--- a/Slides/Slides6.thy Sat Dec 17 16:58:11 2011 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1606 +0,0 @@
-(*<*)
-theory Slides6
-imports "~~/src/HOL/Library/LaTeXsugar" "Nominal"
-begin
-
-declare [[show_question_marks = false]]
-
-notation (latex output)
- set ("_") and
- Cons ("_::/_" [66,65] 65)
-
-(*>*)
-
-text_raw {*
- \renewcommand{\slidecaption}{Hefei, 15.~April 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}<1->[c]
- \frametitle{My Background}
-
- \mbox{}\\[-10mm]
- \begin{itemize}
- \item My background is in theory and programming languages.\bigskip
- \pause
-
- \item But I am also a programmer with a \alert<2>{software system} of around 800 kloc
- (though I am responsible for only appr.~35 kloc),
-
- \item and I write papers.
- \end{itemize}
-
- \only<2>{
- \begin{textblock}{6}(6.5,11.5)
- \begin{tikzpicture}
- \draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
- {\color{darkgray}
- \begin{minipage}{6.5cm}\raggedright
- \begin{tabular}[b]{@ {}p{4.5cm}c@ {}}
- \raggedright
- The software is a theorem prover, called {\bf Isabelle}.
- & \mbox{}\hspace{-5mm}\raisebox{-14mm}{\includegraphics[scale=0.28]{isabelle1.png}}
- \end{tabular}%
- \end{minipage}};
- \end{tikzpicture}
- \end{textblock}}
-
- \only<4>{
- \begin{textblock}{6}(3,11.5)
- \begin{tikzpicture}
- \draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
- {\color{darkgray}
- \begin{minipage}{9.6cm}\raggedright
- So I can experience every day that writing error-free code is {\bf very, very hard}
- and that papers are also {\bf hard} to get correct.
- \end{minipage}};
- \end{tikzpicture}
- \end{textblock}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{3 Points}
- \large
- \begin{itemize}
- \item It is easy to make mistakes.\bigskip
- \item Theorem provers can prevent mistakes, {\bf if} the problem
- is formulated so that it is suitable for theorem provers.\bigskip
- \item This re-formulation can be done, even in domains where
- we do not expect it.
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
-
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}
- \frametitle{Getting Papers Correct}
-
- \begin{minipage}{1.1\textwidth}
- My work over the last 5 years.\\
- {\small (in the fields of programming languages, logic and lambda-calculi)}
- \end{minipage}\bigskip
-
- \only<1>{
- \mbox{}\\[15mm]
- \begin{center}
- \begin{tikzpicture}[node distance=0.5mm]
- \node at (-1.0,-0.3) (proof) [double arrow, fill=gray,text=white, minimum height=2cm]{\bf Proof};
- \node [left=of proof]{\Large\bf Specification};
- \node [right=of proof]{\Large\bf Code};
- \end{tikzpicture}
- \end{center}
- }
- \pause
-
- \begin{tabular}{c@ {\hspace{2mm}}c}
- \begin{tabular}{c}
- \includegraphics[scale=0.09]{harper.jpg}\\[-2mm]
- {\footnotesize Bob Harper}\\[-2.5mm]
- {\footnotesize (CMU)}
- \end{tabular}
- \begin{tabular}{c}
- \includegraphics[scale=0.31]{pfenning.jpg}\\[-2mm]
- {\footnotesize Frank Pfenning}\\[-2.5mm]
- {\footnotesize (CMU)}
- \end{tabular} &
-
- \begin{tabular}{p{6cm}}
- \raggedright\small
- \color{gray}{published a proof in ACM Transactions on Computational Logic (2005),
- $\sim$31pp}
- \end{tabular}\\
-
- \\[-4mm]
-
- \begin{tabular}{c}
- \includegraphics[scale=0.3]{appel.jpg}\\[-2mm]
- {\footnotesize Andrew Appel}\\[-2.5mm]
- {\footnotesize (Princeton)}
- \end{tabular} &
-
- \begin{tabular}{p{6cm}}
- \raggedright\small
- \color{gray}{relied on their proof in a safety critical system (proof carrying code)}
- \end{tabular}
-
- \end{tabular}\medskip
-
- \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 needs to run 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 code developer/ web server/ Apple
- Store\end{tabular}};
-
- \onslide<4->{
- \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<3->{
- \node at (3.8,1.3) [single arrow, fill=red,text=white, minimum height=3cm]{\bf LF proof};
- \node at (3.8,1.9) {\small certificate};
- }
-
- \onslide<2>{\node at (4.0,1.3) [text=red]{\begin{tabular}{c}\bf Highly\\\bf Dangerous!\end{tabular}};}
-
- \end{tikzpicture}
- \end{center}
- \end{block}
- \end{textblock}
-
- \begin{textblock}{15}(2,12)
- \small
- \begin{itemize}
- \item<4-> Appel's checker is $\sim$2700 lines of code (1865 loc of\\ LF definitions;
- 803 loc in C including 2 library functions)\\[-3mm]
- \item<5-> 167 loc in C implement a type-checker
- \end{itemize}
- \end{textblock}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-text {*
- \tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex]
- \tikzstyle{node1}=[rectangle, minimum size=10mm, rounded corners=3mm, very thick,
- draw=black!50, top color=white, bottom color=black!20]
- \tikzstyle{node2}=[rectangle, minimum size=12mm, rounded corners=3mm, very thick,
- draw=red!70, top color=white, bottom color=red!50!black!20]
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<2->[squeeze]
- \frametitle{Type-Checking in LF}
-
- \begin{columns}
-
- \begin{column}{0.8\textwidth}
- \begin{textblock}{0}(1,2)
-
- \begin{tikzpicture}
- \matrix[ampersand replacement=\&,column sep=7mm, row sep=5mm]
- { \&[-10mm]
- \node (def1) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}}; \&
- \node (proof1) [node1] {\large Proof}; \&
- \node (alg1) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}}; \\
-
- \onslide<4->{\node {\begin{tabular}{c}\small 1st\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
- \onslide<4->{\node (def2) [node2] {\large Spec$^\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}
-
- \only<7->{
- \begin{textblock}{14}(0.6,12.8)
- \begin{block}{}
- \small Each time one needs to check $\sim$31pp~of informal paper proofs.
- You have to be able to keep definitions and proofs consistent.
- \end{block}
- \end{textblock}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}<1->[c]
- \frametitle{Lessons Learned}
-
- \begin{itemize}
- \item Theorem provers help with keeping large proofs consistent;
- make them modifiable.\medskip
-
- \item They can ensure that all cases are covered.\medskip
-
- \item Some reasoning can be automated.
- \end{itemize}\bigskip\pause
-
- \begin{minipage}{1.1\textwidth}
- Formal reasoning needs to be ``smooth''.\\
- {\small (ideally as close as possible to reasoning with ``pen-and-paper'')}
- \end{minipage}
-
- \only<2->{
- \begin{textblock}{3}(0.1,9.9)
- \begin{tikzpicture}
- \node at (0,0) [single arrow, shape border rotate=0, fill=red,text=red]{a};
- \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>[c]
- \frametitle{Getting Programs Correct}
-
- \begin{center}
- \begin{tikzpicture}[node distance=0.5mm]
- \node at (-1.0,-0.3) (proof) [double arrow, fill=gray,text=white, minimum height=2cm]{\bf Proof};
- \node [left=of proof]{\Large\bf Specification};
- \node [right=of proof]{\Large\bf Code};
- \end{tikzpicture}
- \end{center}
-
-
-
- \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 (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\
- \bl{matches (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\
- \bl{matches (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\
- \bl{matches (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\
- \bl{matches (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\
- \onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}}\\
- \onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}}\\
- \onslide<2->{\bl{matches 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
-
- \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.\pause
-
- \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 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 States}
-
- \mbox{}\\[3cm]
-
- \begin{itemize}
- \item \smath{\text{final}_L\,X \dn}\\
- \smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_L) \;\wedge\; \forall s \in X.\; s \in L}
- \smallskip
- \item we can prove: \smath{L = \bigcup \{X.\;\text{final}_L\,X\}}
-
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes}
-
- \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 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}. Not trivial, but after a bit
- of thinking, one can prove that if
-
- \begin{center}
- \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm}
- \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}[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}}
-
- \begin{center}
- \bf \underline{Short Bio:}
- \end{center}
- \mbox{}\\[-17mm]\mbox{}\small
- \begin{itemize}
- \item PhD in Cambridge
- \item Emmy-Noether Research Fellowship at the TU Munich
- \item talks at: CMU, Yale, Princeton, MIT,$\ldots$
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Future Research}
-
- My existing strengths:\bigskip
-
- \begin{itemize}
- \item Isabelle (implementation)\bigskip
- \item background in logic, programming languages, formal methods
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Future Research}
-
- I want to have a single logic framework in which I can
- write programs and prove their correctness.\bigskip
-
- \begin{itemize}
- \item extensions of HOL (IO, modules, advanced types)
- \item high-level programming languages
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Future Research}
-
- Compilers\bigskip
-
- \begin{itemize}
- \item the high-level language needs to be compiled to correct machine
- code
- \item compiler verification, machine code verification
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Future Research}
-
- Stronger type-systems\bigskip
-
- \begin{itemize}
- \item ``correct by construction''
- \item GADTs, dependent types
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Future Research}
-
- Proof automation\bigskip
-
- \begin{itemize}
- \item external tools generate ``proof-certificates''
- \item certificates are imported into Isabelle
- \item GPU based external provers
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Future Research}
-
- Large-scale applications\bigskip
-
- \begin{itemize}
- \item verification of Java-Script, Scala,$\ldots$
- \item interesting code (INTEL in Shanghai)
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-(*<*)
-end
-(*>*)
\ No newline at end of file