--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/Slides8.thy Sun May 22 10:20:18 2011 +0200
@@ -0,0 +1,1530 @@
+(*<*)
+theory Slides8
+imports "~~/src/HOL/Library/LaTeXsugar" "Main"
+begin
+
+declare [[show_question_marks = false]]
+
+notation (latex output)
+ set ("_") and
+ Cons ("_::/_" [66,65] 65)
+
+(*>*)
+
+text_raw {*
+ \renewcommand{\slidecaption}{Copenhagen, 23rd~May 2011}
+
+ \newcommand{\abst}[2]{#1.#2}% atom-abstraction
+ \newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing
+ \newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions
+ \newcommand{\unit}{\langle\rangle}% unit
+ \newcommand{\app}[2]{#1\,#2}% application
+ \newcommand{\eqprob}{\mathrel{{\approx}?}}
+ \newcommand{\freshprob}{\mathrel{\#?}}
+ \newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction
+ \newcommand{\id}{\varepsilon}% identity substitution
+
+ \newcommand{\bl}[1]{\textcolor{blue}{#1}}
+ \newcommand{\gr}[1]{\textcolor{gray}{#1}}
+ \newcommand{\rd}[1]{\textcolor{red}{#1}}
+
+ \newcommand{\ok}{\includegraphics[scale=0.07]{ok.png}}
+ \newcommand{\notok}{\includegraphics[scale=0.07]{notok.png}}
+ \newcommand{\largenotok}{\includegraphics[scale=1]{notok.png}}
+
+ \renewcommand{\Huge}{\fontsize{61.92}{77}\selectfont}
+ \newcommand{\veryHuge}{\fontsize{74.3}{93}\selectfont}
+ \newcommand{\VeryHuge}{\fontsize{89.16}{112}\selectfont}
+ \newcommand{\VERYHuge}{\fontsize{107}{134}\selectfont}
+
+ \newcommand{\LL}{$\mathbb{L}\,$}
+
+
+ \pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}%
+ {rgb(0mm)=(0,0,0.9);
+ rgb(0.9mm)=(0,0,0.7);
+ rgb(1.3mm)=(0,0,0.5);
+ rgb(1.4mm)=(1,1,1)}
+
+ \def\myitemi{\begin{pgfpicture}{-1ex}{-0.55ex}{1ex}{1ex}
+ \usebeamercolor[fg]{subitem projected}
+ {\pgftransformscale{0.8}\pgftext{\normalsize\pgfuseshading{bigsphere}}}
+ \pgftext{%
+ \usebeamerfont*{subitem projected}}
+ \end{pgfpicture}}
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[t]
+ \frametitle{%
+ \begin{tabular}{@ {\hspace{-3mm}}c@ {}}
+ \\
+ \LARGE Verifying a Regular Expression\\[-1mm]
+ \LARGE Matcher and Formal Language\\[-1mm]
+ \LARGE Theory\\[5mm]
+ \end{tabular}}
+ \begin{center}
+ Christian Urban\\
+ \small Technical University of Munich, Germany
+ \end{center}
+
+
+ \begin{center}
+ \small joint work with Chunhan Wu and Xingyuan Zhang from the PLA
+ University of Science and Technology in Nanjing
+ \end{center}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{This Talk: 4 Points}
+ \large
+ \begin{itemize}
+ \item It is easy to make mistakes.\medskip
+ \item Theorem provers can prevent mistakes, {\bf if} the problem
+ is formulated so that it is suitable for theorem provers.\medskip
+ \item This re-formulation can be done, even in domains where
+ we least expect it.\medskip
+ \item Where theorem provers are superior to the {\color{gray}{(best)}} human reasoners. ;o)
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{}
+
+ \begin{tabular}{c@ {\hspace{2mm}}c}
+ \\[6mm]
+ \begin{tabular}{c}
+ \includegraphics[scale=0.12]{harper.jpg}\\[-2mm]
+ {\footnotesize Bob Harper}\\[-2.5mm]
+ {\footnotesize (CMU)}
+ \end{tabular}
+ \begin{tabular}{c}
+ \includegraphics[scale=0.36]{pfenning.jpg}\\[-2mm]
+ {\footnotesize Frank Pfenning}\\[-2.5mm]
+ {\footnotesize (CMU)}
+ \end{tabular} &
+
+ \begin{tabular}{p{6cm}}
+ \raggedright
+ \color{gray}{published a proof in\\ {\bf ACM Transactions on Computational Logic} (2005),
+ $\sim$31pp}
+ \end{tabular}\\
+
+ \pause
+ \\[0mm]
+
+ \begin{tabular}{c}
+ \includegraphics[scale=0.36]{appel.jpg}\\[-2mm]
+ {\footnotesize Andrew Appel}\\[-2.5mm]
+ {\footnotesize (Princeton)}
+ \end{tabular} &
+
+ \begin{tabular}{p{6cm}}
+ \raggedright
+ \color{gray}{relied on their proof in a\\ {\bf security} critical application}
+ \end{tabular}
+ \end{tabular}
+
+
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}
+ \frametitle{Proof-Carrying Code}
+
+ \begin{textblock}{10}(2.5,2.2)
+ \begin{block}{Idea:}
+ \begin{center}
+ \begin{tikzpicture}
+ \draw[help lines,cream] (0,0.2) grid (8,4);
+
+ \draw[line width=1mm, red] (5.5,0.6) rectangle (7.5,4);
+ \node[anchor=base] at (6.5,2.8)
+ {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering user: untrusted code\end{tabular}};
+
+ \draw[line width=1mm, red] (0.5,0.6) rectangle (2.5,4);
+ \node[anchor=base] at (1.5,2.3)
+ {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering developer ---\\ web server\end{tabular}};
+
+ \onslide<3->{
+ \draw[line width=1mm, red, fill=red] (5.5,0.6) rectangle (7.5,1.8);
+ \node[anchor=base,white] at (6.5,1.1)
+ {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\bf\centering proof- checker\end{tabular}};}
+
+ \node at (3.8,3.0) [single arrow, fill=red,text=white, minimum height=3cm]{\bf code};
+ \onslide<2->{
+ \node at (3.8,1.3) [single arrow, fill=red,text=white, minimum height=3cm]{\bf certificate};
+ \node at (3.8,1.9) {\small\color{gray}{\mbox{}\hspace{-1mm}a proof in LF}};
+ }
+
+
+ \end{tikzpicture}
+ \end{center}
+ \end{block}
+ \end{textblock}
+
+ %\begin{textblock}{15}(2,12)
+ %\small
+ %\begin{itemize}
+ %\item<4-> Appel's checker is $\sim$2700 lines of code (1865 loc of\\ LF definitions;
+ %803 loc in C including 2 library functions)\\[-3mm]
+ %\item<5-> 167 loc in C implement a type-checker
+ %\end{itemize}
+ %\end{textblock}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text {*
+ \tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex]
+ \tikzstyle{node1}=[rectangle, minimum size=10mm, rounded corners=3mm, very thick,
+ draw=black!50, top color=white, bottom color=black!20]
+ \tikzstyle{node2}=[rectangle, minimum size=12mm, rounded corners=3mm, very thick,
+ draw=red!70, top color=white, bottom color=red!50!black!20]
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<2->[squeeze]
+ \frametitle{}
+
+ \begin{columns}
+
+ \begin{column}{0.8\textwidth}
+ \begin{textblock}{0}(1,2)
+
+ \begin{tikzpicture}
+ \matrix[ampersand replacement=\&,column sep=7mm, row sep=5mm]
+ { \&[-10mm]
+ \node (def1) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}}; \&
+ \node (proof1) [node1] {\large Proof}; \&
+ \node (alg1) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}}; \\
+
+ \onslide<4->{\node {\begin{tabular}{c}\small 1st\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
+ \onslide<4->{\node (def2) [node2] {\large Spec$^\text{+ex}$};} \&
+ \onslide<4->{\node (proof2) [node1] {\large Proof};} \&
+ \onslide<4->{\node (alg2) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
+
+ \onslide<5->{\node {\begin{tabular}{c}\small 2nd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
+ \onslide<5->{\node (def3) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
+ \onslide<5->{\node (proof3) [node1] {\large Proof};} \&
+ \onslide<5->{\node (alg3) [node2] {\large Alg$^\text{-ex}$};} \\
+
+ \onslide<6->{\node {\begin{tabular}{c}\small 3rd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
+ \onslide<6->{\node (def4) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
+ \onslide<6->{\node (proof4) [node2] {\large\hspace{1mm}Proof\hspace{1mm}};} \&
+ \onslide<6->{\node (alg4) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
+ };
+
+ \draw[->,black!50,line width=2mm] (proof1) -- (def1);
+ \draw[->,black!50,line width=2mm] (proof1) -- (alg1);
+
+ \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (def2);}
+ \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (alg2);}
+
+ \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (def3);}
+ \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (alg3);}
+
+ \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (def4);}
+ \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (alg4);}
+
+ \onslide<3->{\draw[white,line width=1mm] (1.1,3.2) -- (0.9,2.85) -- (1.1,2.35) -- (0.9,2.0);}
+ \end{tikzpicture}
+
+ \end{textblock}
+ \end{column}
+ \end{columns}
+
+
+ \begin{textblock}{3}(12,3.6)
+ \onslide<4->{
+ \begin{tikzpicture}
+ \node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h};
+ \end{tikzpicture}}
+ \end{textblock}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+
+(*<*)
+atom_decl name
+
+nominal_datatype lam =
+ Var "name"
+ | App "lam" "lam"
+ | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
+
+nominal_primrec
+ subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]")
+where
+ "(Var x)[y::=s] = (if x=y then s else (Var x))"
+| "(App t\<^isub>1 t\<^isub>2)[y::=s] = App (t\<^isub>1[y::=s]) (t\<^isub>2[y::=s])"
+| "x\<sharp>(y,s) \<Longrightarrow> (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])"
+apply(finite_guess)+
+apply(rule TrueI)+
+apply(simp add: abs_fresh)
+apply(fresh_guess)+
+done
+
+lemma subst_eqvt[eqvt]:
+ fixes pi::"name prm"
+ shows "pi\<bullet>(t1[x::=t2]) = (pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)]"
+by (nominal_induct t1 avoiding: x t2 rule: lam.strong_induct)
+ (auto simp add: perm_bij fresh_atm fresh_bij)
+
+lemma fresh_fact:
+ fixes z::"name"
+ shows "\<lbrakk>z\<sharp>s; (z=y \<or> z\<sharp>t)\<rbrakk> \<Longrightarrow> z\<sharp>t[y::=s]"
+by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
+ (auto simp add: abs_fresh fresh_prod fresh_atm)
+
+lemma forget:
+ assumes asm: "x\<sharp>L"
+ shows "L[x::=P] = L"
+ using asm
+by (nominal_induct L avoiding: x P rule: lam.strong_induct)
+ (auto simp add: abs_fresh fresh_atm)
+(*>*)
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}
+
+ \begin{textblock}{16}(1,1)
+ \renewcommand{\isasymbullet}{$\cdot$}
+ \tiny\color{black}
+*}
+lemma substitution_lemma_not_to_be_tried_at_home:
+ assumes asm: "x\<noteq>y" "x\<sharp>L"
+ shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
+using asm
+proof (induct M arbitrary: x y N L rule: lam.induct)
+ case (Lam z M1)
+ have ih: "\<And>x y N L. \<lbrakk>x\<noteq>y; x\<sharp>L\<rbrakk> \<Longrightarrow> M1[x::=N][y::=L] = M1[y::=L][x::=N[y::=L]]" by fact
+ have "x\<noteq>y" by fact
+ have "x\<sharp>L" by fact
+ obtain z'::"name" where fc: "z'\<sharp>(x,y,z,M1,N,L)" by (rule exists_fresh) (auto simp add: fs_name1)
+ have eq: "Lam [z'].([(z',z)]\<bullet>M1) = Lam [z].M1" using fc
+ by (auto simp add: lam.inject alpha fresh_prod fresh_atm)
+ have fc': "z'\<sharp>N[y::=L]" using fc by (simp add: fresh_fact fresh_prod)
+ have "([(z',z)]\<bullet>x) \<noteq> ([(z',z)]\<bullet>y)" using `x\<noteq>y` by (auto simp add: calc_atm)
+ moreover
+ have "([(z',z)]\<bullet>x)\<sharp>([(z',z)]\<bullet>L)" using `x\<sharp>L` by (simp add: fresh_bij)
+ ultimately
+ have "M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]
+ = M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]]"
+ using ih by simp
+ then have "[(z',z)]\<bullet>(M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]
+ = M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]])"
+ by (simp add: perm_bool)
+ then have ih': "([(z',z)]\<bullet>M1)[x::=N][y::=L] = ([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]]"
+ by (simp add: eqvts perm_swap)
+ show "(Lam [z].M1)[x::=N][y::=L] = (Lam [z].M1)[y::=L][x::=N[y::=L]]" (is "?LHS=?RHS")
+ proof -
+ have "?LHS = (Lam [z'].([(z',z)]\<bullet>M1))[x::=N][y::=L]" using eq by simp
+ also have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[x::=N][y::=L])" using fc by (simp add: fresh_prod)
+ also from ih have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]])" sorry
+ also have "\<dots> = (Lam [z'].([(z',z)]\<bullet>M1))[y::=L][x::=N[y::=L]]" using fc fc' by (simp add: fresh_prod)
+ also have "\<dots> = ?RHS" using eq by simp
+ finally show "?LHS = ?RHS" .
+ qed
+qed (auto simp add: forget)
+text_raw {*
+ \end{textblock}
+ \mbox{}
+
+ \only<2->{
+ \begin{textblock}{11.5}(4,2.3)
+ \begin{minipage}{9.3cm}
+ \begin{block}{}\footnotesize
+*}
+lemma substitution_lemma\<iota>:
+ assumes asm: "x \<noteq> y" "x \<sharp> L"
+ shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
+ using asm
+by (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
+ (auto simp add: forget fresh_fact)
+text_raw {*
+ \end{block}
+ \end{minipage}
+ \end{textblock}}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{Regular Expressions}
+
+ \begin{textblock}{6}(2,4)
+ \begin{tabular}{@ {}rrl}
+ \bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\
+ & \bl{$\mid$} & \bl{[]}\\
+ & \bl{$\mid$} & \bl{c}\\
+ & \bl{$\mid$} & \bl{r$_1$ + r$_2$}\\
+ & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\
+ & \bl{$\mid$} & \bl{r$^*$}\\
+ \end{tabular}
+ \end{textblock}
+
+ \begin{textblock}{6}(8,3.5)
+ \includegraphics[scale=0.35]{Screen1.png}
+ \end{textblock}
+
+ \begin{textblock}{6}(10.2,2.8)
+ \footnotesize Isabelle:
+ \end{textblock}
+
+ \only<2>{
+ \begin{textblock}{9}(3.6,11.8)
+ \bl{matches r s $\;\Longrightarrow\;$ true $\vee$ false}\\[3.5mm]
+
+ \hspace{10mm}\begin{tikzpicture}
+ \coordinate (m1) at (0.4,1);
+ \draw (0,0.3) node (m2) {\small\color{gray}rexp};
+ \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
+
+ \coordinate (s1) at (0.81,1);
+ \draw (1.3,0.3) node (s2) {\small\color{gray} string};
+ \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
+ \end{tikzpicture}
+ \end{textblock}}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{Specification}
+
+ \small
+ \begin{textblock}{6}(0,3.5)
+ \begin{tabular}{r@ {\hspace{0.5mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l}
+ \multicolumn{4}{c}{rexp $\Rightarrow$ set of strings}\bigskip\\
+ &\bl{\LL ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\
+ &\bl{\LL ([])} & \bl{$\dn$} & \bl{\{[]\}}\\
+ &\bl{\LL (c)} & \bl{$\dn$} & \bl{\{c\}}\\
+ &\bl{\LL (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}\\
+ \rd{$\Rightarrow$} &\bl{\LL (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) ;; \LL (r$_2$)}\\
+ \rd{$\Rightarrow$} &\bl{\LL (r$^*$)} & \bl{$\dn$} & \bl{(\LL (r))$^\star$}\\
+ \end{tabular}
+ \end{textblock}
+
+ \begin{textblock}{9}(7.3,3)
+ {\mbox{}\hspace{2cm}\footnotesize Isabelle:\smallskip}
+ \includegraphics[scale=0.325]{Screen3.png}
+ \end{textblock}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{Version 1}
+ \small
+ \mbox{}\\[-8mm]\mbox{}
+
+ \begin{center}\def\arraystretch{1.05}
+ \begin{tabular}{@ {\hspace{-5mm}}l@ {\hspace{2.5mm}}c@ {\hspace{2.5mm}}l@ {}}
+ \bl{match [] []} & \bl{$=$} & \bl{true}\\
+ \bl{match [] (c::s)} & \bl{$=$} & \bl{false}\\
+ \bl{match ($\varnothing$::rs) s} & \bl{$=$} & \bl{false}\\
+ \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
+ \bl{match (c::rs) []} & \bl{$=$} & \bl{false}\\
+ \bl{match (c::rs) (d::s)} & \bl{$=$} & \bl{if c = d then match rs s else false}\\
+ \bl{match (r$_1$ + r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s $\vee$ match (r$_2$::rs) s}\\
+ \bl{match (r$_1$ $\cdot$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\
+ \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
+ \end{tabular}
+ \end{center}
+
+ \begin{textblock}{9}(0.2,1.6)
+ \hspace{10mm}\begin{tikzpicture}
+ \coordinate (m1) at (0.44,-0.5);
+ \draw (0,0.3) node (m2) {\small\color{gray}\mbox{}\hspace{-9mm}list of rexps};
+ \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
+
+ \coordinate (s1) at (0.86,-0.5);
+ \draw (1.5,0.3) node (s2) {\small\color{gray} string};
+ \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
+ \end{tikzpicture}
+ \end{textblock}
+
+ \begin{textblock}{9}(2.8,11.8)
+ \bl{matches$_1$ r s $\;=\;$ match [r] s}
+ \end{textblock}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[c]
+ \frametitle{Testing}
+
+ \small
+ Every good programmer should do thourough tests:
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-20mm}}lcl}
+ \bl{matches$_1$ (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\
+ \bl{matches$_1$ (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\
+ \bl{matches$_1$ (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\
+ \bl{matches$_1$ (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\
+ \bl{matches$_1$ (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\
+ \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}}\\
+ \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}}\\
+ \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x3} & \bl{$\mapsto$} & \bl{false}}
+ \end{tabular}
+ \end{center}
+
+ \onslide<3->
+ {Looks OK \ldots let's ship it to customers\hspace{5mm}
+ \raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[c]
+ \frametitle{Version 1}
+
+ \only<1->{Several hours later\ldots}\pause
+
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{0mm}}lcl}
+ \bl{matches$_1$ []$^*$ s} & \bl{$\mapsto$} & loops\\
+ \onslide<4->{\bl{matches$_1$ ([] + \ldots)$^*$ s} & \bl{$\mapsto$} & loops\\}
+ \end{tabular}
+ \end{center}
+
+ \small
+ \onslide<3->{
+ \begin{center}
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
+ \ldots\\
+ \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
+ \ldots\\
+ \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
+ \end{tabular}
+ \end{center}}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{Testing}
+
+ \begin{itemize}
+ \item While testing is an important part in the process of programming development\pause\ldots
+
+ \item we can only test a {\bf finite} amount of examples.\bigskip\pause
+
+ \begin{center}
+ \colorbox{cream}
+ {\gr{\begin{minipage}{10cm}
+ ``Testing can only show the presence of errors, never their
+ absence.'' (Edsger W.~Dijkstra)
+ \end{minipage}}}
+ \end{center}\bigskip\pause
+
+ \item In a theorem prover we can establish properties that apply to
+ {\bf all} input and {\bf all} output.
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{Version 2}
+ \mbox{}\\[-14mm]\mbox{}
+
+ \small
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
+ \bl{nullable ($\varnothing$)} & \bl{$=$} & \bl{false} &\\
+ \bl{nullable ([])} & \bl{$=$} & \bl{true} &\\
+ \bl{nullable (c)} & \bl{$=$} & \bl{false} &\\
+ \bl{nullable (r$_1$ + r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\vee$ nullable r$_2$} & \\
+ \bl{nullable (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\wedge$ nullable r$_2$} & \\
+ \bl{nullable (r$^*$)} & \bl{$=$} & \bl{true} & \\
+ \end{tabular}\medskip
+
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+ \bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\
+ \bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\
+ \bl{der c (d)} & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\
+ \bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\
+ \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\
+ & & \bl{\;\;\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\
+ \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\
+
+ \bl{derivative r []} & \bl{$=$} & \bl{r} & \\
+ \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
+ \end{tabular}\medskip
+
+ \bl{matches$_2$ r s $=$ nullable (derivative r s)}
+
+ \begin{textblock}{6}(9.5,0.9)
+ \begin{flushright}
+ \color{gray}``if r matches []''
+ \end{flushright}
+ \end{textblock}
+
+ \begin{textblock}{6}(9.5,6.18)
+ \begin{flushright}
+ \color{gray}``derivative w.r.t.~a char''
+ \end{flushright}
+ \end{textblock}
+
+ \begin{textblock}{6}(9.5,12.1)
+ \begin{flushright}
+ \color{gray}``deriv.~w.r.t.~a string''
+ \end{flushright}
+ \end{textblock}
+
+ \begin{textblock}{6}(9.5,13.98)
+ \begin{flushright}
+ \color{gray}``main''
+ \end{flushright}
+ \end{textblock}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{Is the Matcher Error-Free?}
+
+ We expect that
+
+ \begin{center}
+ \begin{tabular}{lcl}
+ \bl{matches$_2$ r s = true} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
+ \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\
+ \bl{matches$_2$ r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
+ \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\
+ \end{tabular}
+ \end{center}
+ \pause\pause\bigskip
+ By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip
+
+ \begin{tabular}{lrcl}
+ Lemmas: & \bl{nullable (r)} & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\
+ & \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\
+ \end{tabular}
+
+ \only<4->{
+ \begin{textblock}{3}(0.9,4.5)
+ \rd{\huge$\forall$\large{}r s.}
+ \end{textblock}}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[c]
+ \frametitle{
+ \begin{tabular}{c}
+ \mbox{}\\[23mm]
+ \LARGE Demo
+ \end{tabular}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+
+ \mbox{}\\[-2mm]
+
+ \small
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
+ \bl{nullable (NULL)} & \bl{$=$} & \bl{false} &\\
+ \bl{nullable (EMPTY)} & \bl{$=$} & \bl{true} &\\
+ \bl{nullable (CHR c)} & \bl{$=$} & \bl{false} &\\
+ \bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \\
+ \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\
+ \bl{nullable (STAR r)} & \bl{$=$} & \bl{true} & \\
+ \end{tabular}\medskip
+
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+ \bl{der c (NULL)} & \bl{$=$} & \bl{NULL} & \\
+ \bl{der c (EMPTY)} & \bl{$=$} & \bl{NULL} & \\
+ \bl{der c (CHR d)} & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \\
+ \bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \\
+ \bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \\
+ & & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\
+ \bl{der c (STAR r)} & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} &\smallskip\\
+
+ \bl{derivative r []} & \bl{$=$} & \bl{r} & \\
+ \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
+ \end{tabular}\medskip
+
+ \bl{matches r s $=$ nullable (derivative r s)}
+
+ \only<2>{
+ \begin{textblock}{8}(1.5,4)
+ \includegraphics[scale=0.3]{approved.png}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{No Automata?}
+
+ You might be wondering why I did not use any automata?
+
+ \begin{itemize}
+ \item {\bf Def.:} A \alert{regular language} is one where there is a DFA that
+ recognises it.\bigskip\pause
+ \end{itemize}
+
+
+ There are many reasons why this is a good definition:\medskip
+ \begin{itemize}
+ \item pumping lemma
+ \item closure properties of regular languages\\ (e.g.~closure under complement)
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{Really Bad News!}
+
+ DFAs are bad news for formalisations in theorem provers. They might
+ be represented as:
+
+ \begin{itemize}
+ \item graphs
+ \item matrices
+ \item partial functions
+ \end{itemize}
+
+ All constructions are messy to reason about.\bigskip\bigskip
+ \pause
+
+ \small
+ \only<2>{
+ Constable et al needed (on and off) 18 months for a 3-person team
+ to formalise automata theory in Nuprl including Myhill-Nerode. There is
+ only very little other formalised work on regular languages I know of
+ in Coq, Isabelle and HOL.}
+ \only<3>{Typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two
+ automata with no inaccessible states \ldots''
+ }
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{}
+ \large
+ \begin{center}
+ \begin{tabular}{p{9cm}}
+ My point:\bigskip\\
+
+ The theory about regular languages can be reformulated
+ to be more\\ suitable for theorem proving.
+ \end{tabular}
+ \end{center}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Myhill-Nerode Theorem}
+
+ \begin{itemize}
+ \item provides necessary and suf\!ficient conditions for a language
+ being regular (pumping lemma only necessary)\medskip
+
+ \item will help with closure properties of regular languages\bigskip\pause
+
+ \item key is the equivalence relation:\smallskip
+ \begin{center}
+ \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
+ \end{center}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Myhill-Nerode Theorem}
+
+ \mbox{}\\[5cm]
+
+ \begin{itemize}
+ \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Equivalence Classes}
+
+ \begin{itemize}
+ \item \smath{L = []}
+ \begin{center}
+ \smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}}
+ \end{center}\bigskip\bigskip
+
+ \item \smath{L = [c]}
+ \begin{center}
+ \smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}}
+ \end{center}\bigskip\bigskip
+
+ \item \smath{L = \varnothing}
+ \begin{center}
+ \smath{\Big\{U\!N\!IV\Big\}}
+ \end{center}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Regular Languages}
+
+ \begin{itemize}
+ \item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M}
+ such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
+
+ \item Myhill-Nerode:
+
+ \begin{center}
+ \begin{tabular}{l}
+ finite $\Rightarrow$ regular\\
+ \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r.\; L = \mathbb{L}(r)}\\[3mm]
+ regular $\Rightarrow$ finite\\
+ \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Final Equiv.~Classes}
+
+ \mbox{}\\[3cm]
+
+ \begin{itemize}
+ \item \smath{\text{finals}\,L \dn
+ \{{\lbrack\mkern-2mu\lbrack{s}\rbrack\mkern-2mu\rbrack}_\approx\;|\; s \in L\}}\\
+ \medskip
+
+ \item we can prove: \smath{L = \bigcup (\text{finals}\,L)}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Transitions between ECs}
+
+ \smath{L = \{[c]\}}
+
+ \begin{tabular}{@ {\hspace{-7mm}}cc}
+ \begin{tabular}{c}
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+
+ %\draw[help lines] (0,0) grid (3,2);
+
+ \node[state,initial] (q_0) {$R_1$};
+ \node[state,accepting] (q_1) [above right of=q_0] {$R_2$};
+ \node[state] (q_2) [below right of=q_0] {$R_3$};
+
+ \path[->] (q_0) edge node {c} (q_1)
+ edge node [swap] {$\Sigma-{c}$} (q_2)
+ (q_2) edge [loop below] node {$\Sigma$} ()
+ (q_1) edge node {$\Sigma$} (q_2);
+ \end{tikzpicture}
+ \end{tabular}
+ &
+ \begin{tabular}[t]{ll}
+ \\[-20mm]
+ \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
+
+ \smath{R_1}: & \smath{\{[]\}}\\
+ \smath{R_2}: & \smath{\{[c]\}}\\
+ \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
+ \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ;; [c] \subseteq Y}}}
+ \end{tabular}
+
+ \end{tabular}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Systems of Equations}
+
+ Inspired by a method of Brzozowski\;'64, we can build an equational system
+ characterising the equivalence classes:
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-20mm}}c}
+ \\[-13mm]
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+
+ %\draw[help lines] (0,0) grid (3,2);
+
+ \node[state,initial] (p_0) {$R_1$};
+ \node[state,accepting] (p_1) [right of=q_0] {$R_2$};
+
+ \path[->] (p_0) edge [bend left] node {a} (p_1)
+ edge [loop above] node {b} ()
+ (p_1) edge [loop above] node {a} ()
+ edge [bend left] node {b} (p_0);
+ \end{tikzpicture}\\
+ \\[-13mm]
+ \end{tabular}
+ \end{center}
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+ & \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
+ & \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
+ \onslide<3->{we can prove}
+ & \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}}
+ & \onslide<3->{\smath{R_1;; \mathbb{L}(b) \,\cup\, R_2;;\mathbb{L}(b) \,\cup\, \{[]\}}}\\
+ & \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}
+ & \onslide<3->{\smath{R_1;; \mathbb{L}(a) \,\cup\, R_2;;\mathbb{L}(a)}}\\
+ \end{tabular}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[t]
+ \small
+
+ \begin{center}
+ \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+ \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; a + R_2; a}}\\
+
+ & & & \onslide<2->{by Arden}\\
+
+ \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
+ & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
+ & \only<2>{\smath{R_1; a + R_2; a}}%
+ \only<3->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<4->{by Arden}\\
+
+ \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<5->{by substitution}\\
+
+ \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<6->{by Arden}\\
+
+ \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<7->{by substitution}\\
+
+ \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
+ \cdot a\cdot a^\star}}\\
+ \end{tabular}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE A Variant of Arden's Lemma}
+
+ {\bf Arden's Lemma:}\smallskip
+
+ If \smath{[] \not\in A} then
+ \begin{center}
+ \smath{X = X; A + \text{something}}
+ \end{center}
+ has the (unique) solution
+ \begin{center}
+ \smath{X = \text{something} ; A^\star}
+ \end{center}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \small
+
+ \begin{center}
+ \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+ \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; a + R_2; a}}\\
+
+ & & & \onslide<2->{by Arden}\\
+
+ \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
+ & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
+ & \only<2>{\smath{R_1; a + R_2; a}}%
+ \only<3->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<4->{by Arden}\\
+
+ \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<5->{by substitution}\\
+
+ \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<6->{by Arden}\\
+
+ \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<7->{by substitution}\\
+
+ \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
+ \cdot a\cdot a^\star}}\\
+ \end{tabular}
+ \end{center}
+
+ \only<8->{
+ \begin{textblock}{6}(2.5,4)
+ \begin{block}{}
+ \begin{minipage}{8cm}\raggedright
+
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+
+ %\draw[help lines] (0,0) grid (3,2);
+
+ \node[state,initial] (p_0) {$R_1$};
+ \node[state,accepting] (p_1) [right of=q_0] {$R_2$};
+
+ \path[->] (p_0) edge [bend left] node {a} (p_1)
+ edge [loop above] node {b} ()
+ (p_1) edge [loop above] node {a} ()
+ edge [bend left] node {b} (p_0);
+ \end{tikzpicture}
+
+ \end{minipage}
+ \end{block}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Equ's Solving Algorithm}
+
+ \begin{itemize}
+ \item The algorithm must terminate: Arden makes one equation smaller;
+ substitution deletes one variable from the right-hand sides.\bigskip
+
+ \item We need to maintain the invariant that Arden is applicable
+ (if \smath{[] \not\in A} then \ldots):\medskip
+
+ \begin{center}\small
+ \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+ \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
+ \smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
+
+ & & & by Arden\\
+
+ \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
+ \smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Other Direction}
+
+ One has to prove
+
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
+ \end{center}
+
+ by induction on \smath{r}. This is straightforward for \\the base cases:\small
+
+ \begin{center}
+ \begin{tabular}{l@ {\hspace{1mm}}l}
+ \smath{U\!N\!IV /\!/ \!\approx_{\emptyset}} & \smath{= \{U\!N\!IV\}}\smallskip\\
+ \smath{U\!N\!IV /\!/ \!\approx_{\{[]\}}} & \smath{\subseteq \{\{[]\}, U\!N\!IV - \{[]\}\}}\smallskip\\
+ \smath{U\!N\!IV /\!/ \!\approx_{\{[c]\}}} & \smath{\subseteq \{\{[]\}, \{[c]\}, U\!N\!IV - \{[], [c]\}\}}
+ \end{tabular}
+ \end{center}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\LARGE The Other Direction}
+
+ More complicated are the inductive cases:\\ one needs to prove that if
+
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{3mm}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
+ \end{center}
+
+ then
+
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\LARGE Helper Lemma}
+
+ \begin{center}
+ \begin{tabular}{p{10cm}}
+ %If \smath{\text{finite} (f\;' A)} and \smath{f} is injective
+ %(on \smath{A}),\\ then \smath{\text{finite}\,A}.
+ Given two equivalence relations \smath{R_1} and \smath{R_2} with
+ \smath{R_1} refining \smath{R_2} (\smath{R_1 \subseteq R_2}).\\
+ Then\medskip\\
+ \smath{\;\;\text{finite} (U\!N\!IV /\!/ R_1) \Rightarrow \text{finite} (U\!N\!IV /\!/ R_2)}
+ \end{tabular}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\Large Derivatives and Left-Quotients}
+ \small
+ Work by Brozowski ('64) and Antimirov ('96):\pause\smallskip
+
+
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+ \multicolumn{4}{@ {}l}{Left-Quotient:}\\
+ \multicolumn{4}{@ {}l}{\bl{$\text{Ders}\;\text{s}\,A \dn \{\text{s'} \;|\; \text{s @ s'} \in A\}$}}\bigskip\\
+
+ \multicolumn{4}{@ {}l}{Derivative:}\\
+ \bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\
+ \bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\
+ \bl{der c (d)} & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\
+ \bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\
+ \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\
+ & & \bl{\;\;\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\
+ \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\
+
+ \bl{ders [] r} & \bl{$=$} & \bl{r} & \\
+ \bl{ders (s @ [c]) r} & \bl{$=$} & \bl{der c (ders s r)} & \\
+ \end{tabular}\pause
+
+ \begin{center}
+ \alert{$\Rightarrow$}\smath{\;\;\text{Ders}\,\text{s}\,(\mathbb{L}(\text{r})) = \mathbb{L} (\text{ders s r})}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Left-Quotients and MN-Rels}
+
+ \begin{itemize}
+ \item \smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}\medskip
+ \item \bl{$\text{Ders}\;s\,A \dn \{s' \;|\; s @ s' \in A\}$}
+ \end{itemize}\bigskip
+
+ \begin{center}
+ \smath{x \approx_A y \Longleftrightarrow \text{Ders}\;x\;A = \text{Ders}\;y\;A}
+ \end{center}\bigskip\pause\small
+
+ which means
+
+ \begin{center}
+ \smath{x \approx_{\mathbb{L}(r)} y \Longleftrightarrow
+ \mathbb{L}(\text{ders}\;x\;r) = \mathbb{L}(\text{ders}\;y\;r)}
+ \end{center}\pause
+
+ \hspace{8.8mm}or
+ \smath{\;x \approx_{\mathbb{L}(r)} y \Longleftarrow
+ \text{ders}\;x\;r = \text{ders}\;y\;r}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Partial Derivatives}
+
+ Antimirov: \bl{pder : rexp $\Rightarrow$ rexp set}\bigskip
+
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+ \bl{pder c ($\varnothing$)} & \bl{$=$} & \bl{\{$\varnothing$\}} & \\
+ \bl{pder c ([])} & \bl{$=$} & \bl{\{$\varnothing$\}} & \\
+ \bl{pder c (d)} & \bl{$=$} & \bl{if c = d then \{[]\} else \{$\varnothing$\}} & \\
+ \bl{pder c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(pder c r$_1$) $\cup$ (pder c r$_2$)} & \\
+ \bl{pder c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{(pder c r$_1$) $\odot$ r$_2$} & \\
+ & & \bl{\hspace{-10mm}$\cup$ (if nullable r$_1$ then pder c r$_2$ else $\varnothing$)}\\
+ \bl{pder c (r$^*$)} & \bl{$=$} & \bl{(pder c r) $\odot$ r$^*$} &\smallskip\\
+ \end{tabular}
+
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+ \bl{pders [] r} & \bl{$=$} & \bl{r} & \\
+ \bl{pders (s @ [c]) r} & \bl{$=$} & \bl{pder c (pders s r)} & \\
+ \end{tabular}\pause
+
+ \begin{center}
+ \alert{$\Rightarrow$}\smath{\;\;\text{Ders}\,\text{s}\,(\mathbb{L}(\text{r})) = \bigcup (\mathbb{L}\;`\; (\text{pders s r}))}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\LARGE Final Result}
+
+ \mbox{}\\[7mm]
+
+ \begin{itemize}
+ \item \alt<1>{\smath{\text{pders x r \mbox{$=$} pders y r}}}
+ {\smath{\underbrace{\text{pders x r \mbox{$=$} pders y r}}_{R_1}}}
+ refines \bl{x $\approx_{\mathbb{L}(\text{r})}$ y}\pause
+ \item \smath{\text{finite} (U\!N\!IV /\!/ R_1)} \bigskip\pause
+ \item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}. Qed.
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE What Have We Achieved?}
+
+ \begin{itemize}
+ \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
+ \bigskip\pause
+ \item regular languages are closed under complementation; this is now easy\medskip
+ \begin{center}
+ \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
+ \end{center}
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Examples}
+
+ \begin{itemize}
+ \item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular
+ \begin{quote}\small
+ \begin{tabular}{lcl}
+ \smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\
+ \smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\
+ \smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\
+ \smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\
+ \end{tabular}
+ \end{quote}
+
+ \item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular
+ \begin{quote}\small
+ \begin{tabular}{lcl}
+ \smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\
+ \smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\
+ \smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\
+ \smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\
+ & \smath{\vdots} &\\
+ \end{tabular}
+ \end{quote}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE What We Have Not Achieved}
+
+ \begin{itemize}
+ \item regular expressions are not good if you look for a minimal
+ one for a language (DFAs have this notion)\pause\bigskip
+
+ \item Is there anything to be said about context free languages:\medskip
+
+ \begin{quote}
+ A context free language is where every string can be recognised by
+ a pushdown automaton.\bigskip
+ \end{quote}
+ \end{itemize}
+
+ \textcolor{gray}{\footnotesize Yes. Derivatives also work for c-f grammars. Ongoing work.}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Conclusion}
+
+ \begin{itemize}
+ \item We formalised the Myhill-Nerode theorem based on
+ regular expressions only (DFAs are difficult to deal with in a theorem prover).\smallskip
+
+ \item Seems to be a common theme: algorithms need to be reformulated
+ to better suit formal treatment.\smallskip
+
+ \item The most interesting aspect is that we are able to
+ implement the matcher directly inside the theorem prover
+ (ongoing work).\smallskip
+
+ \item Parsing is a vast field which seem to offer new results.
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[b]
+ \frametitle{
+ \begin{tabular}{c}
+ \mbox{}\\[13mm]
+ \alert{\LARGE Thank you very much!}\\
+ \alert{\Large Questions?}
+ \end{tabular}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+
+(*<*)
+end
+(*>*)
\ No newline at end of file