \begin{tikzpicture}
-\begin{axis}[
-    xlabel={\pcode{a}s},
+  \begin{axis}[
+    xlabel={$n$},
+    x label style={at={(1.1,0.05)}},
     ylabel={time in secs},
     enlargelimits=false,
-    xtick={0,3000,...,12000},
-    xmax=12000,
+    xtick={0,4000,...,12000},
+    xmax=13500,
     ymax=35,
     ytick={0,5,...,30},
     scaled ticks=false,
     axis lines=left,
-    width=5.5cm,
+    width=5cm,
     height=4.5cm
 ]
-\addplot[green,mark=square*,mark options={fill=white}] table {re2b.data};
+\addplot[green,mark=square*,mark options={fill=white}] table {re2.data};
 \addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
 \end{axis}
 \end{tikzpicture}
 \end{tabular}
 \end{center}
 
+\small
+In the handouts is a similar graph with \bl{$(a^*)^* \cdot b$} for Java.
 
 \end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -108,7 +111,7 @@
 
 \begin{itemize}
 
-\item A \alert{\bf language} is a set of strings, for example\medskip
+\item A \alert{\bf Language} is a set of strings, for example\medskip
 \begin{center}
 \bl{$\{[], hello, \textit{foobar}, a, abc\}$}
 \end{center}\bigskip
@@ -228,23 +231,23 @@
 \bigskip
 
 \begin{center}
-\bl{$Der\,c\,A \dn \{ s \;|\;  c\!::\!s \in A\}$ } 
+\bl{$\Der\,c\,A \dn \{ s \;|\;  c\!::\!s \in A\}$ } 
 \end{center}\bigskip\bigskip\bigskip
 
 For \bl{$A = \{\textit{foo}, \textit{bar}, \textit{frak}\}$} then
 
 \begin{center}
 \bl{\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l}
-$Der\,f\,A$ & $=$ & $\{\textit{oo}, \textit{rak}\}$\\
-$Der\,b\,A$ & $=$ &  $\{\textit{ar}\}$\\  
-$Der\,a\,A$ & $=$ & $\{\}$\\\pause
+$\Der\,f\,A$ & $=$ & $\{\textit{oo}, \textit{rak}\}$\\
+$\Der\,b\,A$ & $=$ &  $\{\textit{ar}\}$\\  
+$\Der\,a\,A$ & $=$ & $\{\}$\\\pause
 \end{tabular}}
 \end{center}
 
 \small
 We can extend this definition to strings
 \[
-\bl{Ders\,s\,A = \{s'\;|\;s\,@\,s' \in A\}}
+\bl{\Ders\,s\,A = \{s'\;|\;s\,@\,s' \in A\}}
 \]
 
 \end{itemize}
@@ -404,22 +407,24 @@
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[c]
-\frametitle{\bl{$(a^{?\{n\}}) \cdot a^{\{n\}}$}}
+\frametitle{\bl{$(a^{?\{n\}}) \cdot a^{\{n\}}$} and \bl{$(a^*)^* \cdot b$}}
 
-\begin{center}
+\begin{center}\footnotesize
+\begin{tabular}{@{}c@{\hspace{-2mm}}c@{}}
 \begin{tikzpicture}
 \begin{axis}[
-    xlabel={\pcode{a}s},
+    xlabel={$n$},
+    x label style={at={(1.05,0.0)}},
     ylabel={time in secs},
     enlargelimits=false,
     xtick={0,5,...,30},
-    xmax=30,
+    xmax=33,
     ymax=35,
     ytick={0,5,...,30},
     scaled ticks=false,
     axis lines=left,
-    width=9cm,
-    height=7cm, 
+    width=5.5cm,
+    height=4.5cm, 
     legend entries={Python,Ruby},  
     legend pos=north west,
     legend cell align=left  
@@ -430,6 +435,28 @@
   table {re-ruby.data};  
 \end{axis}
 \end{tikzpicture}
+&
+\begin{tikzpicture}
+\begin{axis}[
+    xlabel={$n$},
+    x label style={at={(1.05,0.0)}},
+    ylabel={time in secs},
+    enlargelimits=false,
+    xtick={0,5,...,30},
+    xmax=33,
+    ymax=35,
+    ytick={0,5,...,30},
+    scaled ticks=false,
+    axis lines=left,
+    width=5.5cm,
+    height=4.5cm, 
+    legend entries={Java},  
+    legend pos=north west,
+    legend cell align=left]
+\addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data};
+\end{axis}
+\end{tikzpicture}
+\end{tabular}
 \end{center}
 
 \end{frame}
@@ -444,10 +471,10 @@
 \item Evil regular expressions\medskip
 \begin{itemize}
 \item \bl{$(a^{?\{n\}}) \cdot a^{\{n\}}$}
-\item \bl{$(a^+)^+$}
+\item \bl{$(a^*)^*$}
 \item \bl{$([a$\,-\,$z]^+)^*$}
-\item \bl{$(a + a \cdot a)^+$}
-\item \bl{$(a + a?)^+$}
+\item \bl{$(a + a \cdot a)^*$}
+\item \bl{$(a + a?)^*$}
 \end{itemize}
 \end{itemize}
 
@@ -484,7 +511,7 @@
 expression that matches just \bl{$s$}?\bigskip\bigskip\bigskip\bigskip
 
 \small
-\bl{$der\,c\,r$} gives the answer, Brzozowski 1964
+\bl{$\der\,c\,r$} gives the answer, Brzozowski 1964
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
@@ -494,17 +521,17 @@
 
 \begin{center}
 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
-  \bl{$der\, c\, (\ZERO)$}      & \bl{$\dn$} & \bl{$\ZERO$} & \\
-  \bl{$der\, c\, (\ONE)$}           & \bl{$\dn$} & \bl{$\ZERO$} & \\
-  \bl{$der\, c\, (d)$}                     & \bl{$\dn$} & \bl{if $c = d$ then $\ONE$ else $\ZERO$} & \\
-  \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{$(der\,c\,r) \cdot (r^*)$} &\bigskip\\\pause
+  \bl{$\der\, c\, (\ZERO)$}      & \bl{$\dn$} & \bl{$\ZERO$} & \\
+  \bl{$\der\, c\, (\ONE)$}           & \bl{$\dn$} & \bl{$\ZERO$} & \\
+  \bl{$\der\, c\, (d)$}                     & \bl{$\dn$} & \bl{if $c = d$ then $\ONE$ else $\ZERO$} & \\
+  \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{$(\der\,c\,r) \cdot (r^*)$} &\bigskip\\\pause
 
-  \bl{$\textit{ders}\, []\, r$}     & \bl{$\dn$} & \bl{$r$} & \\
-  \bl{$\textit{ders}\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$\textit{ders}\,s\,(der\,c\,r)$} & \\
+  \bl{$\ders\, []\, r$}     & \bl{$\dn$} & \bl{$r$} & \\
+  \bl{$\ders\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$\ders\,s\,(\der\,c\,r)$} & \\
   \end{tabular}
 \end{center}
 
@@ -519,9 +546,9 @@
 
 \begin{center}
 \begin{tabular}{l}
-\bl{$der\,a\,r =\;?$}\\
-\bl{$der\,b\,r =\;?$}\\
-\bl{$der\,c\,r =\;?$}
+\bl{$\der\,a\,r =\;?$}\\
+\bl{$\der\,b\,r =\;?$}\\
+\bl{$\der\,c\,r =\;?$}
 \end{tabular}
 \end{center}
 
@@ -533,8 +560,8 @@
 \frametitle{The Algorithm}
 
 \begin{center}
-\begin{tabular}{l}
-\bl{$\textit{matches}\,r\,s \dn \textit{nullable}(\textit{ders}\,r\,s)$}  
+\begin{tabular}{l} 
+\bl{$\textit{matches}\,r\,s \dn \textit{nullable}(\ders\,r\,s)$}  
 \end{tabular}
 \end{center}
 
@@ -544,16 +571,16 @@
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[t]
-\frametitle{The Algorithm}
+\frametitle{An Example}
 
+Does \bl{$r_1$} match \bl{$abc$}?
 \begin{center}
-\begin{tabular}{@{}rll@{}}
-Input: & \bl{$r_1$}, \bl{$abc$}\medskip\\
-Step 1: & build derivative of \bl{$a$} and \bl{$r_1$} & \bl{$(r_2 = der\,a\,r_1)$}\smallskip\\
-Step 2: & build derivative of \bl{$b$} and \bl{$r_2$} & \bl{$(r_3 = der\,b\,r_2)$}\smallskip\\
-Step 3: & build derivative of \bl{$c$} and \bl{$r_3$} & \bl{$(r_4 = der\,b\,r_3)$}\smallskip\\
-Step 4: & the string is exhausted; test & (\bl{$nullable(r_4)$})\\
-        & whether \bl{$r_4$} can recognise\\
+\begin{tabular}{@{}rl@{}l@{}}
+Step 1: & build derivative of \bl{$a$} and \bl{$r_1$} & \bl{$(r_2 = \der\,a\,r_1)$}\smallskip\\
+Step 2: & build derivative of \bl{$b$} and \bl{$r_2$} & \bl{$(r_3 = \der\,b\,r_2)$}\smallskip\\
+Step 3: & build derivative of \bl{$c$} and \bl{$r_3$} & \bl{$(r_4 = \der\,c\,r_3)$}\smallskip\\
+Step 4: & the string is exhausted: & \bl{($nullable(r_4))$}\\
+        & test whether \bl{$r_4$} can recognise\\
         & the empty string\smallskip\\
 Output: & result of the test\\
         & $\Rightarrow \bl{\textit{true}} \,\text{or}\, 
@@ -573,11 +600,11 @@
 expression \bl{$r_1$} then\medskip
 
 \begin{enumerate}
-\item \bl{$Der\,a\,(L(r_1))$}\pause
-\item \bl{$Der\,b\,(Der\,a\,(L(r_1)))$}\pause
-\item \bl{$Der\,c\,(Der\,b\,(Der\,a\,(L(r_1))))$}\bigskip
+\item \bl{$\Der\,a\,(L(r_1))$}\pause
+\item \bl{$\Der\,b\,(\Der\,a\,(L(r_1)))$}\pause
+\item \bl{$\Der\,c\,(\Der\,b\,(\Der\,a\,(L(r_1))))$}\bigskip
 \item finally we test whether the empty string is in this 
-set; same for  \bl{$Ders\,abc\,(L(r_1))$}.\medskip
+set; same for  \bl{$\Ders\,abc\,(L(r_1))$}.\medskip
 \end{enumerate}
 
 The matching algorithm works similarly, just over regular expressions instead of sets.
@@ -587,16 +614,17 @@
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[c]
-\frametitle{\bl{$(a^{?\{n\}}) \cdot a^{\{n\}}$}}
+\frametitle{Oops\ldots\bl{$(a^{?\{n\}}) \cdot a^{\{n\}}$}}
 
 \begin{center}
 \begin{tikzpicture}
 \begin{axis}[
-    xlabel={\pcode{a}s},
+    xlabel={$n$},
+    x label style={at={(1.05,0.0)}},
     ylabel={time in secs},
     enlargelimits=false,
     xtick={0,5,...,30},
-    xmax=30,
+    xmax=31,
     ytick={0,5,...,30},
     scaled ticks=false,
     axis lines=left,
@@ -639,7 +667,7 @@
 \end{center}
 
 This problem is aggravated with \bl{$a^?$} being represented 
-as \bl{$\ONE + a$}.
+as \bl{$a + \ONE$}.
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
@@ -657,7 +685,8 @@
 \end{tabular}
 \end{center}
 
-What is their meaning? What are the cases for \bl{$nullable$} and \bl{$der$}?
+What is their meaning?\\
+What are the cases for \bl{$nullable$} and \bl{$\der$}?
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
@@ -669,11 +698,12 @@
 \begin{center}
 \begin{tikzpicture}
 \begin{axis}[
-    xlabel={\pcode{a}s},
+    xlabel={$n$},
+    x label style={at={(1.05,0.0)}},   
     ylabel={time in secs},
     enlargelimits=false,
     xtick={0,200,...,1000},
-    xmax=1000,
+    xmax=1100,
     ytick={0,5,...,30},
     scaled ticks=false,
     axis lines=left,
@@ -707,9 +737,9 @@
 
 \begin{center}
 \begin{tabular}{l}
-\bl{$der\,a\,r = ((\ONE \cdot b) + \ZERO) \cdot r$}\\
-\bl{$der\,b\,r = ((\ZERO \cdot b) + \ONE)\cdot r$}\\
-\bl{$der\,c\,r = ((\ZERO \cdot b) + \ZERO)\cdot r$}
+\bl{$\der\,a\,r = ((\ONE \cdot b) + \ZERO) \cdot r$}\\
+\bl{$\der\,b\,r = ((\ZERO \cdot b) + \ONE)\cdot r$}\\
+\bl{$\der\,c\,r = ((\ZERO \cdot b) + \ZERO)\cdot r$}
 \end{tabular}
 \end{center}
 
@@ -719,7 +749,6 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[t]
 \frametitle{\bl{$(a^{?\{n\}}) \cdot a^{\{n\}}$}}
@@ -727,17 +756,20 @@
 \begin{center}
 \begin{tikzpicture}
 \begin{axis}[
-    xlabel={\pcode{a}s},
+    xlabel={$n$},
+    x label style={at={(1.05,0.0)}},     
     ylabel={time in secs},
     enlargelimits=false,
     xtick={0,3000,...,12000},
-    xmax=12000,
+    xmax=14000,
     ymax=35,
     ytick={0,5,...,30},
     scaled ticks=false,
     axis lines=left,
     width=9cm,
-    height=7cm
+    height=7cm,
+    legend entries={Scala V2,Scala V3},  
+    legend pos=north east
 ]
 \addplot[green,mark=square*,mark options={fill=white}] table {re2b.data};
 \addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
@@ -749,6 +781,40 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{\bl{$(a^*)^* \cdot b$}}
+
+\begin{center}
+\begin{tikzpicture}
+  \begin{axis}[
+    xlabel={$n$},
+    x label style={at={(1.09,0.0)}},
+    ylabel={time in secs},
+    enlargelimits=false,
+    xmax=5000,
+    xtick={0,2000,...,6000},
+    ytick={0,5,...,10},
+    ymax=15,
+    scaled ticks=false,
+    axis lines=left,
+    width=9cm,
+    height=5cm, 
+    legend entries={Scala V3},
+    legend pos=north west,
+    legend cell align=left]
+\addplot[green,mark=square*,mark options={fill=white}] table {re2a.data};
+%%where is 2nd graph
+%%\addplot[black,mark=square*,mark options={fill=white}] table {re3.data};
+\end{axis}
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[t]
 \frametitle{What is good about this Alg.}
 
@@ -857,17 +923,17 @@
 \begin{center}
 \begin{tabular}{rp{4cm}}
   & \bl{$s \in L(r)$}\medskip\\
-\bl{$\Leftrightarrow$} & \bl{$[] \in Ders\,s\,(L(r))$}\pause  
+\bl{$\Leftrightarrow$} & \bl{$[] \in \Ders\,s\,(L(r))$}\pause  
 \end{tabular}
 \end{center}
 
-\item if we can show \bl{$Ders\, s\,(L(r)) = L(ders\,s\,r)$} we 
+\item if we can show \bl{$\Ders\, s\,(L(r)) = L(\ders\,s\,r)$} we 
 have
 
 \begin{center}
 \begin{tabular}{rp{4cm}}
-\bl{$\Leftrightarrow$} & \bl{$[] \in L(ders\,s\,r)$}\medskip\\
-\bl{$\Leftrightarrow$} & \bl{$nullable(ders\,s\,r)$}\medskip\\
+\bl{$\Leftrightarrow$} & \bl{$[] \in L(\ders\,s\,r)$}\medskip\\
+\bl{$\Leftrightarrow$} & \bl{$nullable(\ders\,s\,r)$}\medskip\\
 \bl{$\dn$} & \bl{$matches\,s\,r$}
 \end{tabular}
 \end{center}
@@ -884,13 +950,13 @@
 Let \bl{$Der\,c\,A$} be the set defined as
 
 \begin{center}
-\bl{$Der\,c\,A \dn \{ s \;|\;  c\!::\!s \in A\}$ } 
+\bl{$\Der\,c\,A \dn \{ s \;|\;  c\!::\!s \in A\}$ } 
 \end{center}
 
 We can prove
 
 \begin{center}
-\bl{$L(der\,c\,r) = Der\,c\,(L(r))$}
+\bl{$L(\der\,c\,r) = \Der\,c\,(L(r))$}
 \end{center}
 
 by induction on \bl{$r$}.
@@ -919,7 +985,7 @@
 We can then prove
 
 \begin{center}
-\bl{$Ders\,s\,(L(r)) = L(ders\,s\,r)$}  
+\bl{$\Ders\,s\,(L(r)) = L(\ders\,s\,r)$}  
 \end{center}
 
 
diff -r 55be90b2a642 -r c08290ee4f1f style.sty
--- a/style.sty	Sat Oct 01 23:34:37 2016 +0100
+++ b/style.sty	Sun Oct 02 14:07:42 2016 +0100
@@ -13,7 +13,10 @@
 \newcommand{\ZERO}{\mbox{\bf 0}}
 \newcommand{\ONE}{\mbox{\bf 1}}
 \newcommand{\Left}{\textit{Left}}
-
+\newcommand{\Der}{\textit{Der}}
+\newcommand{\der}{\textit{der}}
+\newcommand{\Ders}{\textit{Ders}}
+\newcommand{\ders}{\textit{ders}}
 
 %%% for trees
 %% http://anorien.csc.warwick.ac.uk/mirrors/CTAN/graphics/pgf/contrib/forest/forest.pdf