(*<*)+ −
theory Slides3+ −
imports "~~/src/HOL/Library/LaTeXsugar" "Nominal"+ −
begin+ −
+ −
declare [[show_question_marks = false]]+ −
+ −
notation (latex output)+ −
set ("_") and+ −
Cons ("_::/_" [66,65] 65) + −
+ −
(*>*)+ −
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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}}+ −
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*}+ −
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text_raw {*+ −
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\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}}+ −
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+ −
*}+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
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text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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 {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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 {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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 {*+ −
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\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 {*+ −
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\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 {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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}}+ −
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*}+ −
+ −
text_raw {*+ −
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\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 {*+ −
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\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 {*+ −
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\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+ −
(*>*)+ −