diff -r a6f3e1b08494 -r b6873d123f9b Slides/Slides3.thy --- a/Slides/Slides3.thy Sat May 12 21:05:59 2012 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1243 +0,0 @@ -(*<*) -theory Slides3 -imports "~~/src/HOL/Library/LaTeXsugar" "Nominal" -begin - -declare [[show_question_marks = false]] - -notation (latex output) - set ("_") and - Cons ("_::/_" [66,65] 65) - -(*>*) - -text_raw {* - \renewcommand{\slidecaption}{UNIF, Edinburgh, 14.~July 2010} - - \newcommand{\abst}[2]{#1.#2}% atom-abstraction - \newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing - \newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions - \newcommand{\unit}{\langle\rangle}% unit - \newcommand{\app}[2]{#1\,#2}% application - \newcommand{\eqprob}{\mathrel{{\approx}?}} - \newcommand{\freshprob}{\mathrel{\#?}} - \newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction - \newcommand{\id}{\varepsilon}% identity substitution - - \pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}% - {rgb(0mm)=(0,0,0.9); - rgb(0.9mm)=(0,0,0.7); - rgb(1.3mm)=(0,0,0.5); - rgb(1.4mm)=(1,1,1)} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[c] - \frametitle{Quiz} - - Assuming that \smath{a} and \smath{b} are distinct variables,\\ - is it possible to find $\lambda$-terms \smath{M_1} to \smath{M_7} - that make the following pairs \alert{$\alpha$-equivalent}? - - \begin{tabular}{@ {\hspace{14mm}}p{12cm}} - \begin{itemize} - \item \smath{\lambda a.\lambda b. (M_1\,b)\;} and - \smath{\lambda b.\lambda a. (a\,M_1)\;} - - \item \smath{\lambda a.\lambda b. (M_2\,b)\;} and - \smath{\lambda b.\lambda a. (a\,M_3)\;} - - \item \smath{\lambda a.\lambda b. (b\,M_4)\;} and - \smath{\lambda b.\lambda a. (a\,M_5)\;} - - \item \smath{\lambda a.\lambda b. (b\,M_6)\;} and - \smath{\lambda a.\lambda a. (a\,M_7)\;} - \end{itemize} - \end{tabular} - - If there is one solution for a pair, can you describe all its solutions? - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[t] - \frametitle{% - \begin{tabular}{@ {\hspace{-3mm}}c@ {}} - \\ - \huge Nominal Unification\\[-2mm] - \Large Hitting a Sweet Spot\\[5mm] - \end{tabular}} - \begin{center} - Christian Urban - \end{center} - \begin{center} - \small initial spark from Roy Dyckhoff in November 2001\\[0mm] - \small joint work with Andy Pitts and Jamie Gabbay\\[0mm] - \end{center} - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -*} -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-4>[c] - \frametitle{One Motivation} - - \onslide<2->{Typing implemented in Prolog \textcolor{darkgray}{(from a textbook)}}\bigskip\\ - - \onslide<3->{\color{darkgray} - \begin{tabular}{l} - type (Gamma, var(X), T) :- member (X,T) Gamma.\smallskip\medskip\\ - - type (Gamma, app(M, N), T') :-\\ - \hspace{3cm}type (Gamma, M, arrow(T, T')),\\ - \hspace{3cm}type (Gamma, N, T).\smallskip\medskip\\ - - type (Gamma, lam(X, M), arrow(T, T')) :-\\ - \hspace{3cm}type ((X, T)::Gamma, M, T').\smallskip\medskip\\ - - member X X::Tail.\\ - member X Y::Tail :- member X Tail.\\ - \end{tabular}} - - \only<4>{ - \begin{textblock}{6}(2.5,2) - \begin{tikzpicture} - \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] - {\color{darkgray} - \begin{minipage}{8cm}\raggedright - The problem is that \smath{\lambda x.\lambda x. (x\;x)} - will have the types - \begin{center} - \begin{tabular}{l} - \smath{T\rightarrow (T\rightarrow S) \rightarrow S} and\\ - \smath{(T\rightarrow S)\rightarrow T \rightarrow S}\\ - \end{tabular} - \end{center} - \end{minipage}}; - \end{tikzpicture} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[c] - \frametitle{Higher-Order Unification} - - State of the art at the time: - - \begin{itemize} - \item Lambda Prolog with full Higher-Order Unification\\ - \textcolor{darkgray}{(no mgus, undecidable, modulo $\alpha\beta$)}\bigskip - \item Higher-Order Pattern Unification\\ - \textcolor{darkgray}{(has mgus, decidable, some restrictions, modulo $\alpha\beta_0$)} - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-10>[t] - \frametitle{Underlying Ideas} - - \begin{itemize} - \item<1-> Unification (\alert{only}) up to $\alpha$ - - \item<2-> Swappings / Permutations - - \only<2-5>{ - \begin{center} - \begin{tabular}{r@ {\hspace{1mm}}l@ {\hspace{12mm}}r@ {\hspace{1mm}}l} - \\ - \only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}% - \only<3>{\smath{[b\!:=\!a]}}% - \only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} & - \onslide<2-5>{\smath{\lambda a.b}} & - - \only<2>{\smath{\textcolor{white}{[b\!:=\!a]}}}% - \only<3>{\smath{[b\!:=\!a]}}% - \only<4-5>{\smath{\alert{\swap{a}{b}\,\act}}} & - \onslide<2-5>{\smath{\lambda c.b}}\\ - - \onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda a.a}}\only<4-5>{\smath{\lambda b.a}} & - \onslide<3-5>{\smath{=}} & \only<3>{\smath{\lambda c.a}}\only<4-5>{\smath{\lambda c.a}}\\ - \end{tabular} - \end{center}\bigskip - - \onslide<4-5>{ - \begin{center} - \begin{tikzpicture} - \draw (0,0) node[inner sep=0mm,fill=cream, ultra thick, draw=cream] - {\begin{minipage}{8cm} - \begin{tabular}{r@ {\hspace{3mm}}l} - \smath{\swap{a}{b}\act t} $\;\dn$ & \alert{swap} {\bf all} occurrences of\\ - & \smath{b} and \smath{a} in \smath{t} - \end{tabular} - \end{minipage}}; - \end{tikzpicture} - \end{center}}\bigskip - - \onslide<5>{ - Unlike for \smath{[b\!:=\!a]\act(-)}, for \smath{\swap{a}{b}\act (-)} we do - have if \smath{t =_\alpha t'} then \smath{\pi \act t =_\alpha \pi \act t'.}}} - - \item<6-> Variables (or holes)\bigskip - - \begin{center} - \onslide<7->{\mbox{}\hspace{-25mm}\smath{\lambda x\hspace{-0.5mm}s .}} - \onslide<8-9>{\raisebox{-1.7mm}{\huge\smath{(}}}\raisebox{-4mm}{\begin{tikzpicture} - \fill[blue] (0, 0) circle (5mm); - \end{tikzpicture}} - \onslide<8-9>{\smath{y\hspace{-0.5mm}s}{\raisebox{-1.7mm}{\huge\smath{)}}}}\bigskip - \end{center} - - \only<8-9>{\smath{y\hspace{-0.5mm}s} are the parameters the hole can depend on\onslide<9->{, but - then you need $\beta_0$-reduction\medskip - \begin{center} - \smath{(\lambda x. t) y \longrightarrow_{\beta_0} t[x:=y]} - \end{center}}} - - \only<10>{we will record the information about which parameters a hole - \alert{\bf cannot} depend on} - - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-4>[c] - \frametitle{Terms} - - \begin{tabular}{lll @ {\hspace{10mm}}lll} - - \onslide<1->{\pgfuseshading{smallbluesphere}} & - \onslide<1->{\colorbox{cream}{\smath{\unit}}} & - \onslide<1->{Units} & - - \onslide<2->{\pgfuseshading{smallbluesphere}} & - \onslide<2->{\colorbox{cream}{\smath{a}}} & - \onslide<2->{Atoms} \\[5mm] - - \onslide<1->{\pgfuseshading{smallbluesphere}} & - \onslide<1->{\colorbox{cream}{\smath{\pair{t}{t'}}}} & - \onslide<1->{Pairs} & - - \onslide<3->{\pgfuseshading{smallbluesphere}} & - \onslide<3->{\colorbox{cream}{\smath{\abst{a}{t}}}} & - \onslide<3->{Abstractions}\\[5mm] - - \onslide<1->{\pgfuseshading{smallbluesphere}} & - \onslide<1->{\colorbox{cream}{\smath{\app{F}{t}}}} & - \onslide<1->{Funct.} & - - \onslide<4->{\pgfuseshading{smallbluesphere}} & - \onslide<4->{\colorbox{cream}{\smath{\pi\susp X}}} & - \onslide<4->{Suspensions} - \end{tabular} - - \only<2>{ - \begin{textblock}{13}(1.5,12) - \small Atoms are constants \textcolor{darkgray}{(infinitely many of them)} - \end{textblock}} - - \only<3>{ - \begin{textblock}{13}(1.5,12) - \small \smath{\ulcorner \lambda\abst{a}{a}\urcorner \mapsto \text{fn\ }\abst{a}{a}}\\ - \small constructions like \smath{\text{fn\ }\abst{X}{X}} are not allowed - \end{textblock}} - - \only<4>{ - \begin{textblock}{13}(1.5,12) - \small \smath{X} is a variable standing for a term\\ - \small \smath{\pi} is an explicit permutation \smath{\swap{a_1}{b_1}\ldots\swap{a_n}{b_n}}, - waiting to be applied to the term that is substituted for \smath{X} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-3>[c] - \frametitle{Permutations} - - a permutation applied to a term - - \begin{center} - \begin{tabular}{lrcl} - \pgfuseshading{smallbluesphere} & - \smath{[]\act c} & \smath{\dn} & \smath{c} \\ - - \pgfuseshading{smallbluesphere} & - \smath{\swap{a}{b}\!::\!\pi\act c} & \smath{\dn} & - \smath{\begin{cases} - a & \text{if}\;\pi\act c = b\\ - b & \text{if}\;\pi\act c = a\\ - \pi\act c & \text{otherwise} - \end{cases}}\\ - - \onslide<2->{\pgfuseshading{smallbluesphere}} & - \onslide<2->{\smath{\pi\act\abst{a}{t}}} & \onslide<2->{\smath{\dn}} & - \onslide<2->{\smath{\abst{\pi\act a}{\pi\act t}}}\\ - - \onslide<3->{\pgfuseshading{smallbluesphere}} & - \onslide<3->{\smath{\pi\act\pi'\act X}} & \onslide<3->{\smath{\dn}} & - \onslide<3->{\smath{(\pi @ \pi')\act X}}\\ - \end{tabular} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-3>[c] - \frametitle{Freshness Constraints} - - Recall \smath{\lambda a. \raisebox{-0.7mm}{\tikz \fill[blue] (0, 0) circle (2.5mm);}} - \bigskip\pause - - We therefore will identify - - \begin{center} - \smath{\text{fn\ } a. X \;\approx\; \text{fn\ } b. \alert<3->{\swap{a}{b}}\act X} - \end{center} - - provided that `\smath{b} is fresh for \smath{X} --- (\smath{b\fresh X})', - i.e., does not occur freely in any ground term that might be substituted for - \smath{X}.\bigskip\pause - - If we know more about \smath{X}, e.g., if we knew that \smath{a\fresh X} and - \smath{b\fresh X}, then we can replace\\ \smath{\swap{a}{b}\act X} by - \smath{X}. - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-4>[c] - \frametitle{Equivalence Judgements} - - \alt<1>{Our equality is {\bf not} just}{but judgements} - - \begin{center} - \begin{tabular}{rl} - \colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} t \approx t'}} & \alert{$\alpha$-equivalence}\\[1mm] - \onslide<4->{\colorbox{cream}{\smath{\onslide<2->{\nabla \vdash} a \fresh t}}} & - \onslide<4->{\alert{freshness}} - \end{tabular} - \end{center} - - \onslide<2->{ - where - \begin{center} - \smath{\nabla = \{a_1\fresh X_1,\ldots, a_n\fresh X_n\}} - \end{center} - is a finite set of \alert{freshness assumptions}.} - - \onslide<3->{ - \begin{center} - \smath{\{a\fresh X,b\fresh X\} \vdash \text{fn\ } a. X \approx \text{fn\ } b. X} - \end{center}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[c] - \frametitle{Rules for Equivalence} - - \begin{center} - \begin{tabular}{c} - Excerpt\\ - (i.e.~only the interesting rules) - \end{tabular} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[c] - \frametitle{Rules for Equivalence} - - \begin{center} - \begin{tabular}{c} - \colorbox{cream}{\smath{\infer{\nabla \vdash a \approx a}{}}}\\[8mm] - - \colorbox{cream}{% - \smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{a}{t'}} - {\nabla \vdash t \approx t'}}}\\[8mm] - - \colorbox{cream}{% - \smath{\infer{\nabla \vdash \abst{a}{t} \approx \abst{b}{t'}} - {a\not=b\;\; & \nabla \vdash t \approx \swap{a}{b}\act t'\;\;& \nabla \vdash a\fresh t'}}} - \end{tabular} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-3>[c] - \frametitle{Rules for Equivalence} - - \begin{center} - \colorbox{cream}{% - \smath{% - \infer{\nabla \vdash \pi\act X \approx \pi'\act X} - {\begin{array}{c} - (a\fresh X)\in\nabla\\ - \text{for all}\; a \;\text{with}\;\pi\act a \not= \pi'\act a - \end{array} - }}} - \end{center} - - \onslide<2->{ - for example\\[4mm] - - \alt<2>{% - \begin{center} - \smath{\{a\fresh\!X, b\fresh\!X\} \vdash X \approx \swap{a}{b}\act X} - \end{center}} - {% - \begin{center} - \smath{\{a\fresh\!X, c\fresh\!X\} \vdash \swap{a}{c}\swap{a}{b}\act X \approx \swap{b}{c}\act X} - \end{center}} - - \onslide<3->{ - \begin{tabular}{@ {}lllll@ {}} - because & - \smath{\swap{a}{c}\swap{a}{b}}: & - \smath{a\mapsto b} & - \smath{\swap{b}{c}}: & - \smath{a\mapsto a}\\ - & & \smath{b\mapsto c} & & \smath{b\mapsto c}\\ - & & \smath{c\mapsto a} & & \smath{c\mapsto b}\\ - \end{tabular} - disagree at \smath{a} and \smath{c}.} - } - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[c] - \frametitle{Rules for Freshness} - - \begin{center} - \begin{tabular}{c} - Excerpt\\ - (i.e.~only the interesting rules) - \end{tabular} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[c] - \frametitle{Rules for Freshness} - - \begin{center} - \begin{tabular}{c} - \colorbox{cream}{% - \smath{\infer{\nabla \vdash a\fresh b}{a\not= b}}}\\[5mm] - - \colorbox{cream}{% - \smath{\infer{\nabla \vdash a\fresh\abst{a}{t}}{}}}\hspace{7mm} - \colorbox{cream}{% - \smath{\infer{\nabla \vdash a\fresh\abst{b}{t}} - {a\not= b\;\; & \nabla \vdash a\fresh t}}}\\[5mm] - - \colorbox{cream}{% - \smath{\infer{\nabla \vdash a\fresh \pi\act X} - {(\pi^{-1}\act a\fresh X)\in\nabla}}} - \end{tabular} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-4>[t] - \frametitle{$\approx$ is an Equivalence} - \mbox{}\\[5mm] - - \begin{center} - \colorbox{cream}{\alert{Theorem:} - $\approx$ is an equivalence relation.} - \end{center}\bigskip - - \only<1>{% - \begin{tabular}{ll} - (Reflexivity) & $\smath{\nabla\vdash t\approx t}$\\[2mm] - (Symmetry) & if $\smath{\nabla\vdash t_1\approx t_2}\;$ - then $\;\smath{\nabla\vdash t_2\approx t_1}$\\[2mm] - (Transitivity) & if $\smath{\nabla\vdash t_1\approx t_2}\;$ and - $\;\smath{\nabla\vdash t_2\approx t_3}$\\ - & then $\smath{\nabla\vdash t_1\approx t_3}$\\ - \end{tabular}} - - \only<2->{% - \begin{itemize} - \item<2-> \smath{\nabla \vdash t\approx t'} then \smath{\nabla \vdash \pi\act t\approx \pi\act t'} - - \item<2-> \smath{\nabla \vdash a\fresh t} then - \smath{\nabla \vdash \pi\act a\fresh \pi\act t} - - \item<3-> \smath{\nabla \vdash t\approx \pi\act t'} then - \smath{\nabla \vdash (\pi^{-1})\act t\approx t'} - - \item<3-> \smath{\nabla \vdash a\fresh \pi\act t} then - \smath{\nabla \vdash (\pi^{-1})\act a\fresh t} - - \item<4-> \smath{\nabla \vdash a\fresh t} and \smath{\nabla \vdash t\approx t'} then - \smath{\nabla \vdash a\fresh t'} - \end{itemize} - } - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-4> - \frametitle{Comparison $=_\alpha$} - - Traditionally \smath{=_\alpha} is defined as - - \begin{center} - \colorbox{cream}{% - \begin{minipage}{9cm} - \raggedright least congruence which identifies \smath{\abst{a}{t}} - with \smath{\abst{b}{[a:=b]t}} provided \smath{b} is not free - in \smath{t} - \end{minipage}} - \end{center} - - where \smath{[a:=b]t} replaces all free occurrences of\\ - \smath{a} by \smath{b} in \smath{t}. - \bigskip - - \only<2>{% - \begin{textblock}{13}(1.2,10) - For \alert{ground} terms: - - \begin{center} - \colorbox{cream}{% - \begin{minipage}{9.0cm} - \begin{tabular}{@ {}rl} - \underline{Theorem:} - & \smath{t=_\alpha t'\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset \vdash t\approx t'}\\[2mm] - & \smath{a\not\in F\hspace{-0.9mm}A(t)\;\;} if\hspace{-0.5mm}f~\smath{\;\;\emptyset\vdash a\fresh t} - \end{tabular} - \end{minipage}} - \end{center} - \end{textblock}} - - \only<3>{% - \begin{textblock}{13}(1.2,10) - In general \smath{=_\alpha} and \smath{\approx} are distinct! - \begin{center} - \colorbox{cream}{% - \begin{minipage}{6.0cm} - \smath{\abst{a}{X}=_\alpha \abst{b}{X}\;} but not\\[2mm] - \smath{\emptyset \vdash \abst{a}{X} \approx \abst{b}{X}\;} (\smath{a\not=b}) - \end{minipage}} - \end{center} - \end{textblock}} - - \only<4>{ - \begin{textblock}{6}(1,2) - \begin{tikzpicture} - \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] - {\color{darkgray} - \begin{minipage}{10cm}\raggedright - That is a crucial point: if we had\\[-2mm] - \[\smath{\emptyset \vdash \abst{a}{X}\approx \abst{b}{X}}\mbox{,}\] - then applying $\smath{[X:=a]}$, $\smath{[X:=b]}$, $\ldots$\\ - give two terms that are {\bf not} $\alpha$-equivalent.\\[3mm] - The freshness constraints $\smath{a\fresh X}$ and $\smath{b\fresh X}$ - rule out the problematic substitutions. Therefore - - \[\smath{\{a\fresh X,b\fresh X\} \vdash \abst{a}{X}\approx \abst{b}{X}}\] - - does hold. - \end{minipage}}; - \end{tikzpicture} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-9> - \frametitle{Substitution} - - \begin{tabular}{l@ {\hspace{8mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l@ {}} - \pgfuseshading{smallbluesphere} & - \smath{\sigma(\abst{a}{t})} & \smath{\dn} & \smath{\abst{a}{\sigma(t)}}\\[2mm] - - \pgfuseshading{smallbluesphere} & - \smath{\sigma(\pi\act X)} & \smath{\dn} & - \smath{\begin{cases}% - \pi\;\act\;\sigma(X) & \!\!\text{if\ } \sigma(X)\not=X\\ - \pi\act X & \!\!\text{otherwise}% - \end{cases}}\\[6mm] - \end{tabular}\bigskip\bigskip - - \pause - \only<2-5>{ - \only<2->{for example} - \def\arraystretch{1.3} - \begin{tabular}{@ {\hspace{14mm}}l@ {\hspace{3mm}}l} - \onslide<2->{\textcolor{white}{$\Rightarrow$}} & - \onslide<2->{\alt<3>{\smath{\underline{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}} - {\smath{\abst{a}{\swap{a}{b}\act X}\;\,[X:=\pair{b}{Y}]}}}\\ - \onslide<3->{\smath{\Rightarrow}} & - \onslide<3->{\alt<3,4>{\smath{\abst{a}{\underline{\swap{a}{b}\act X[X:=\pair{b}{Y}]}}}} - {\smath{\abst{a}{\swap{a}{b}\act X}[X:=\pair{b}{Y}]}}}\\ - \onslide<4->{\smath{\Rightarrow}} & - \onslide<4->{\alt<4>{\smath{\abst{a}{\swap{a}{b}\act \underline{\pair{b}{Y}}}}} - {\smath{\abst{a}{\underline{\swap{a}{b}}\act \pair{b}{Y}}}}}\\ - \onslide<5->{\smath{\Rightarrow}} & - \onslide<5->{\smath{\abst{a}{\pair{a}{\swap{a}{b}\act Y}}}} - \end{tabular}} - - \only<6-> - {\begin{tabular}{l@ {\hspace{8mm}}l@ {}} - \pgfuseshading{smallbluesphere} & - if \smath{\nabla\vdash t\approx t'} and\hspace{-2mm}\mbox{} - \raisebox{-2.7mm}{ - \alt<7>{\begin{tikzpicture} - \draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=3mm] - {\smath{\;\nabla'\vdash\sigma(\nabla)\;}}; - \end{tikzpicture}} - {\begin{tikzpicture} - \draw (0,0) node[inner sep=1mm,fill=white, very thick, draw=white, rounded corners=3mm] - {\smath{\;\nabla'\vdash\sigma(\nabla)\;}}; - \end{tikzpicture}}}\\ - & then \smath{\nabla'\vdash\sigma(t)\approx\sigma(t')} - \end{tabular}} - - \only<9> - {\begin{tabular}{l@ {\hspace{8mm}}l@ {}} - \\[-4mm] - \pgfuseshading{smallbluesphere} & - \smath{\sigma(\pi\act t)=\pi\act\sigma(t)} - \end{tabular}} - - - \only<7>{ - \begin{textblock}{6}(10,10.5) - \begin{tikzpicture} - \draw (0,0) node[inner sep=1mm,fill=cream, very thick, draw=red, rounded corners=2mm] - {\color{darkgray} - \begin{minipage}{3.8cm}\raggedright - this means\\[1mm] - \smath{\nabla'\vdash a\fresh\sigma(X)}\\[1mm] - holds for all\\[1mm] - \smath{(a\fresh X)\in\nabla} - \end{minipage}}; - \end{tikzpicture} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-> - \frametitle{Equational Problems} - - An equational problem - \[ - \colorbox{cream}{\smath{t \eqprob t'}} - \] - is \alert{solved} by - - \begin{center} - \begin{tabular}{ll} - \pgfuseshading{smallbluesphere} & a substitution \smath{\sigma} (terms for variables)\\[3mm] - \pgfuseshading{smallbluesphere} & {\bf and} a set of freshness assumptions \smath{\nabla} - \end{tabular} - \end{center} - - so that \smath{\nabla\vdash \sigma(t)\approx \sigma(t')}. - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-> - - Unifying equations may entail solving - \alert{freshness problems}. - - \bigskip - - E.g.~assuming that \smath{a\not=a'}, then - \[ - \smath{\abst{a}{t}\eqprob \abst{a'}{t'}} - \] - can only be solved if - \[ - \smath{t\eqprob \swap{a}{a'}\act t'} \quad\text{\emph{and}}\quad - \smath{a\freshprob t'} - \] - can be solved. - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-> - \frametitle{Freshness Problems} - - A freshness problem - \[ - \colorbox{cream}{\smath{a \freshprob t}} - \] - is \alert{solved} by - - \begin{center} - \begin{tabular}{ll} - \pgfuseshading{smallbluesphere} & a substitution \smath{\sigma}\\[3mm] - \pgfuseshading{smallbluesphere} & and a set of freshness assumptions \smath{\nabla} - \end{tabular} - \end{center} - - so that \smath{\nabla\vdash a \fresh \sigma(t)}. - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-3> - \frametitle{Existence of MGUs} - - \underline{Theorem}: There is an algorithm which, given a nominal - unification problem \smath{P}, decides whether\\ - or not it has a solution \smath{(\sigma,\nabla)}, and returns a \\ - \alert{most general} one if it does.\bigskip\bigskip - - \only<3>{ - Proof: one can reduce all the equations to `solved form' - first (creating a substitution), and then solve the freshness - problems (easy).} - - \only<2>{ - \begin{textblock}{6}(2.5,9.5) - \begin{tikzpicture} - \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] - {\color{darkgray} - \begin{minipage}{8cm}\raggedright - \alert{most general:}\\ - straightforward definition\\ - ``if\hspace{-0.5mm}f there exists a \smath{\tau} such that \ldots'' - \end{minipage}}; - \end{tikzpicture} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1> - \frametitle{Remember the Quiz?} - - \textcolor{gray}{Assuming that $a$ and $b$ are distinct variables,\\ - is it possible to find $\lambda$-terms $M_1$ to $M_7$ - that make the following pairs $\alpha$-equivalent?} - - \begin{tabular}{@ {\hspace{14mm}}p{12cm}} - \begin{itemize} - \item \smath{\lambda a.\lambda b. (M_1\,b)\;} and - \smath{\lambda b.\lambda a. (a\,M_1)\;} - - \item \textcolor{gray}{$\lambda a.\lambda b. (M_2\,b)\;$ and - $\lambda b.\lambda a. (a\,M_3)\;$} - - \item \textcolor{gray}{$\lambda a.\lambda b. (b\,M_4)\;$ and - $\lambda b.\lambda a. (a\,M_5)\;$} - - \item \smath{\lambda a.\lambda b. (b\,M_6)\;} and - \smath{\lambda a.\lambda a. (a\,M_7)\;} - \end{itemize} - \end{tabular} - - \textcolor{gray}{If there is one solution for a pair, can you - describe all its solutions?} - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-> - \frametitle{Answers to the Quiz} - \small - \def\arraystretch{1.6} - \begin{tabular}{c@ {\hspace{2mm}}l} - & \only<1>{\smath{\lambda a.\lambda b. (M_1\,b)\;} and \smath{\;\lambda b.\lambda a. (a\,M_1)}}% - \only<2->{\smath{\abst{a}{\abst{b}{\pair{M_1}{b}}} \;\eqprob\; \abst{b}{\abst{a}{\pair{a}{M_1}}}}}\\ - - \onslide<3->{\smath{\redu{\id}}} & - \only<3>{\smath{\abst{b}{\pair{M_1}{b}} \eqprob - \alert{\swap{a}{b}} \act \abst{a}{\pair{a}{M_1}}\;,\;a\freshprob \abst{a}{\pair{a}{M_1}}}}% - \only<4->{\smath{\abst{b}{\pair{M_1}{b}} \eqprob \abst{b}{\pair{b}{\swap{a}{b}\act M_1}}\;,\ - a\freshprob \abst{a}{\pair{a}{M_1}}}}\\ - - \onslide<5->{\smath{\redu{\id}}} & - \only<5->{\smath{\pair{M_1}{b} \eqprob \pair{b}{\swap{a}{b}\act M_1}\;,\;% - a\freshprob \abst{a}{\pair{a}{M_1}}}}\\ - - \onslide<6->{\smath{\redu{\id}}} & - \only<6->{\smath{M_1 \eqprob b \;,\; b \eqprob \swap{a}{b}\act M_1\;,\;% - a\freshprob \abst{a}{\pair{a}{M_1}}}}\\ - - \onslide<7->{\smath{\redu{[M_1:=b]}}} & - \only<7>{\smath{b \eqprob \swap{a}{b}\act \alert{b}\;,\;% - a\freshprob \abst{a}{\pair{a}{\alert{b}}}}}% - \only<8->{\smath{b \eqprob a\;,\; a\freshprob \abst{a}{\pair{a}{b}}}}\\ - - \onslide<9->{\smath{\redu{}}} & - \only<9->{\smath{F\hspace{-0.5mm}AIL}} - \end{tabular} - - \only<10>{ - \begin{textblock}{6}(2,11) - \begin{tikzpicture} - \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] - {\color{darkgray} - \begin{minipage}{9cm}\raggedright - \smath{\lambda a.\lambda b. (M_1\,b)} \smath{=_\alpha} - \smath{\lambda b.\lambda a. (a\,M_1)} has no solution - \end{minipage}}; - \end{tikzpicture} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-> - \frametitle{Answers to the Quiz} - \small - \def\arraystretch{1.6} - \begin{tabular}{c@ {\hspace{2mm}}l} - & \only<1>{\smath{\lambda a.\lambda b. (b\,M_6)\;} and \smath{\;\lambda a.\lambda a. (a\,M_7)}}% - \only<2->{\smath{\abst{a}{\abst{b}{\pair{b}{M_6}}} \;\eqprob\; \abst{a}{\abst{a}{\pair{a}{M_7}}}}}\\ - - \onslide<3->{\smath{\redu{\id}}} & - \only<3->{\smath{\abst{b}{\pair{b}{M_6}} \eqprob \abst{a}{\pair{a}{M_7}}}}\\ - - \onslide<4->{\smath{\redu{\id}}} & - \only<4->{\smath{\pair{b}{M_6} \eqprob \pair{b}{\swap{b}{a}\act M_7}\;,\;b\freshprob\pair{a}{M_7}}}\\ - - \onslide<5->{\smath{\redu{\id}}} & - \only<5->{\smath{b\eqprob b\;,\; M_6 \eqprob \swap{b}{a}\act M_7\;,\;% - b\freshprob \pair{a}{M_7}}}\\ - - \onslide<6->{\smath{\redu{\id}}} & - \only<6->{\smath{M_6 \eqprob \swap{b}{a}\act M_7\;,\;% - b\freshprob \pair{a}{M_7}}}\\ - - \onslide<7->{\makebox[0mm]{\smath{\redu{[M_6:=\swap{b}{a}\act M_7]}}}} & - \only<7->{\smath{\qquad b\freshprob \pair{a}{M_7}}}\\ - - \onslide<8->{\smath{\redu{\varnothing}}} & - \only<8->{\smath{b\freshprob a\;,\;b\freshprob M_7}}\\ - - \onslide<9->{\smath{\redu{\varnothing}}} & - \only<9->{\smath{b\freshprob M_7}}\\ - - \onslide<10->{\makebox[0mm]{\smath{\redu{\{b\fresh M_7\}}}}} & - \only<10->{\smath{\;\;\varnothing}}\\ - - \end{tabular} - - \only<10>{ - \begin{textblock}{6}(6,9) - \begin{tikzpicture} - \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] - {\color{darkgray} - \begin{minipage}{7cm}\raggedright - \smath{\lambda a.\lambda b. (b\,M_6)\;} \smath{=_\alpha} - \smath{\;\lambda a.\lambda a. (a\,M_7)}\\[2mm] - we can take \smath{M_7} to be any $\lambda$-term that does not - contain free occurrences of \smath{b}, so long as we take \smath{M_6} to - be the result of swapping all occurrences of \smath{b} and \smath{a} - throughout \smath{M_7} - \end{minipage}}; - \end{tikzpicture} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-> - \frametitle{Properties} - - \begin{itemize} - \item An interesting feature of nominal unification is that it - does not need to create new atoms.\bigskip - - \begin{center}\small - \colorbox{cream}{ - \smath{\{a.t \eqprob b.t'\}\cup P \redu{\id} \{t \eqprob \swap{a}{b}\act t', a \freshprob t'\} \cup P}} - \end{center}\bigskip\bigskip - \pause - - \item The alternative rule - - \begin{center}\small - \colorbox{cream}{ - \begin{tabular}{@ {}l@ {}} - \smath{\{a.t \eqprob b.t'\}\cup P \redu{\id}}\\ - \mbox{}\hspace{2cm}\smath{\{\swap{a}{c}\act t \eqprob - \swap{b}{c}\act t', c \freshprob t, c \freshprob t'\} \cup P} - \end{tabular}} - \end{center} - - leads to a more complicated notion of mgu.\medskip\pause - - \footnotesize - \smath{\{a.X \eqprob b.Y\} \redu{} (\{a\fresh Y, c\fresh Y\}, [X:=\swap{a}{c}\swap{b}{c}\act Y])} - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-3> - \frametitle{Is it Useful?} - - Yes. $\alpha$Prolog by James Cheney (main developer)\bigskip\bigskip - - \color{darkgray} - \begin{tabular}{@ {}l} - type (Gamma, var(X), T) :- member (X,T) Gamma.\smallskip\medskip\\ - - type (Gamma, app(M, N), T') :-\\ - \hspace{3cm}type (Gamma, M, arrow(T, T')),\\ - \hspace{3cm}type (Gamma, N, T).\smallskip\medskip\\ - - type (Gamma, lam(\alert{x.M}), arrow(T, T')) / \alert{x \# Gamma} :-\\ - \hspace{3cm}type ((x, T)::Gamma, M, T').\smallskip\medskip\\ - - member X X::Tail.\\ - member X Y::Tail :- member X Tail.\\ - \end{tabular} - - \only<2->{ - \begin{textblock}{6}(1.5,0.5) - \begin{tikzpicture} - \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] - {\color{darkgray} - \begin{minipage}{9cm}\raggedright - {\bf One problem:} If we ask whether - - \begin{center} - ?- type ([(x, T')], lam(x.Var(x)), T) - \end{center} - - is typable, we expect an answer for T.\bigskip - - \onslide<3>{Solution: Before back-chaining freshen all variables and atoms - in a program (clause).} - \end{minipage}}; - \end{tikzpicture} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-> - \frametitle{Equivariant Unification} - - James Cheney proposed - - \begin{center} - \colorbox{cream}{ - \smath{t \eqprob t' \redu{\nabla, \sigma, \pi} - \nabla \vdash \sigma(t) \approx \pi \act \sigma(t')}} - \end{center}\bigskip\bigskip - \pause - - But he also showed this problem is undecidable\\ in general. :( - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-> - \frametitle{Taking Atoms as Variables} - - Instead of \smath{a.X}, have \smath{A.X}.\bigskip - \pause - - Unfortunately this breaks the mgu-property: - - \begin{center} - \smath{a.Z \eqprob X.Y.v(a)} - \end{center} - - can be solved by - - \begin{center} - \smath{[X:=a, Z:=Y.v(a)]} and - \smath{[Y:=a, Z:=Y.v(Y)]} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[c] - \frametitle{HOPU vs. NOMU} - - \begin{itemize} - \item James Cheney showed\bigskip - \begin{center} - \colorbox{cream}{\smath{HOPU \Rightarrow NOMU}} - \end{center}\bigskip - - \item Jordi Levy and Mateu Villaret established\bigskip - \begin{center} - \colorbox{cream}{\smath{HOPU \Leftarrow NOMU}} - \end{center}\bigskip - \end{itemize} - - The translations `explode' the problems quadratically. - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1> - \small\tt - - \begin{minipage}{13cm} - \begin{tabular}{@ {\hspace{-2mm}}p{11.5cm}} - \\ - From: Zhenyu Qian \\ - To: Christian Urban \\ - Subject: RE: Linear Higher-Order Pattern Unification\\ - Date: Mon, 14 Apr 2008 09:56:47 +0800\\ - \\ - Hi Christian,\\ - \\ - Thanks for your interests and asking. I know that that paper is complex. As - I told Tobias when we met last time, I have raised the question to myself - many times whether the proof could have some flaws, and so making it through - a theorem prover would definitely bring piece to my mind (no matter what - the result would be). The only problem for me is the time.\\ - \ldots\\ - - Thanks/Zhenyu - \end{tabular} - \end{minipage} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1> - \frametitle{Complexity} - - \begin{itemize} - \item Christiopher Calves and Maribel Fernandez showed first that - it is polynomial and then also quadratic - - \item Jordi Levy and Mateu Villaret showed that it is quadratic - by a translation into a subset of NOMU and using ideas from - Martelli/Montenari. - - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[c] - \frametitle{Conclusion} - - \begin{itemize} - \item Nominal Unification is a completely first-order - language, but implements unification modulo $\alpha$. - \textcolor{gray}{(verification\ldots Ramana Kumar and Michael Norrish)} - \medskip\pause - - \item NOMU has been applied in term-rewriting and - logic programming. \textcolor{gray}{(Maribel Fernandez et - al has a KB-completion procedure.)} - I hope it will also be used in typing - systems.\medskip\pause - - \item NOMU and HOPU are `equivalent' (it took a long time - and considerable research to find this out).\medskip\pause - - \item The question about complexity is still an ongoing - story.\medskip - \end{itemize} - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[c] - \frametitle{ - \begin{tabular}{c} - \mbox{}\\[23mm] - \alert{\LARGE Thank you very much!}\\ - \alert{\Large Questions?} - \end{tabular}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1-3> - \frametitle{Most General Unifiers} - - \underline{Definition}: For a unification problem - \smath{P}, a solution \smath{(\sigma_1,\nabla_1)} is - \alert{more general} than another solution - \smath{(\sigma_2,\nabla_2)}, iff~there exists a substitution - \smath{\tau} with - - \begin{center} - \begin{tabular}{ll} - \pgfuseshading{smallbluesphere} & - \alt<2>{\smath{\alert{\nabla_2\vdash\tau(\nabla_1)}}} - {\smath{\nabla_2\vdash\tau(\nabla_1)}}\\ - \pgfuseshading{smallbluesphere} & - \alt<3>{\smath{\alert{\nabla_2\vdash\sigma_2\approx \tau\circ\sigma_1}}} - {\smath{\nabla_2\vdash\sigma_2\approx \tau\circ\sigma_1}} - \end{tabular} - \end{center} - - \only<2>{ - \begin{textblock}{13}(1.5,10.5) - \smath{\nabla_2\vdash a\fresh \sigma(X)} holds for all - \smath{(a\fresh X)\in\nabla_1} - \end{textblock}} - - \only<3>{ - \begin{textblock}{11}(1.5,10.5) - \smath{\nabla_2\vdash \sigma_2(X)\approx - \sigma(\sigma_1(X))} - holds for all - \smath{X\in\text{dom}(\sigma_2)\cup\text{dom}(\sigma\circ\sigma_1)} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -(*<*) -end -(*>*) \ No newline at end of file