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theory Slides
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imports "LaTeXsugar"
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begin
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notation (latex output)
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set ("_") and
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Cons ("_::/_" [66,65] 65)
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(*>*)
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text_raw {*
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%\renewcommand{\slidecaption}{Cambridge, 9 November 2010}
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\renewcommand{\slidecaption}{Munich, 17 November 2010}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\LARGE A Formalisation of the\\[-3mm]
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\LARGE Myhill-Nerode Theorem\\[-3mm]
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\LARGE based on Regular Expressions\\[-3mm]
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\large \onslide<2>{\alert{or, Regular Languages Done Right}}\\
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\end{tabular}}
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\begin{center}
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Christian Urban
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\end{center}
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\begin{center}
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joint work with Chunhan Wu and Xingyuan Zhang from the PLA
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University of Science and Technology in Nanjing
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{In Most Textbooks\ldots}
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\begin{itemize}
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\item A \alert{regular language} is one where there is a DFA that
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recognises it.\bigskip\pause
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\end{itemize}
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I can think of three reasons why this is a good definition:\medskip
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\begin{itemize}
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\item string matching via DFAs (yacc)
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\item pumping lemma
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\item closure properties of regular languages (closed under complement)
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{Really Bad News!}
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DFAs are bad news for formalisations in theorem provers. They might
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be represented as:
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\begin{itemize}
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\item graphs
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\item matrices
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\item partial functions
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\end{itemize}
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All constructions are messy to reason about.\bigskip\bigskip
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\pause
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\small
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\only<2>{Alexander and Tobias: ``\ldots automata theory \ldots does not come for free \ldots''}
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\only<3>{
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Constable et al needed (on and off) 18 months for a 3-person team
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to formalise automata theory in Nuprl including Myhill-Nerode. There is
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only very little other formalised work on regular languages I know of
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in Coq, Isabelle and HOL.}
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\only<4>{typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two
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automata with no inaccessible states \ldots''
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}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{Regular Expressions}
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\ldots are a simple datatype:
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\only<1>{
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\begin{center}\color{blue}
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\begin{tabular}{rcl}
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rexp & $::=$ & NULL\\
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& $\mid$ & EMPTY\\
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& $\mid$ & CHR c\\
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& $\mid$ & ALT rexp rexp\\
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& $\mid$ & SEQ rexp rexp\\
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& $\mid$ & STAR rexp
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\end{tabular}
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\end{center}}
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\only<2->{
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\begin{center}
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\begin{tabular}{rcl}
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\smath{r} & \smath{::=} & \smath{0} \\
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& \smath{\mid} & \smath{[]}\\
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& \smath{\mid} & \smath{c}\\
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& \smath{\mid} & \smath{r_1 + r_2}\\
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& \smath{\mid} & \smath{r_1 \cdot r_2}\\
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& \smath{\mid} & \smath{r^\star}
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\end{tabular}
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\end{center}}
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\only<3->{Induction and recursion principles come for free.}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{Semantics of Rexps}
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\begin{center}
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\begin{tabular}{rcl}
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\smath{\mathbb{L}(0)} & \smath{=} & \smath{\varnothing}\\
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\smath{\mathbb{L}([])} & \smath{=} & \smath{\{[]\}}\\
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\smath{\mathbb{L}(c)} & \smath{=} & \smath{\{[c]\}}\\
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\smath{\mathbb{L}(r_1 + r_2)} & \smath{=} & \smath{\mathbb{L}(r_1) \cup \mathbb{L}(r_2)}\\
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\smath{\mathbb{L}(r_1 \cdot r_2)} & \smath{=} & \smath{\mathbb{L}(r_1)\; ;\; \mathbb{L} (r_2)}\\
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\smath{\mathbb{L}(r^\star)} & \smath{=} & \smath{\mathbb{L}(r)^\star}
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\end{tabular}
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\end{center}
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\small
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\begin{center}
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\begin{tabular}{rcl}
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\smath{L_1 ; L_2} & \smath{\dn} & \smath{\{ s_1 @ s_2 \mid s_1 \in L_1 \wedge s_2 \in L_2\}}\bigskip\\
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\multicolumn{3}{c}{
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\smath{\infer{[] \in L^\star}{}} \hspace{10mm}
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\smath{\infer{s_1 @ s_2 \in L^\star}{s_1 \in L & s_2 \in L^\star}}
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}
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\end{tabular}
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE Regular Expression Matching}
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\begin{itemize}
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\item Harper in JFP'99: ``Functional Pearl: Proof- Directed Debugging''\medskip
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\item Yi in JFP'06: ``Educational Pearl: `Proof-Directed Debugging' revisited
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for a first-order version''\medskip
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\item Owens et al in JFP'09: ``Regular-expression derivatives re-examined''\bigskip\pause
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\begin{quote}\small
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``Unfortunately, regular expression derivatives have been lost in the
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sands of time, and few computer scientists are aware of them.''
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\end{quote}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\begin{center}
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\huge\bf Demo
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\end{center}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE The Myhill-Nerode Theorem}
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\begin{itemize}
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\item provides necessary and suf\!ficient conditions for a language
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being regular (pumping lemma only necessary)\medskip
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\item will help with closure properties of regular languages\bigskip\pause
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\item key is the equivalence relation:\smallskip
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\begin{center}
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\smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
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\end{center}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE The Myhill-Nerode Theorem}
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\mbox{}\\[5cm]
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\begin{itemize}
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\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE Equivalence Classes}
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\begin{itemize}
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\item \smath{L = []}
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\begin{center}
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\smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}}
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\end{center}\bigskip\bigskip
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\item \smath{L = [c]}
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\begin{center}
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\smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}}
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\end{center}\bigskip\bigskip
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\item \smath{L = \varnothing}
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\begin{center}
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\smath{\Big\{U\!N\!IV\Big\}}
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\end{center}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE Regular Languages}
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\begin{itemize}
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\item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M}
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such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
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\item Myhill-Nerode:
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\begin{center}
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\begin{tabular}{l}
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finite $\Rightarrow$ regular\\
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\;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r. L = \mathbb{L}(r)}\\[3mm]
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regular $\Rightarrow$ finite\\
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\;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
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\end{tabular}
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\end{center}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE Final States}
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\mbox{}\\[3cm]
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\begin{itemize}
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\item \smath{\text{final}_L\,X \dn}\\
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\smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_L) \;\wedge\; \forall s \in X.\; s \in L}
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\smallskip
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\item we can prove: \smath{L = \bigcup \{X.\;\text{final}_L\,X\}}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes}
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\smath{L = \{[c]\}}
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\begin{tabular}{@ {\hspace{-7mm}}cc}
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\begin{tabular}{c}
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\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
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\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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%\draw[help lines] (0,0) grid (3,2);
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\node[state,initial] (q_0) {$R_1$};
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\node[state,accepting] (q_1) [above right of=q_0] {$R_2$};
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\node[state] (q_2) [below right of=q_0] {$R_3$};
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\path[->] (q_0) edge node {c} (q_1)
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edge node [swap] {$\Sigma-{c}$} (q_2)
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(q_2) edge [loop below] node {$\Sigma$} ()
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(q_1) edge node {$\Sigma$} (q_2);
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\end{tikzpicture}
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\end{tabular}
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&
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\begin{tabular}[t]{ll}
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\\[-20mm]
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\multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
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\smath{R_1}: & \smath{\{[]\}}\\
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\smath{R_2}: & \smath{\{[c]\}}\\
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\smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
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\multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ; [c] \subseteq Y}}}
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\end{tabular}
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\end{tabular}
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\end{frame}}
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE Systems of Equations}
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Inspired by a method of Brzozowski\;'64, we can build an equational system
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characterising the equivalence classes:
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\begin{center}
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\begin{tabular}{@ {\hspace{-20mm}}c}
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\\[-13mm]
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\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
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\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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%\draw[help lines] (0,0) grid (3,2);
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\node[state,initial] (p_0) {$R_1$};
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\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
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\path[->] (p_0) edge [bend left] node {a} (p_1)
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edge [loop above] node {b} ()
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(p_1) edge [loop above] node {a} ()
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edge [bend left] node {b} (p_0);
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\end{tikzpicture}\\
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\\[-13mm]
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\end{tabular}
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\end{center}
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404 |
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\begin{center}
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\begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
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407 |
& \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
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& \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
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\onslide<3->{we can prove}
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& \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}}
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& \onslide<3->{\smath{R_1; \mathbb{L}(b) \,\cup\, R_2;\mathbb{L}(b) \,\cup\, \{[]\};\{[]\}}}\\
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412 |
& \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}
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& \onslide<3->{\smath{R_1; \mathbb{L}(a) \,\cup\, R_2;\mathbb{L}(a)}}\\
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414 |
\end{tabular}
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\end{center}
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\end{frame}}
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1>[t]
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\small
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\begin{center}
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428 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
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429 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
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430 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
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\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
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& \onslide<1->{\smath{R_1; a + R_2; a}}\\
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433 |
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& & & \onslide<2->{by Arden}\\
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435 |
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436 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
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437 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
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438 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
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& \only<2>{\smath{R_1; a + R_2; a}}%
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440 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
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441 |
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24
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442 |
& & & \onslide<4->{by Arden}\\
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21
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443 |
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444 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
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24
|
445 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
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21
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446 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
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447 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
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448 |
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24
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449 |
& & & \onslide<5->{by substitution}\\
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21
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450 |
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451 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
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24
|
452 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
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21
|
453 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
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|
454 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
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455 |
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24
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456 |
& & & \onslide<6->{by Arden}\\
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457 |
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458 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
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& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
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|
460 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
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24
|
461 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
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462 |
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|
463 |
& & & \onslide<7->{by substitution}\\
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464 |
|
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|
465 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
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|
466 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
467 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
|
468 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
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21
|
469 |
\cdot a\cdot a^\star}}\\
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|
470 |
\end{tabular}
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471 |
\end{center}
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472 |
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473 |
\end{frame}}
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474 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
475 |
*}
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476 |
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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479 |
\mode<presentation>{
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|
480 |
\begin{frame}[c]
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|
481 |
\frametitle{\LARGE A Variant of Arden's Lemma}
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|
482 |
|
|
24
|
483 |
{\bf Arden's Lemma:}\smallskip
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|
21
|
484 |
|
|
|
485 |
If \smath{[] \not\in A} then
|
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|
486 |
\begin{center}
|
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|
487 |
\smath{X = X; A + \text{something}}
|
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|
488 |
\end{center}
|
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|
489 |
has the (unique) solution
|
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|
490 |
\begin{center}
|
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|
491 |
\smath{X = \text{something} ; A^\star}
|
|
|
492 |
\end{center}
|
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493 |
|
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|
494 |
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|
495 |
\end{frame}}
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496 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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497 |
*}
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498 |
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499 |
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500 |
text_raw {*
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501 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
502 |
\mode<presentation>{
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|
503 |
\begin{frame}<1->[t]
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|
504 |
\small
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505 |
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|
506 |
\begin{center}
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|
507 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
508 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
|
509 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
510 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
|
|
511 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
|
|
512 |
|
|
|
513 |
& & & \onslide<2->{by Arden}\\
|
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|
514 |
|
|
|
515 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
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|
516 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
517 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
|
|
518 |
& \only<2>{\smath{R_1; a + R_2; a}}%
|
|
|
519 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
520 |
|
|
24
|
521 |
& & & \onslide<4->{by Arden}\\
|
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21
|
522 |
|
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|
523 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
24
|
524 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
21
|
525 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
|
526 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
527 |
|
|
24
|
528 |
& & & \onslide<5->{by substitution}\\
|
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21
|
529 |
|
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|
530 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
|
24
|
531 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
21
|
532 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
|
533 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
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|
534 |
|
|
24
|
535 |
& & & \onslide<6->{by Arden}\\
|
|
21
|
536 |
|
|
|
537 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
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|
538 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
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|
539 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
24
|
540 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
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|
541 |
|
|
|
542 |
& & & \onslide<7->{by substitution}\\
|
|
|
543 |
|
|
|
544 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
|
545 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
546 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
|
547 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
21
|
548 |
\cdot a\cdot a^\star}}\\
|
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|
549 |
\end{tabular}
|
|
|
550 |
\end{center}
|
|
|
551 |
|
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|
552 |
\only<8->{
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|
553 |
\begin{textblock}{6}(2.5,4)
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554 |
\begin{block}{}
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555 |
\begin{minipage}{8cm}\raggedright
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556 |
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\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
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558 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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559 |
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|
560 |
%\draw[help lines] (0,0) grid (3,2);
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|
561 |
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|
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\node[state,initial] (p_0) {$R_1$};
|
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|
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\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
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|
564 |
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\path[->] (p_0) edge [bend left] node {a} (p_1)
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566 |
edge [loop above] node {b} ()
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567 |
(p_1) edge [loop above] node {a} ()
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edge [bend left] node {b} (p_0);
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569 |
\end{tikzpicture}
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570 |
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571 |
\end{minipage}
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572 |
\end{block}
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573 |
\end{textblock}}
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574 |
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575 |
\end{frame}}
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576 |
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577 |
*}
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578 |
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|
579 |
text_raw {*
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|
581 |
\mode<presentation>{
|
|
|
582 |
\begin{frame}[c]
|
|
|
583 |
\frametitle{\LARGE The Equ's Solving Algorithm}
|
|
|
584 |
|
|
|
585 |
\begin{itemize}
|
|
|
586 |
\item The algorithm must terminate: Arden makes one equation smaller;
|
|
|
587 |
substitution deletes one variable from the right-hand sides.\bigskip
|
|
|
588 |
|
|
24
|
589 |
\item We need to maintain the invariant that Arden is applicable
|
|
|
590 |
(if \smath{[] \not\in A} then \ldots):\medskip
|
|
|
591 |
|
|
|
592 |
\begin{center}\small
|
|
|
593 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
594 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
|
|
595 |
\smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
|
|
|
596 |
|
|
|
597 |
& & & by Arden\\
|
|
|
598 |
|
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|
599 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
|
|
600 |
\smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
|
|
|
601 |
\end{tabular}
|
|
|
602 |
\end{center}
|
|
|
603 |
|
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|
604 |
\end{itemize}
|
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|
605 |
|
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|
606 |
|
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|
607 |
\end{frame}}
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608 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
609 |
*}
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610 |
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|
611 |
text_raw {*
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612 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
|
613 |
\mode<presentation>{
|
|
|
614 |
\begin{frame}[c]
|
|
|
615 |
\frametitle{\LARGE The Equ's Solving Algorithm}
|
|
|
616 |
|
|
|
617 |
\begin{itemize}
|
|
|
618 |
\item The algorithm is still a bit hairy to formalise because of our set-representation
|
|
21
|
619 |
for equations:
|
|
|
620 |
|
|
|
621 |
\begin{center}
|
|
|
622 |
\begin{tabular}{ll}
|
|
|
623 |
\smath{\big\{ (X, \{(Y_1, r_1), (Y_2, r_2), \ldots\}),}\\
|
|
|
624 |
\mbox{}\hspace{5mm}\smath{\ldots}\\
|
|
|
625 |
& \smath{\big\}}
|
|
|
626 |
\end{tabular}
|
|
24
|
627 |
\end{center}\bigskip\pause
|
|
21
|
628 |
|
|
|
629 |
\small
|
|
24
|
630 |
they are generated from \smath{U\!N\!IV /\!/ \approx_L}
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|
21
|
631 |
|
|
|
632 |
\end{itemize}
|
|
|
633 |
|
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|
634 |
|
|
|
635 |
\end{frame}}
|
|
|
636 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
637 |
*}
|
|
|
638 |
|
|
24
|
639 |
|
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640 |
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text_raw {*
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642 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
643 |
\mode<presentation>{
|
|
|
644 |
\begin{frame}[c]
|
|
|
645 |
\frametitle{\LARGE Other Direction}
|
|
|
646 |
|
|
|
647 |
One has to prove
|
|
|
648 |
|
|
|
649 |
\begin{center}
|
|
|
650 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
|
|
|
651 |
\end{center}
|
|
|
652 |
|
|
|
653 |
by induction on \smath{r}. Not trivial, but after a bit
|
|
|
654 |
of thinking (by Chunhan), one can prove that if
|
|
|
655 |
|
|
|
656 |
\begin{center}
|
|
|
657 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm}
|
|
|
658 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
|
|
|
659 |
\end{center}
|
|
|
660 |
|
|
|
661 |
then
|
|
|
662 |
|
|
|
663 |
\begin{center}
|
|
|
664 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
|
|
|
665 |
\end{center}
|
|
|
666 |
|
|
|
667 |
|
|
|
668 |
|
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|
669 |
\end{frame}}
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|
670 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
671 |
*}
|
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|
672 |
|
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|
673 |
text_raw {*
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|
674 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
675 |
\mode<presentation>{
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|
676 |
\begin{frame}[c]
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|
677 |
\frametitle{\LARGE What Have We Achieved?}
|
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|
678 |
|
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|
679 |
\begin{itemize}
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|
680 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
|
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|
681 |
\bigskip\pause
|
|
24
|
682 |
\item regular languages are closed under complementation; this is easy
|
|
21
|
683 |
\begin{center}
|
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|
684 |
\smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
|
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|
685 |
\end{center}\pause\bigskip
|
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|
686 |
|
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24
|
687 |
\item if you want to do regular expression matching (see Scott's paper)\pause\bigskip
|
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21
|
688 |
|
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24
|
689 |
\item I cannot yet give definite numbers
|
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21
|
690 |
\end{itemize}
|
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|
691 |
|
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|
692 |
\only<2>{
|
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|
693 |
\begin{textblock}{10}(4,14)
|
|
|
694 |
\small
|
|
|
695 |
\smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
|
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|
696 |
\end{textblock}
|
|
|
697 |
}
|
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|
698 |
|
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24
|
699 |
|
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|
700 |
|
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21
|
701 |
\end{frame}}
|
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|
702 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
703 |
*}
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|
704 |
|
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|
705 |
text_raw {*
|
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|
706 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
707 |
\mode<presentation>{
|
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|
708 |
\begin{frame}[c]
|
|
24
|
709 |
\frametitle{\LARGE Examples}
|
|
|
710 |
|
|
|
711 |
\begin{itemize}
|
|
|
712 |
\item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular
|
|
|
713 |
\begin{quote}\small
|
|
|
714 |
\begin{tabular}{lcl}
|
|
|
715 |
\smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\
|
|
|
716 |
\smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\
|
|
|
717 |
\smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\
|
|
|
718 |
\smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\
|
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|
719 |
\end{tabular}
|
|
|
720 |
\end{quote}
|
|
|
721 |
|
|
|
722 |
\item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular
|
|
|
723 |
\begin{quote}\small
|
|
|
724 |
\begin{tabular}{lcl}
|
|
|
725 |
\smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\
|
|
|
726 |
\smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\
|
|
|
727 |
\smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\
|
|
|
728 |
\smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\
|
|
|
729 |
& \smath{\vdots} &\\
|
|
|
730 |
\end{tabular}
|
|
|
731 |
\end{quote}
|
|
|
732 |
\end{itemize}
|
|
|
733 |
|
|
|
734 |
\end{frame}}
|
|
|
735 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
736 |
*}
|
|
|
737 |
|
|
|
738 |
|
|
|
739 |
text_raw {*
|
|
|
740 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
741 |
\mode<presentation>{
|
|
|
742 |
\begin{frame}[c]
|
|
|
743 |
\frametitle{\LARGE What We Have Not Achieved}
|
|
|
744 |
|
|
|
745 |
\begin{itemize}
|
|
|
746 |
\item regular expressions are not good if you look for a minimal
|
|
|
747 |
one for a language (DFAs have this notion)\pause\bigskip
|
|
|
748 |
|
|
|
749 |
\item Is there anything to be said about context free languages:\medskip
|
|
|
750 |
|
|
|
751 |
\begin{quote}
|
|
|
752 |
A context free language is where every string can be recognised by
|
|
|
753 |
a pushdown automaton.
|
|
|
754 |
\end{quote}
|
|
|
755 |
\end{itemize}
|
|
|
756 |
|
|
|
757 |
\end{frame}}
|
|
|
758 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
759 |
*}
|
|
|
760 |
|
|
|
761 |
|
|
|
762 |
text_raw {*
|
|
|
763 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
764 |
\mode<presentation>{
|
|
|
765 |
\begin{frame}[c]
|
|
21
|
766 |
\frametitle{\LARGE Conclusion}
|
|
|
767 |
|
|
|
768 |
\begin{itemize}
|
|
24
|
769 |
\item on balance regular expression are superior
|
|
|
770 |
to DFAs, in my opinion\bigskip
|
|
21
|
771 |
|
|
|
772 |
\item I cannot think of a reason to not teach regular languages
|
|
24
|
773 |
to students this way (!?)\bigskip
|
|
21
|
774 |
|
|
|
775 |
\item I have never ever seen a proof of Myhill-Nerode based on
|
|
|
776 |
regular expressions\bigskip
|
|
|
777 |
|
|
24
|
778 |
\item no application, but lots of fun\bigskip
|
|
21
|
779 |
|
|
|
780 |
\item great source of examples
|
|
|
781 |
|
|
|
782 |
\end{itemize}
|
|
|
783 |
|
|
|
784 |
\end{frame}}
|
|
|
785 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
786 |
*}
|
|
|
787 |
|
|
16
|
788 |
(*<*)
|
|
|
789 |
end
|
|
|
790 |
(*>*) |