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(*<*)
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theory Slides3
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imports "LaTeXsugar" "Nominal"
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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}{UNIF, Edinburgh, 14.~July 2010}
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\newcommand{\abst}[2]{#1.#2}% atom-abstraction
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\newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing
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\newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions
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\newcommand{\unit}{\langle\rangle}% unit
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\newcommand{\app}[2]{#1\,#2}% application
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\newcommand{\eqprob}{\mathrel{{\approx}?}}
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\newcommand{\freshprob}{\mathrel{\#?}}
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\newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction
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\newcommand{\id}{\varepsilon}% identity substitution
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\pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}%
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{rgb(0mm)=(0,0,0.9);
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rgb(0.9mm)=(0,0,0.7);
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rgb(1.3mm)=(0,0,0.5);
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rgb(1.4mm)=(1,1,1)}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[c]
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\frametitle{Quiz}
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Assuming that \smath{a} and \smath{b} are distinct variables,\\
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is it possible to find $\lambda$-terms \smath{M_1} to \smath{M_7}
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that make the following pairs \alert{$\alpha$-equivalent}?
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\begin{tabular}{@ {\hspace{14mm}}p{12cm}}
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\begin{itemize}
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\item \smath{\lambda a.\lambda b. (M_1\,b)\;} and
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\smath{\lambda b.\lambda a. (a\,M_1)\;}
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\item \smath{\lambda a.\lambda b. (M_2\,b)\;} and
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\smath{\lambda b.\lambda a. (a\,M_3)\;}
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\item \smath{\lambda a.\lambda b. (b\,M_4)\;} and
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\smath{\lambda b.\lambda a. (a\,M_5)\;}
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\item \smath{\lambda a.\lambda b. (b\,M_6)\;} and
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\smath{\lambda a.\lambda a. (a\,M_7)\;}
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\end{itemize}
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\end{tabular}
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If there is one solution for a pair, can you describe all its solutions?
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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}<1>[t]
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\frametitle{%
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\begin{tabular}{@ {\hspace{-3mm}}c@ {}}
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\\
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\huge Nominal Unification\\[-2mm]
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\Large Hitting a Sweet Spot\\[5mm]
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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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\small initial spark from Roy Dyckhoff in November 2001\\[0mm]
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\small joint work with Andy Pitts and Jamie Gabbay\\[0mm]
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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}<1-4>[c]
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\frametitle{One Motivation}
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\onslide<2->{Typing implemented in Prolog \textcolor{darkgray}{(from a textbook)}}\bigskip\\
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\onslide<3->{\color{darkgray}
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\begin{tabular}{l}
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type (Gamma, var(X), T) :- member (X,T) Gamma.\smallskip\medskip\\
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type (Gamma, app(M, N), T') :-\\
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\hspace{3cm}type (Gamma, M, arrow(T, T')),\\
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\hspace{3cm}type (Gamma, N, T).\smallskip\medskip\\
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type (Gamma, lam(X, M), arrow(T, T')) :-\\
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\hspace{3cm}type ((X, T)::Gamma, M, T').\smallskip\medskip\\
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member X X::Tail.\\
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member X Y::Tail :- member X Tail.\\
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\end{tabular}}
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\only<4>{
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\begin{textblock}{6}(2.5,2)
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
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{\color{darkgray}
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\begin{minipage}{8cm}\raggedright
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The problem is that \smath{\lambda x.\lambda x. (x\;x)}
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will have the types
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\begin{center}
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\begin{tabular}{l}
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\smath{T\rightarrow (T\rightarrow S) \rightarrow S} and\\
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\smath{(T\rightarrow S)\rightarrow T \rightarrow S}\\
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\end{tabular}
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\end{center}
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\end{minipage}};
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\end{tikzpicture}
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\end{textblock}}
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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}<1>[c]
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\frametitle{Higher-Order Unification}
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State of the art:
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\begin{itemize}
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\item Lambda Prolog with full Higher-Order Unification\\
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\textcolor{darkgray}{(no mgus, undecidable, modulo $\alpha\beta$)}\bigskip
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\item Higher-Order Pattern Unification\\
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\textcolor{darkgray}{(has mgus, decidable, some restrictions, modulo $\alpha\beta_0$)}
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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}<1-10>[t]
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\frametitle{Underlying Ideas}
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\begin{itemize}
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\item<1-> Unification (\alert{only}) up to $\alpha$
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\item<2-> Swappings / Permutations
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\only<2-5>{
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\begin{center}
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\begin{tabular}{r@ {\hspace{1mm}}l@ {\hspace{12mm}}r@ {\hspace{1mm}}l}
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\\
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\only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}%
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\only<3>{\smath{[b\!:=\!a]}}%
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\only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} &
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\onslide<2-5>{\smath{\lambda a.b}} &
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\only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}%
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\only<3>{\smath{[b\!:=\!a]}}%
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\only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} &
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\onslide<2-5>{\smath{\lambda c.b}}\\
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\onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda a.a}}\only<4-5>{\smath{\lambda b.a}} &
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\onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda c.a}}\only<4-5>{\smath{\lambda c.a}}\\
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\end{tabular}
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\end{center}\bigskip
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\onslide<4-5>{
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\begin{center}
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=0mm,fill=cream, ultra thick, draw=cream]
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{\begin{minipage}{8cm}
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\begin{tabular}{r@ {\hspace{3mm}}l}
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\smath{\swap{a}{b}\act t} $\;\dn$ & \alert{swap} {\bf all} occurrences of\\
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& \smath{b} and \smath{a} in \smath{t}
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\end{tabular}
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\end{minipage}};
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\end{tikzpicture}
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\end{center}}\bigskip
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\onslide<5>{
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Unlike for \smath{[b\!:=\!a]\act(-)}, for \smath{\swap{a}{b}\act (-)} we do
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have if \smath{t =_\alpha t'} then \smath{\pi \act t =_\alpha \pi \act t'.}}}
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\item<6-> Variables (or holes)\bigskip
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\begin{center}
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\onslide<7->{\mbox{}\hspace{-25mm}\smath{\lambda x\hspace{-0.5mm}s .}}
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\onslide<8-9>{\raisebox{-1.7mm}{\huge\smath{(}}}\raisebox{-4mm}{\begin{tikzpicture}
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\fill[blue] (0, 0) circle (5mm);
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\end{tikzpicture}}
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\onslide<8-9>{\smath{y\hspace{-0.5mm}s}{\raisebox{-1.7mm}{\huge\smath{)}}}}\bigskip
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\end{center}
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\only<8-9>{\smath{y\hspace{-0.5mm}s} are the parameters the hole can depend on\onslide<9->{, but
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then you need $\beta_0$-reduction\medskip
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\begin{center}
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\smath{(\lambda x. t) y \longrightarrow_{\beta_0} t[x:=y]}
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\end{center}}}
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\only<10>{we will record the information about which parameters a hole
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\alert{\bf cannot} depend on}
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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}<1-4>[c]
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\frametitle{Terms}
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\begin{tabular}{lll @ {\hspace{10mm}}lll}
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\onslide<1->{\pgfuseshading{smallbluesphere}} &
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\onslide<1->{\colorbox{cream}{\smath{\unit}}} &
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\onslide<1->{Units} &
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\onslide<2->{\pgfuseshading{smallbluesphere}} &
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\onslide<2->{\colorbox{cream}{\smath{a}}} &
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\onslide<2->{Atoms} \\[5mm]
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\onslide<1->{\pgfuseshading{smallbluesphere}} &
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\onslide<1->{\colorbox{cream}{\smath{\pair{t}{t'}}}} &
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\onslide<1->{Pairs} &
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\onslide<3->{\pgfuseshading{smallbluesphere}} &
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\onslide<3->{\colorbox{cream}{\smath{\abst{a}{t}}}} &
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\onslide<3->{Abstractions}\\[5mm]
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\onslide<1->{\pgfuseshading{smallbluesphere}} &
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\onslide<1->{\colorbox{cream}{\smath{\app{F}{t}}}} &
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\onslide<1->{Funct.} &
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\onslide<4->{\pgfuseshading{smallbluesphere}} &
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\onslide<4->{\colorbox{cream}{\smath{\pi\susp X}}} &
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\onslide<4->{Suspensions}
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\end{tabular}
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\only<2>{
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\begin{textblock}{13}(1.5,12)
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\small Atoms are constants \textcolor{darkgray}{(infinitely many of them)}
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\end{textblock}}
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\only<3>{
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\begin{textblock}{13}(1.5,12)
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\small \smath{\ulcorner \lambda\abst{a}{a}\urcorner \mapsto \text{fn\ }\abst{a}{a}}\\
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\small constructions like \smath{\text{fn\ }\abst{X}{X}} are not allowed
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\end{textblock}}
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\only<4>{
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\begin{textblock}{13}(1.5,12)
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\small \smath{X} is a variable standing for a term\\
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\small \smath{\pi} is an explicit permutation \smath{\swap{a_1}{b_1}\ldots\swap{a_n}{b_n}},
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waiting to be applied to the term that is substituted for \smath{X}
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\end{textblock}}
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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}<1-3>[c]
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\frametitle{Permutations}
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a permutation applied to a term
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\begin{center}
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\begin{tabular}{lrcl}
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\pgfuseshading{smallbluesphere} &
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\smath{[]\act c} & \smath{\dn} & \smath{c} \\
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\pgfuseshading{smallbluesphere} &
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\smath{\swap{a}{b}\!::\!\pi\act c} & \smath{\dn} &
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\smath{\begin{cases}
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a & \text{if}\;\pi\act c = b\\
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b & \text{if}\;\pi\act c = a\\
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\pi\act c & \text{otherwise}
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\end{cases}}\\
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\onslide<2->{\pgfuseshading{smallbluesphere}} &
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\onslide<2->{\smath{\pi\act\abst{a}{t}}} & \onslide<2->{\smath{\dn}} &
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\onslide<2->{\smath{\abst{\pi\act a}{\pi\act t}}}\\
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\onslide<3->{\pgfuseshading{smallbluesphere}} &
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\onslide<3->{\smath{\pi\act\pi'\act X}} & \onslide<3->{\smath{\dn}} &
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\onslide<3->{\smath{(\pi @ \pi')\act X}}\\
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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}<1-3>[c]
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\frametitle{Freshness Constraints}
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Recall \smath{\lambda a. \raisebox{-0.7mm}{\tikz \fill[blue] (0, 0) circle (2.5mm);}}
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\bigskip\pause
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We therefore will identify
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\begin{center}
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\smath{\text{fn\ } a. X \;\approx\; \text{fn\ } b. \alert<3->{\swap{a}{b}}\act X}
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\end{center}
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provided that `\smath{b} is fresh for \smath{X} --- (\smath{b\fresh X})',
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i.e., does not occur freely in any ground term that might be substituted for
|
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\smath{X}.\bigskip\pause
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If we know more about \smath{X}, e.g., if we knew that \smath{a\fresh X} and
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\smath{b\fresh X}, then we can replace\\ \smath{\swap{a}{b}\act X} by
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\smath{X}.
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1-4>[c]
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\frametitle{Equivalence Judgements}
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\alt<1>{Our equality is {\bf not} just}{but judgements}
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\begin{center}
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345 |
\begin{tabular}{rl}
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\colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} t \approx t'}} & \alert{$\alpha$-equivalence}\\[1mm]
|
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347 |
\onslide<4->{\colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} a \fresh t}}} &
|
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\onslide<4->{\alert{freshness}}
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\end{tabular}
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\end{center}
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\onslide<2->{
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where
|
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\begin{center}
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\smath{\nabla = \{a_1\fresh X_1,\ldots, a_n\fresh X_n\}}
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\end{center}
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is a finite set of \alert{freshness assumptions}.}
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\onslide<3->{
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360 |
\begin{center}
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\smath{\{a\fresh X,b\fresh X\} \vdash \text{fn\ } a. X \approx \text{fn\ } b. X}
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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>[c]
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\frametitle{Rules for Equivalence}
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373 |
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\begin{center}
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375 |
\begin{tabular}{c}
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Excerpt\\
|
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377 |
(i.e.~only the interesting rules)
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378 |
\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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\mode<presentation>{
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\begin{frame}<1>[c]
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389 |
\frametitle{Rules for Equivalence}
|
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390 |
|
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\begin{center}
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392 |
\begin{tabular}{c}
|
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\colorbox{cream}{\smath{\infer{\nabla \vdash a \approx a}{}}}\\[8mm]
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394 |
|
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395 |
\colorbox{cream}{%
|
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396 |
\smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{a}{t'}}
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{\nabla \vdash t \approx t'}}}\\[8mm]
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398 |
|
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399 |
\colorbox{cream}{%
|
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|
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\smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{b}{t'}}
|
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401 |
{a\not=b\;\; & \nabla \vdash t \approx \swap{a}{b}\act t'\;\;& \nabla \vdash a\fresh t'}}}
|
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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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|
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text_raw {*
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\mode<presentation>{
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\begin{frame}<1-3>[c]
|
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\frametitle{Rules for Equivalence}
|
|
2351
|
414 |
|
|
|
415 |
\begin{center}
|
|
2356
|
416 |
\colorbox{cream}{%
|
|
|
417 |
\smath{%
|
|
|
418 |
\infer{\nabla \vdash \pi\act X \approx \pi'\act X}
|
|
|
419 |
{\begin{array}{c}
|
|
|
420 |
(a\fresh X)\in\nabla\\
|
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|
421 |
\text{for all}\; a \;\text{with}\;\pi\act a \not= \pi'\act a
|
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|
422 |
\end{array}
|
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|
423 |
}}}
|
|
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|
424 |
\end{center}
|
|
|
425 |
|
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|
426 |
\onslide<2->{
|
|
|
427 |
for example\\[4mm]
|
|
|
428 |
|
|
|
429 |
\alt<2>{%
|
|
|
430 |
\begin{center}
|
|
|
431 |
\smath{\{a\fresh\!X, b\fresh\!X\} \vdash X \approx \swap{a}{b}\act X}
|
|
|
432 |
\end{center}}
|
|
|
433 |
{%
|
|
2351
|
434 |
\begin{center}
|
|
2356
|
435 |
\smath{\{a\fresh\!X, c\fresh\!X\} \vdash \swap{a}{c}\swap{a}{b}\act X \approx \swap{b}{c}\act X}
|
|
|
436 |
\end{center}}
|
|
2351
|
437 |
|
|
2356
|
438 |
\onslide<3->{
|
|
|
439 |
\begin{tabular}{@ {}lllll@ {}}
|
|
|
440 |
because &
|
|
|
441 |
\smath{\swap{a}{c}\swap{a}{b}}: &
|
|
|
442 |
\smath{a\mapsto b} &
|
|
|
443 |
\smath{\swap{b}{c}}: &
|
|
|
444 |
\smath{a\mapsto a}\\
|
|
|
445 |
& & \smath{b\mapsto c} & & \smath{b\mapsto c}\\
|
|
|
446 |
& & \smath{c\mapsto a} & & \smath{c\mapsto b}\\
|
|
|
447 |
\end{tabular}
|
|
|
448 |
disagree at \smath{a} and \smath{c}.}
|
|
|
449 |
}
|
|
|
450 |
|
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|
451 |
\end{frame}}
|
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452 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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453 |
*}
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454 |
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455 |
text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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457 |
\mode<presentation>{
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|
458 |
\begin{frame}<1>[c]
|
|
|
459 |
\frametitle{Rules for Freshness}
|
|
|
460 |
|
|
2351
|
461 |
\begin{center}
|
|
2356
|
462 |
\begin{tabular}{c}
|
|
|
463 |
Excerpt\\
|
|
|
464 |
(i.e.~only the interesting rules)
|
|
2351
|
465 |
\end{tabular}
|
|
2356
|
466 |
\end{center}
|
|
2351
|
467 |
|
|
|
468 |
\end{frame}}
|
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469 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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470 |
*}
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471 |
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472 |
text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
474 |
\mode<presentation>{
|
|
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|
475 |
\begin{frame}<1>[c]
|
|
|
476 |
\frametitle{Rules for Freshness}
|
|
2351
|
477 |
|
|
2356
|
478 |
\begin{center}
|
|
|
479 |
\begin{tabular}{c}
|
|
|
480 |
\colorbox{cream}{%
|
|
|
481 |
\smath{\infer{\nabla \vdash a\fresh b}{a\not= b}}}\\[5mm]
|
|
2351
|
482 |
|
|
2356
|
483 |
\colorbox{cream}{%
|
|
|
484 |
\smath{\infer{\nabla \vdash a\fresh\abst{a}{t}}{}}}\hspace{7mm}
|
|
|
485 |
\colorbox{cream}{%
|
|
|
486 |
\smath{\infer{\nabla \vdash a\fresh\abst{b}{t}}
|
|
|
487 |
{a\not= b\;\; & \nabla \vdash a\fresh t}}}\\[5mm]
|
|
|
488 |
|
|
|
489 |
\colorbox{cream}{%
|
|
|
490 |
\smath{\infer{\nabla \vdash a\fresh \pi\act X}
|
|
|
491 |
{(\pi^{-1}\act a\fresh X)\in\nabla}}}
|
|
2351
|
492 |
\end{tabular}
|
|
|
493 |
\end{center}
|
|
|
494 |
|
|
|
495 |
\end{frame}}
|
|
|
496 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
497 |
*}
|
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|
498 |
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|
499 |
text_raw {*
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|
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|
|
501 |
\mode<presentation>{
|
|
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|
502 |
\begin{frame}<1-4>[t]
|
|
|
503 |
\frametitle{$\approx$ is an Equivalence}
|
|
|
504 |
\mbox{}\\[5mm]
|
|
2351
|
505 |
|
|
|
506 |
\begin{center}
|
|
2356
|
507 |
\colorbox{cream}{\alert{Theorem:}
|
|
|
508 |
$\approx$ is an equivalence relation.}
|
|
|
509 |
\end{center}\bigskip
|
|
2351
|
510 |
|
|
2356
|
511 |
\only<1>{%
|
|
|
512 |
\begin{tabular}{ll}
|
|
|
513 |
(Reflexivity) & $\smath{\nabla\vdash t\approx t}$\\[2mm]
|
|
|
514 |
(Symmetry) & if $\smath{\nabla\vdash t_1\approx t_2}\;$
|
|
|
515 |
then $\;\smath{\nabla\vdash t_2\approx t_1}$\\[2mm]
|
|
|
516 |
(Transitivity) & if $\smath{\nabla\vdash t_1\approx t_2}\;$ and
|
|
|
517 |
$\;\smath{\nabla\vdash t_2\approx t_3}$\\
|
|
|
518 |
& then $\smath{\nabla\vdash t_1\approx t_3}$\\
|
|
|
519 |
\end{tabular}}
|
|
2351
|
520 |
|
|
2356
|
521 |
\only<2->{%
|
|
|
522 |
\begin{itemize}
|
|
|
523 |
\item<2-> \smath{\nabla \vdash t\approx t'} then \smath{\nabla \vdash \pi\act t\approx \pi\act t'}
|
|
2351
|
524 |
|
|
2356
|
525 |
\item<2-> \smath{\nabla \vdash a\fresh t} then
|
|
|
526 |
\smath{\nabla \vdash \pi\act a\fresh \pi\act t}
|
|
2351
|
527 |
|
|
2356
|
528 |
\item<3-> \smath{\nabla \vdash t\approx \pi\act t'} then
|
|
|
529 |
\smath{\nabla \vdash (\pi^{-1})\act t\approx t'}
|
|
|
530 |
|
|
|
531 |
\item<3-> \smath{\nabla \vdash a\fresh \pi\act t} then
|
|
|
532 |
\smath{\nabla \vdash (\pi^{-1})\act a\fresh t}
|
|
2351
|
533 |
|
|
2356
|
534 |
\item<4-> \smath{\nabla \vdash a\fresh t} and \smath{\nabla \vdash t\approx t'} then
|
|
|
535 |
\smath{\nabla \vdash a\fresh t'}
|
|
|
536 |
\end{itemize}
|
|
|
537 |
}
|
|
|
538 |
|
|
2351
|
539 |
\end{frame}}
|
|
|
540 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
541 |
*}
|
|
|
542 |
|
|
|
543 |
text_raw {*
|
|
|
544 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
545 |
\mode<presentation>{
|
|
2356
|
546 |
\begin{frame}<1-4>
|
|
|
547 |
\frametitle{Comparison $=_\alpha$}
|
|
2351
|
548 |
|
|
2356
|
549 |
Traditionally \smath{=_\alpha} is defined as
|
|
2351
|
550 |
|
|
|
551 |
\begin{center}
|
|
2356
|
552 |
\colorbox{cream}{%
|
|
|
553 |
\begin{minipage}{9cm}
|
|
|
554 |
\raggedright least congruence which identifies \smath{\abst{a}{t}}
|
|
|
555 |
with \smath{\abst{b}{[a:=b]t}} provided \smath{b} is not free
|
|
|
556 |
in \smath{t}
|
|
|
557 |
\end{minipage}}
|
|
|
558 |
\end{center}
|
|
2351
|
559 |
|
|
2356
|
560 |
where \smath{[a:=b]t} replaces all free occurrences of\\
|
|
|
561 |
\smath{a} by \smath{b} in \smath{t}.
|
|
|
562 |
\bigskip
|
|
2351
|
563 |
|
|
2356
|
564 |
\only<2>{%
|
|
|
565 |
\begin{textblock}{13}(1.2,10)
|
|
|
566 |
For \alert{ground} terms:
|
|
2351
|
567 |
|
|
2356
|
568 |
\begin{center}
|
|
|
569 |
\colorbox{cream}{%
|
|
|
570 |
\begin{minipage}{9.0cm}
|
|
|
571 |
\begin{tabular}{@ {}rl}
|
|
|
572 |
\underline{Theorem:}
|
|
|
573 |
& \smath{t=_\alpha t'\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset \vdash t\approx t'}\\[2mm]
|
|
2357
|
574 |
& \smath{a\not\in F\hspace{-0.9mm}A(t)\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset\vdash a\fresh t}
|
|
2351
|
575 |
\end{tabular}
|
|
2356
|
576 |
\end{minipage}}
|
|
|
577 |
\end{center}
|
|
2351
|
578 |
\end{textblock}}
|
|
2356
|
579 |
|
|
|
580 |
\only<3>{%
|
|
|
581 |
\begin{textblock}{13}(1.2,10)
|
|
|
582 |
In general \smath{=_\alpha} and \smath{\approx} are distinct!
|
|
|
583 |
\begin{center}
|
|
|
584 |
\colorbox{cream}{%
|
|
|
585 |
\begin{minipage}{6.0cm}
|
|
|
586 |
\smath{\abst{a}{X}=_\alpha \abst{b}{X}\;} but not\\[2mm]
|
|
|
587 |
\smath{\emptyset \vdash \abst{a}{X} \approx \abst{b}{X}\;} (\smath{a\not=b})
|
|
|
588 |
\end{minipage}}
|
|
|
589 |
\end{center}
|
|
|
590 |
\end{textblock}}
|
|
|
591 |
|
|
|
592 |
\only<4>{
|
|
|
593 |
\begin{textblock}{6}(1,2)
|
|
2351
|
594 |
\begin{tikzpicture}
|
|
2356
|
595 |
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
|
|
|
596 |
{\color{darkgray}
|
|
|
597 |
\begin{minipage}{10cm}\raggedright
|
|
|
598 |
That is a crucial point: if we had\\[-2mm]
|
|
|
599 |
\[\smath{\emptyset \vdash \abst{a}{X}\approx \abst{b}{X}}\mbox{,}\]
|
|
|
600 |
then applying $\smath{[X:=a]}$, $\smath{[X:=b]}$, $\ldots$\\
|
|
|
601 |
give two terms that are {\bf not} $\alpha$-equivalent.\\[3mm]
|
|
|
602 |
The freshness constraints $\smath{a\fresh X}$ and $\smath{b\fresh X}$
|
|
|
603 |
rule out the problematic substitutions. Therefore
|
|
|
604 |
|
|
|
605 |
\[\smath{\{a\fresh X,b\fresh X\} \vdash \abst{a}{X}\approx \abst{b}{X}}\]
|
|
|
606 |
|
|
|
607 |
does hold.
|
|
2351
|
608 |
\end{minipage}};
|
|
|
609 |
\end{tikzpicture}
|
|
|
610 |
\end{textblock}}
|
|
|
611 |
|
|
|
612 |
\end{frame}}
|
|
|
613 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
614 |
*}
|
|
|
615 |
|
|
|
616 |
text_raw {*
|
|
|
617 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
618 |
\mode<presentation>{
|
|
2356
|
619 |
\begin{frame}<1-9>
|
|
|
620 |
\frametitle{Substitution}
|
|
|
621 |
|
|
|
622 |
\begin{tabular}{l@ {\hspace{8mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l@ {}}
|
|
|
623 |
\pgfuseshading{smallbluesphere} &
|
|
|
624 |
\smath{\sigma(\abst{a}{t})} & \smath{\dn} & \smath{\abst{a}{\sigma(t)}}\\[2mm]
|
|
2351
|
625 |
|
|
2356
|
626 |
\pgfuseshading{smallbluesphere} &
|
|
|
627 |
\smath{\sigma(\pi\act X)} & \smath{\dn} &
|
|
|
628 |
\smath{\begin{cases}%
|
|
|
629 |
\pi\;\act\;\sigma(X) & \!\!\text{if\ } \sigma(X)\not=X\\
|
|
|
630 |
\pi\act X & \!\!\text{otherwise}%
|
|
|
631 |
\end{cases}}\\[6mm]
|
|
|
632 |
\end{tabular}\bigskip\bigskip
|
|
2351
|
633 |
|
|
2356
|
634 |
\pause
|
|
|
635 |
\only<2-5>{
|
|
|
636 |
\only<2->{for example}
|
|
|
637 |
\def\arraystretch{1.3}
|
|
|
638 |
\begin{tabular}{@ {\hspace{14mm}}l@ {\hspace{3mm}}l}
|
|
|
639 |
\onslide<2->{\textcolor{white}{$\Rightarrow$}} &
|
|
|
640 |
\onslide<2->{\alt<3>{\smath{\underline{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}}
|
|
|
641 |
{\smath{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}}\\
|
|
|
642 |
\onslide<3->{\smath{\Rightarrow}} &
|
|
|
643 |
\onslide<3->{\alt<3,4>{\smath{\abst{a}{\underline{\swap{a}{b}\act X[X:=\pair{b}{Y}]}}}}
|
|
|
644 |
{\smath{\abst{a}{\swap{a}{b}\act X}[X:=\pair{b}{Y}]}}}\\
|
|
|
645 |
\onslide<4->{\smath{\Rightarrow}} &
|
|
|
646 |
\onslide<4->{\alt<4>{\smath{\abst{a}{\swap{a}{b}\act \underline{\pair{b}{Y}}}}}
|
|
|
647 |
{\smath{\abst{a}{\underline{\swap{a}{b}}\act \pair{b}{Y}}}}}\\
|
|
|
648 |
\onslide<5->{\smath{\Rightarrow}} &
|
|
|
649 |
\onslide<5->{\smath{\abst{a}{\pair{a}{\swap{a}{b}\act Y}}}}
|
|
|
650 |
\end{tabular}}
|
|
2351
|
651 |
|
|
2356
|
652 |
\only<6->
|
|
|
653 |
{\begin{tabular}{l@ {\hspace{8mm}}l@ {}}
|
|
|
654 |
\pgfuseshading{smallbluesphere} &
|
|
|
655 |
if \smath{\nabla\vdash t\approx t'} and\hspace{-2mm}\mbox{}
|
|
|
656 |
\raisebox{-2.7mm}{
|
|
|
657 |
\alt<7>{\begin{tikzpicture}
|
|
|
658 |
\draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=3mm]
|
|
|
659 |
{\smath{\;\nabla'\vdash\sigma(\nabla)\;}};
|
|
|
660 |
\end{tikzpicture}}
|
|
|
661 |
{\begin{tikzpicture}
|
|
|
662 |
\draw (0,0) node[inner sep=1mm,fill=white, very thick, draw=white, rounded corners=3mm]
|
|
|
663 |
{\smath{\;\nabla'\vdash\sigma(\nabla)\;}};
|
|
|
664 |
\end{tikzpicture}}}\\
|
|
|
665 |
& then \smath{\nabla'\vdash\sigma(t)\approx\sigma(t')}
|
|
|
666 |
\end{tabular}}
|
|
2351
|
667 |
|
|
2356
|
668 |
\only<9>
|
|
|
669 |
{\begin{tabular}{l@ {\hspace{8mm}}l@ {}}
|
|
|
670 |
\\[-4mm]
|
|
|
671 |
\pgfuseshading{smallbluesphere} &
|
|
|
672 |
\smath{\sigma(\pi\act t)=\pi\act\sigma(t)}
|
|
|
673 |
\end{tabular}}
|
|
2351
|
674 |
|
|
|
675 |
|
|
2356
|
676 |
\only<7>{
|
|
|
677 |
\begin{textblock}{6}(10,10.5)
|
|
2351
|
678 |
\begin{tikzpicture}
|
|
2356
|
679 |
\draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=2mm]
|
|
|
680 |
{\color{darkgray}
|
|
|
681 |
\begin{minipage}{3.8cm}\raggedright
|
|
|
682 |
this means\\[1mm]
|
|
|
683 |
\smath{\nabla'\vdash a\fresh\sigma(X)}\\[1mm]
|
|
|
684 |
holds for all\\[1mm]
|
|
|
685 |
\smath{(a\fresh X)\in\nabla}
|
|
2351
|
686 |
\end{minipage}};
|
|
|
687 |
\end{tikzpicture}
|
|
|
688 |
\end{textblock}}
|
|
|
689 |
|
|
|
690 |
\end{frame}}
|
|
|
691 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
692 |
*}
|
|
|
693 |
|
|
|
694 |
text_raw {*
|
|
|
695 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
696 |
\mode<presentation>{
|
|
2356
|
697 |
\begin{frame}<1->
|
|
|
698 |
\frametitle{Equational Problems}
|
|
2351
|
699 |
|
|
2356
|
700 |
An equational problem
|
|
|
701 |
\[
|
|
|
702 |
\colorbox{cream}{\smath{t \eqprob t'}}
|
|
|
703 |
\]
|
|
|
704 |
is \alert{solved} by
|
|
2351
|
705 |
|
|
|
706 |
\begin{center}
|
|
2356
|
707 |
\begin{tabular}{ll}
|
|
|
708 |
\pgfuseshading{smallbluesphere} & a substitution \smath{\sigma} (terms for variables)\\[3mm]
|
|
|
709 |
\pgfuseshading{smallbluesphere} & {\bf and} a set of freshness assumptions \smath{\nabla}
|
|
|
710 |
\end{tabular}
|
|
2351
|
711 |
\end{center}
|
|
|
712 |
|
|
2356
|
713 |
so that \smath{\nabla\vdash \sigma(t)\approx \sigma(t')}.
|
|
2351
|
714 |
|
|
|
715 |
|
|
|
716 |
\end{frame}}
|
|
|
717 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
718 |
*}
|
|
|
719 |
|
|
|
720 |
text_raw {*
|
|
|
721 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
722 |
\mode<presentation>{
|
|
|
723 |
\begin{frame}<1->
|
|
2357
|
724 |
|
|
|
725 |
Unifying equations may entail solving
|
|
|
726 |
\alert{freshness problems}.
|
|
|
727 |
|
|
|
728 |
\bigskip
|
|
|
729 |
|
|
|
730 |
E.g.~assuming that \smath{a\not=a'}, then
|
|
|
731 |
\[
|
|
|
732 |
\smath{\abst{a}{t}\eqprob \abst{a'}{t'}}
|
|
|
733 |
\]
|
|
|
734 |
can only be solved if
|
|
|
735 |
\[
|
|
|
736 |
\smath{t\eqprob \swap{a}{a'}\act t'} \quad\text{\emph{and}}\quad
|
|
|
737 |
\smath{a\freshprob t'}
|
|
|
738 |
\]
|
|
|
739 |
can be solved.
|
|
|
740 |
|
|
|
741 |
\end{frame}}
|
|
|
742 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
743 |
*}
|
|
|
744 |
|
|
|
745 |
text_raw {*
|
|
|
746 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
747 |
\mode<presentation>{
|
|
|
748 |
\begin{frame}<1->
|
|
|
749 |
\frametitle{Freshness Problems}
|
|
|
750 |
|
|
|
751 |
A freshness problem
|
|
|
752 |
\[
|
|
|
753 |
\colorbox{cream}{\smath{a \freshprob t}}
|
|
|
754 |
\]
|
|
|
755 |
is \alert{solved} by
|
|
|
756 |
|
|
|
757 |
\begin{center}
|
|
|
758 |
\begin{tabular}{ll}
|
|
|
759 |
\pgfuseshading{smallbluesphere} & a substitution \smath{\sigma}\\[3mm]
|
|
|
760 |
\pgfuseshading{smallbluesphere} & and a set of freshness assumptions \smath{\nabla}
|
|
|
761 |
\end{tabular}
|
|
|
762 |
\end{center}
|
|
|
763 |
|
|
|
764 |
so that \smath{\nabla\vdash a \fresh \sigma(t)}.
|
|
|
765 |
|
|
|
766 |
\end{frame}}
|
|
|
767 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
768 |
*}
|
|
|
769 |
|
|
|
770 |
text_raw {*
|
|
|
771 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
772 |
\mode<presentation>{
|
|
|
773 |
\begin{frame}<1-3>
|
|
|
774 |
\frametitle{Existence of MGUs}
|
|
|
775 |
|
|
|
776 |
\underline{Theorem}: There is an algorithm which, given a nominal
|
|
|
777 |
unification problem \smath{P}, decides whether\\
|
|
|
778 |
or not it has a solution \smath{(\sigma,\nabla)}, and returns a \\
|
|
|
779 |
\alert{most general} one if it does.\bigskip\bigskip
|
|
|
780 |
|
|
|
781 |
\only<3>{
|
|
|
782 |
Proof: one can reduce all the equations to `solved form'
|
|
|
783 |
first (creating a substitution), and then solve the freshness
|
|
|
784 |
problems (easy).}
|
|
|
785 |
|
|
|
786 |
\only<2>{
|
|
|
787 |
\begin{textblock}{6}(2.5,9.5)
|
|
|
788 |
\begin{tikzpicture}
|
|
|
789 |
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
|
|
|
790 |
{\color{darkgray}
|
|
|
791 |
\begin{minipage}{8cm}\raggedright
|
|
|
792 |
\alert{most general:}\\
|
|
|
793 |
straightforward definition\\
|
|
|
794 |
``if\hspace{-0.5mm}f there exists a \smath{\tau} such that \ldots''
|
|
|
795 |
\end{minipage}};
|
|
|
796 |
\end{tikzpicture}
|
|
|
797 |
\end{textblock}}
|
|
|
798 |
|
|
|
799 |
\end{frame}}
|
|
|
800 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
801 |
*}
|
|
|
802 |
|
|
|
803 |
text_raw {*
|
|
|
804 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
805 |
\mode<presentation>{
|
|
|
806 |
\begin{frame}<1>
|
|
|
807 |
\frametitle{Remember the Quiz?}
|
|
|
808 |
|
|
|
809 |
\textcolor{gray}{Assuming that $a$ and $b$ are distinct variables,\\
|
|
|
810 |
is it possible to find $\lambda$-terms $M_1$ to $M_7$
|
|
|
811 |
that make the following pairs $\alpha$-equivalent?}
|
|
|
812 |
|
|
|
813 |
\begin{tabular}{@ {\hspace{14mm}}p{12cm}}
|
|
|
814 |
\begin{itemize}
|
|
|
815 |
\item \smath{\lambda a.\lambda b. (M_1\,b)\;} and
|
|
|
816 |
\smath{\lambda b.\lambda a. (a\,M_1)\;}
|
|
|
817 |
|
|
|
818 |
\item \textcolor{gray}{$\lambda a.\lambda b. (M_2\,b)\;$ and
|
|
|
819 |
$\lambda b.\lambda a. (a\,M_3)\;$}
|
|
|
820 |
|
|
|
821 |
\item \textcolor{gray}{$\lambda a.\lambda b. (b\,M_4)\;$ and
|
|
|
822 |
$\lambda b.\lambda a. (a\,M_5)\;$}
|
|
|
823 |
|
|
|
824 |
\item \smath{\lambda a.\lambda b. (b\,M_6)\;} and
|
|
|
825 |
\smath{\lambda a.\lambda a. (a\,M_7)\;}
|
|
|
826 |
\end{itemize}
|
|
|
827 |
\end{tabular}
|
|
|
828 |
|
|
|
829 |
\textcolor{gray}{If there is one solution for a pair, can you
|
|
|
830 |
describe all its solutions?}
|
|
|
831 |
|
|
|
832 |
|
|
|
833 |
\end{frame}}
|
|
|
834 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
835 |
*}
|
|
|
836 |
|
|
|
837 |
text_raw {*
|
|
|
838 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
839 |
\mode<presentation>{
|
|
|
840 |
\begin{frame}<1->
|
|
|
841 |
\frametitle{Answers to the Quiz}
|
|
|
842 |
\small
|
|
|
843 |
\def\arraystretch{1.6}
|
|
|
844 |
\begin{tabular}{c@ {\hspace{2mm}}l}
|
|
|
845 |
& \only<1>{\smath{\lambda a.\lambda b. (M_1\,b)\;} and \smath{\;\lambda b.\lambda a. (a\,M_1)}}%
|
|
|
846 |
\only<2->{\smath{\abst{a}{\abst{b}{\pair{M_1}{b}}} \;\eqprob\; \abst{b}{\abst{a}{\pair{a}{M_1}}}}}\\
|
|
|
847 |
|
|
|
848 |
\onslide<3->{\smath{\redu{\id}}} &
|
|
|
849 |
\only<3>{\smath{\abst{b}{\pair{M_1}{b}} \eqprob
|
|
|
850 |
\alert{\swap{a}{b}} \act \abst{a}{\pair{a}{M_1}}\;,\;a\freshprob \abst{a}{\pair{a}{M_1}}}}%
|
|
|
851 |
\only<4->{\smath{\abst{b}{\pair{M_1}{b}} \eqprob \abst{b}{\pair{b}{\swap{a}{b}\act M_1}}\;,\
|
|
|
852 |
a\freshprob \abst{a}{\pair{a}{M_1}}}}\\
|
|
|
853 |
|
|
|
854 |
\onslide<5->{\smath{\redu{\id}}} &
|
|
|
855 |
\only<5->{\smath{\pair{M_1}{b} \eqprob \pair{b}{\swap{a}{b}\act M_1}\;,\;%
|
|
|
856 |
a\freshprob \abst{a}{\pair{a}{M_1}}}}\\
|
|
|
857 |
|
|
|
858 |
\onslide<6->{\smath{\redu{\id}}} &
|
|
|
859 |
\only<6->{\smath{M_1 \eqprob b \;,\; b \eqprob \swap{a}{b}\act M_1\;,\;%
|
|
|
860 |
a\freshprob \abst{a}{\pair{a}{M_1}}}}\\
|
|
|
861 |
|
|
|
862 |
\onslide<7->{\smath{\redu{[M_1:=b]}}} &
|
|
|
863 |
\only<7>{\smath{b \eqprob \swap{a}{b}\act \alert{b}\;,\;%
|
|
|
864 |
a\freshprob \abst{a}{\pair{a}{\alert{b}}}}}%
|
|
|
865 |
\only<8->{\smath{b \eqprob a\;,\; a\freshprob \abst{a}{\pair{a}{b}}}}\\
|
|
|
866 |
|
|
|
867 |
\onslide<9->{\smath{\redu{}}} &
|
|
|
868 |
\only<9->{\smath{F\hspace{-0.5mm}AIL}}
|
|
|
869 |
\end{tabular}
|
|
|
870 |
|
|
|
871 |
\only<10>{
|
|
|
872 |
\begin{textblock}{6}(2,11)
|
|
|
873 |
\begin{tikzpicture}
|
|
|
874 |
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
|
|
|
875 |
{\color{darkgray}
|
|
|
876 |
\begin{minipage}{9cm}\raggedright
|
|
|
877 |
\smath{\lambda a.\lambda b. (M_1\,b)} \smath{=_\alpha}
|
|
|
878 |
\smath{\lambda b.\lambda a. (a\,M_1)} has no solution
|
|
|
879 |
\end{minipage}};
|
|
|
880 |
\end{tikzpicture}
|
|
|
881 |
\end{textblock}}
|
|
|
882 |
|
|
|
883 |
\end{frame}}
|
|
|
884 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
885 |
*}
|
|
|
886 |
|
|
|
887 |
text_raw {*
|
|
|
888 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
889 |
\mode<presentation>{
|
|
|
890 |
\begin{frame}<1->
|
|
|
891 |
\frametitle{Answers to the Quiz}
|
|
|
892 |
\small
|
|
|
893 |
\def\arraystretch{1.6}
|
|
|
894 |
\begin{tabular}{c@ {\hspace{2mm}}l}
|
|
|
895 |
& \only<1>{\smath{\lambda a.\lambda b. (b\,M_6)\;} and \smath{\;\lambda a.\lambda a. (a\,M_7)}}%
|
|
|
896 |
\only<2->{\smath{\abst{a}{\abst{b}{\pair{b}{M_6}}} \;\eqprob\; \abst{a}{\abst{a}{\pair{a}{M_7}}}}}\\
|
|
|
897 |
|
|
|
898 |
\onslide<3->{\smath{\redu{\id}}} &
|
|
|
899 |
\only<3->{\smath{\abst{b}{\pair{b}{M_6}} \eqprob \abst{a}{\pair{a}{M_7}}}}\\
|
|
|
900 |
|
|
|
901 |
\onslide<4->{\smath{\redu{\id}}} &
|
|
|
902 |
\only<4->{\smath{\pair{b}{M_6} \eqprob \pair{b}{\swap{b}{a}\act M_7}\;,\;b\freshprob\pair{a}{M_7}}}\\
|
|
|
903 |
|
|
|
904 |
\onslide<5->{\smath{\redu{\id}}} &
|
|
|
905 |
\only<5->{\smath{b\eqprob b\;,\; M_6 \eqprob \swap{b}{a}\act M_7\;,\;%
|
|
|
906 |
b\freshprob \pair{a}{M_7}}}\\
|
|
|
907 |
|
|
|
908 |
\onslide<6->{\smath{\redu{\id}}} &
|
|
|
909 |
\only<6->{\smath{M_6 \eqprob \swap{b}{a}\act M_7\;,\;%
|
|
|
910 |
b\freshprob \pair{a}{M_7}}}\\
|
|
|
911 |
|
|
|
912 |
\onslide<7->{\makebox[0mm]{\smath{\redu{[M_6:=\swap{b}{a}\act M_7]}}}} &
|
|
|
913 |
\only<7->{\smath{\qquad b\freshprob \pair{a}{M_7}}}\\
|
|
|
914 |
|
|
|
915 |
\onslide<8->{\smath{\redu{\varnothing}}} &
|
|
|
916 |
\only<8->{\smath{b\freshprob a\;,\;b\freshprob M_7}}\\
|
|
|
917 |
|
|
|
918 |
\onslide<9->{\smath{\redu{\varnothing}}} &
|
|
|
919 |
\only<9->{\smath{b\freshprob M_7}}\\
|
|
|
920 |
|
|
|
921 |
\onslide<10->{\makebox[0mm]{\smath{\redu{\{b\fresh M_7\}}}}} &
|
|
|
922 |
\only<10->{\smath{\;\;\varnothing}}\\
|
|
|
923 |
|
|
|
924 |
\end{tabular}
|
|
|
925 |
|
|
|
926 |
\only<10>{
|
|
|
927 |
\begin{textblock}{6}(6,9)
|
|
|
928 |
\begin{tikzpicture}
|
|
|
929 |
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
|
|
|
930 |
{\color{darkgray}
|
|
|
931 |
\begin{minipage}{7cm}\raggedright
|
|
|
932 |
\smath{\lambda a.\lambda b. (b\,M_6)\;} \smath{=_\alpha}
|
|
|
933 |
\smath{\;\lambda a.\lambda a. (a\,M_7)}\\[2mm]
|
|
|
934 |
we can take \smath{M_7} to be any $\lambda$-term that does not
|
|
|
935 |
contain free occurrences of \smath{b}, so long as we take \smath{M_6} to
|
|
|
936 |
be the result of swapping all occurrences of \smath{b} and \smath{a}
|
|
|
937 |
throughout \smath{M_7}
|
|
|
938 |
\end{minipage}};
|
|
|
939 |
\end{tikzpicture}
|
|
|
940 |
\end{textblock}}
|
|
|
941 |
|
|
|
942 |
\end{frame}}
|
|
|
943 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
944 |
*}
|
|
|
945 |
|
|
|
946 |
text_raw {*
|
|
|
947 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
948 |
\mode<presentation>{
|
|
|
949 |
\begin{frame}<1->
|
|
|
950 |
\frametitle{Properties}
|
|
|
951 |
|
|
|
952 |
\begin{itemize}
|
|
|
953 |
\item An interesting feature of nominal unification is that it
|
|
|
954 |
does not need to create new atoms.\bigskip
|
|
|
955 |
|
|
|
956 |
\begin{center}\small
|
|
|
957 |
\colorbox{cream}{
|
|
|
958 |
\smath{\{a.t \eqprob b.t'\}\cup P \redu{\id} \{t \eqprob \swap{a}{b}\act t', a \freshprob t'\} \cup P}}
|
|
|
959 |
\end{center}\bigskip\bigskip
|
|
|
960 |
\pause
|
|
|
961 |
|
|
|
962 |
\item The alternative rule
|
|
|
963 |
|
|
|
964 |
%\begin{center}\small
|
|
|
965 |
%\colorbox{cream}{
|
|
|
966 |
%\smath{\{a.t \eqprob b.t'\}\cup P \redu{\id} \{\swap{a}{c}\act t \eqprob \swap{b}{c}\act t', a \freshprob t'\} \cup P}}
|
|
|
967 |
%\end{center}
|
|
|
968 |
|
|
|
969 |
|
|
|
970 |
leads to a more complicated notion of mgu.
|
|
|
971 |
\end{itemize}
|
|
|
972 |
|
|
|
973 |
\end{frame}}
|
|
|
974 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
975 |
*}
|
|
|
976 |
|
|
|
977 |
text_raw {*
|
|
|
978 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
979 |
\mode<presentation>{
|
|
|
980 |
\begin{frame}<1-3>
|
|
|
981 |
\frametitle{Is it Useful?}
|
|
|
982 |
|
|
|
983 |
Yes. $\alpha$Prolog by James Cheney (main developer)\bigskip\bigskip
|
|
|
984 |
|
|
|
985 |
\color{darkgray}
|
|
|
986 |
\begin{tabular}{@ {}l}
|
|
|
987 |
type (Gamma, var(X), T) :- member (X,T) Gamma.\smallskip\medskip\\
|
|
|
988 |
|
|
|
989 |
type (Gamma, app(M, N), T') :-\\
|
|
|
990 |
\hspace{3cm}type (Gamma, M, arrow(T, T')),\\
|
|
|
991 |
\hspace{3cm}type (Gamma, N, T).\smallskip\medskip\\
|
|
|
992 |
|
|
|
993 |
type (Gamma, lam(\alert{x.M}), arrow(T, T')) / \alert{x \# Gamma} :-\\
|
|
|
994 |
\hspace{3cm}type ((x, T)::Gamma, M, T').\smallskip\medskip\\
|
|
|
995 |
|
|
|
996 |
member X X::Tail.\\
|
|
|
997 |
member X Y::Tail :- member X Tail.\\
|
|
|
998 |
\end{tabular}
|
|
|
999 |
|
|
|
1000 |
\only<2->{
|
|
|
1001 |
\begin{textblock}{6}(1.5,0.5)
|
|
|
1002 |
\begin{tikzpicture}
|
|
|
1003 |
\draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
|
|
|
1004 |
{\color{darkgray}
|
|
|
1005 |
\begin{minipage}{9cm}\raggedright
|
|
|
1006 |
{\bf Problem:} If we ask whether
|
|
|
1007 |
|
|
|
1008 |
\begin{center}
|
|
|
1009 |
?- type ([(x, T')], lam(x.Var(x)), T)
|
|
|
1010 |
\end{center}
|
|
|
1011 |
|
|
|
1012 |
is typable, we expect an answer for T.\bigskip
|
|
|
1013 |
|
|
|
1014 |
\onslide<3>{Solution: Before back-chaining freshen all variables and atoms
|
|
|
1015 |
in a program (clause).}
|
|
|
1016 |
\end{minipage}};
|
|
|
1017 |
\end{tikzpicture}
|
|
|
1018 |
\end{textblock}}
|
|
|
1019 |
|
|
|
1020 |
\end{frame}}
|
|
|
1021 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1022 |
*}
|
|
|
1023 |
|
|
|
1024 |
text_raw {*
|
|
|
1025 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1026 |
\mode<presentation>{
|
|
|
1027 |
\begin{frame}<1->
|
|
|
1028 |
\frametitle{Equivariant Unification}
|
|
|
1029 |
|
|
|
1030 |
James Cheney
|
|
|
1031 |
|
|
|
1032 |
\begin{center}
|
|
|
1033 |
\colorbox{cream}{
|
|
|
1034 |
\smath{t \eqprob t' \redu{\nabla, \sigma, \pi}
|
|
|
1035 |
\nabla \vdash \sigma(t) \approx \pi \act \sigma(t')}}
|
|
|
1036 |
\end{center}\bigskip\bigskip
|
|
|
1037 |
\pause
|
|
|
1038 |
|
|
|
1039 |
He also showed that this is undecidable in general. :(
|
|
|
1040 |
|
|
|
1041 |
\end{frame}}
|
|
|
1042 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1043 |
*}
|
|
|
1044 |
|
|
|
1045 |
text_raw {*
|
|
|
1046 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1047 |
\mode<presentation>{
|
|
|
1048 |
\begin{frame}<1->
|
|
|
1049 |
\frametitle{Taking Atoms as Variables}
|
|
|
1050 |
|
|
|
1051 |
Instead of \smath{a.X}, have \smath{A.X}.\bigskip
|
|
|
1052 |
\pause
|
|
|
1053 |
|
|
|
1054 |
Unfortunately this breaks the mgu-property:
|
|
|
1055 |
|
|
|
1056 |
\begin{center}
|
|
|
1057 |
|
|
|
1058 |
\end{center}
|
|
|
1059 |
|
|
|
1060 |
\end{frame}}
|
|
|
1061 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1062 |
*}
|
|
|
1063 |
|
|
|
1064 |
text_raw {*
|
|
|
1065 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1066 |
\mode<presentation>{
|
|
|
1067 |
\begin{frame}<1>
|
|
|
1068 |
\frametitle{HOPU vs. NOMU}
|
|
|
1069 |
|
|
|
1070 |
\begin{itemize}
|
|
|
1071 |
\item James Cheney showed\bigskip
|
|
|
1072 |
\begin{center}
|
|
|
1073 |
\colorbox{cream}{\smath{HOPU \Rightarrow NOMU}}
|
|
|
1074 |
\end{center}\bigskip
|
|
|
1075 |
|
|
|
1076 |
\item Levi\bigskip
|
|
|
1077 |
\begin{center}
|
|
|
1078 |
\colorbox{cream}{\smath{HOPU \Leftarrow NOMU}}
|
|
|
1079 |
\end{center}\bigskip
|
|
|
1080 |
\end{itemize}
|
|
|
1081 |
|
|
|
1082 |
The translations `explode' the problems quadratically.
|
|
|
1083 |
|
|
|
1084 |
\end{frame}}
|
|
|
1085 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1086 |
*}
|
|
|
1087 |
|
|
|
1088 |
text_raw {*
|
|
|
1089 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1090 |
\mode<presentation>{
|
|
|
1091 |
\begin{frame}<1>
|
|
|
1092 |
\small\tt
|
|
|
1093 |
|
|
|
1094 |
\begin{minipage}{13cm}
|
|
|
1095 |
\begin{tabular}{@ {\hspace{-2mm}}p{11.5cm}}
|
|
|
1096 |
\\
|
|
|
1097 |
From: Zhenyu Qian <zhqian@microsoft.com>\\
|
|
|
1098 |
To: Christian Urban <urbanc@in.tum.de>\\
|
|
|
1099 |
Subject: RE: Linear Higher-Order Pattern Unification\\
|
|
|
1100 |
Date: Mon, 14 Apr 2008 09:56:47 +0800\\
|
|
|
1101 |
\\
|
|
|
1102 |
Hi Christian,\\
|
|
|
1103 |
\\
|
|
|
1104 |
Thanks for your interests and asking. I know that that paper is complex. As
|
|
|
1105 |
I told Tobias when we met last time, I have raised the question to myself
|
|
|
1106 |
many times whether the proof could have some flaws, and so making it through
|
|
|
1107 |
a theorem prover would definitely bring piece to my mind (no matter what
|
|
|
1108 |
the result would be). The only problem for me is the time.\\
|
|
|
1109 |
\ldots\\
|
|
|
1110 |
|
|
|
1111 |
Thanks/Zhenyu
|
|
|
1112 |
\end{tabular}
|
|
|
1113 |
\end{minipage}
|
|
|
1114 |
|
|
|
1115 |
\end{frame}}
|
|
|
1116 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1117 |
*}
|
|
|
1118 |
|
|
|
1119 |
text_raw {*
|
|
|
1120 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1121 |
\mode<presentation>{
|
|
|
1122 |
\begin{frame}<1>
|
|
|
1123 |
\frametitle{Complexity}
|
|
|
1124 |
|
|
|
1125 |
\begin{itemize}
|
|
|
1126 |
\item Maribel Fernandez
|
|
|
1127 |
|
|
|
1128 |
\item Levi
|
|
|
1129 |
|
|
|
1130 |
|
|
|
1131 |
\end{itemize}
|
|
|
1132 |
|
|
|
1133 |
\end{frame}}
|
|
|
1134 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1135 |
*}
|
|
|
1136 |
|
|
|
1137 |
|
|
|
1138 |
text_raw {*
|
|
|
1139 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1140 |
\mode<presentation>{
|
|
|
1141 |
\begin{frame}<1->[c]
|
|
2356
|
1142 |
\frametitle{Conclusion}
|
|
2351
|
1143 |
|
|
|
1144 |
\begin{itemize}
|
|
2357
|
1145 |
\item Nominal Unification is a completely first-order
|
|
|
1146 |
language, but implements unification modulo $\alpha$.\medskip\pause
|
|
2351
|
1147 |
|
|
2357
|
1148 |
\item NOMU has been applied in term-rewriting and
|
|
|
1149 |
logic programming. I hope it will also be used in typing
|
|
|
1150 |
systems.\medskip\pause
|
|
|
1151 |
|
|
|
1152 |
\item NOMU and HOPU are `equivalent' (it took a long time
|
|
|
1153 |
and considerable time to find this out).\medskip\pause
|
|
|
1154 |
|
|
|
1155 |
\item The question about complexity is still an ongoing
|
|
|
1156 |
story.\medskip
|
|
2351
|
1157 |
\end{itemize}
|
|
|
1158 |
|
|
|
1159 |
|
|
|
1160 |
\end{frame}}
|
|
|
1161 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1162 |
*}
|
|
|
1163 |
|
|
2357
|
1164 |
text_raw {*
|
|
|
1165 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1166 |
\mode<presentation>{
|
|
|
1167 |
\begin{frame}<1>[c]
|
|
|
1168 |
\frametitle{
|
|
|
1169 |
\begin{tabular}{c}
|
|
|
1170 |
\mbox{}\\[23mm]
|
|
|
1171 |
\alert{\LARGE Thank you very much}\\
|
|
|
1172 |
\alert{\Large Questions?}
|
|
|
1173 |
\end{tabular}}
|
|
|
1174 |
|
|
|
1175 |
\end{frame}}
|
|
|
1176 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1177 |
*}
|
|
|
1178 |
|
|
|
1179 |
text_raw {*
|
|
|
1180 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1181 |
\mode<presentation>{
|
|
|
1182 |
\begin{frame}<1-3>
|
|
|
1183 |
\frametitle{Most General Unifiers}
|
|
|
1184 |
|
|
|
1185 |
\underline{Definition}: For a unification problem
|
|
|
1186 |
\smath{P}, a solution \smath{(\sigma_1,\nabla_1)} is
|
|
|
1187 |
\alert{more general} than another solution
|
|
|
1188 |
\smath{(\sigma_2,\nabla_2)}, iff~there exists a substitution
|
|
|
1189 |
\smath{\tau} with
|
|
|
1190 |
|
|
|
1191 |
\begin{center}
|
|
|
1192 |
\begin{tabular}{ll}
|
|
|
1193 |
\pgfuseshading{smallbluesphere} &
|
|
|
1194 |
\alt<2>{\smath{\alert{\nabla_2\vdash\tau(\nabla_1)}}}
|
|
|
1195 |
{\smath{\nabla_2\vdash\tau(\nabla_1)}}\\
|
|
|
1196 |
\pgfuseshading{smallbluesphere} &
|
|
|
1197 |
\alt<3>{\smath{\alert{\nabla_2\vdash\sigma_2\approx \tau\circ\sigma_1}}}
|
|
|
1198 |
{\smath{\nabla_2\vdash\sigma_2\approx \tau\circ\sigma_1}}
|
|
|
1199 |
\end{tabular}
|
|
|
1200 |
\end{center}
|
|
|
1201 |
|
|
|
1202 |
\only<2>{
|
|
|
1203 |
\begin{textblock}{13}(1.5,10.5)
|
|
|
1204 |
\smath{\nabla_2\vdash a\fresh \sigma(X)} holds for all
|
|
|
1205 |
\smath{(a\fresh X)\in\nabla_1}
|
|
|
1206 |
\end{textblock}}
|
|
|
1207 |
|
|
|
1208 |
\only<3>{
|
|
|
1209 |
\begin{textblock}{11}(1.5,10.5)
|
|
|
1210 |
\smath{\nabla_2\vdash \sigma_2(X)\approx
|
|
|
1211 |
\sigma(\sigma_1(X))}
|
|
|
1212 |
holds for all
|
|
|
1213 |
\smath{X\in\text{dom}(\sigma_2)\cup\text{dom}(\sigma\circ\sigma_1)}
|
|
|
1214 |
\end{textblock}}
|
|
|
1215 |
|
|
|
1216 |
\end{frame}}
|
|
|
1217 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1218 |
*}
|
|
|
1219 |
|
|
2351
|
1220 |
(*<*)
|
|
|
1221 |
end
|
|
|
1222 |
(*>*) |