--- 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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \begin{frame}<1>
- \small\tt
-
- \begin{minipage}{13cm}
- \begin{tabular}{@ {\hspace{-2mm}}p{11.5cm}}
- \\
- From: Zhenyu Qian <zhqian@microsoft.com>\\
- To: Christian Urban <urbanc@in.tum.de>\\
- 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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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<presentation>{
- \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