author  paulson 
Fri, 31 Jan 1997 17:13:19 +0100  
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parent 1144  5a62ecf80126 
permissions  rwrr 
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%process by latex inddefsslides; dvips Plime inddefsslides 
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% $Id$ 

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\documentclass[a4,slidesonly,semlayer]{seminar} 

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\usepackage{fancybox} 
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\usepackage{semhelv} 

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\usepackage{epsf} 

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\def\printlandscape{\special{landscape}} % Works with dvips. 
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\slidesmag{5}\articlemag{2} %the difference is 3 instead of 4! 
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\extraslideheight{30pt} 
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\renewcommand\slidefuzz{6pt} 
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\sloppy\hfuzz2pt %sloppy defines \hfuzz0.5pt but it's mainly for text 
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\newcommand\sbs{\subseteq} 
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\newcommand\Pow{{\cal P}} 
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\newcommand\lfp{\hbox{\tt lfp}} 
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\newcommand\gfp{\hbox{\tt gfp}} 
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\newcommand\lst{\hbox{\tt list}} 
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\newcommand\term{\hbox{\tt term}} 
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\newcommand\heading[1]{% 
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\begin{center}\large\bf\shadowbox{#1}\end{center} 
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\vspace{1ex minus 1ex}} 
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\newpagestyle{mine}% 
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{L. Paulson\hfil A Fixedpoint Approach to (Co)Inductive Definitions\hfil 
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\thepage}% 
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{\hfil\special{psfile=cuarms.eps hscale=20 vscale=20 voffset=6 
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hoffset=14}\hfil} 
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\pagestyle{mine} 
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\begin{document} 
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\slidefonts 
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\begin{slide}\centering 
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\shadowbox{% 
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\begin{Bcenter} 
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{\Large\bf A Fixedpoint Approach to}\\[2ex] 
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{\Large\bf (Co)Inductive Definitions} 
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\end{Bcenter}} 
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\bigskip 
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\begin{Bcenter} 
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Lawrence C. Paulson\\ 
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Computer Laboratory\\ 
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University of Cambridge\\ 
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England\\[1ex] 
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\verblcp@cl.cam.ac.uk 
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\end{Bcenter} 
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\bigskip 
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{\footnotesize Thanks: SERC grants GR/G53279, GR/H40570; ESPRIT Project 6453 
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`Types'} 
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\end{slide} 
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\begin{slide} 
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\heading{Inductive Definitions} 
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\begin{itemize} 
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\item {\bf datatypes} 
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\begin{itemize} 
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\item finite lists, trees 
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\item syntax of expressions, \ldots 
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\end{itemize} 
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\item {\bf inference systems} 
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\begin{itemize} 
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\item transitive closure of a relation 
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\item transition systems 
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\item structural operational semantics 
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\end{itemize} 
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\end{itemize} 
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Supported by Boyer/Moore, HOL, Coq, \ldots, Isabelle/ZF 
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\end{slide} 
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\begin{slide} 
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\heading{Coinductive Definitions} 
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\begin{itemize} 
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\item {\bf codatatypes} 
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\begin{itemize} 
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\item {\it infinite\/} lists, trees 
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\item syntax of {\it infinite\/} expressions, \ldots 
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\end{itemize} 
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\item {\bf bisimulation relations} 
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\begin{itemize} 
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\item process equivalence 
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\item uses in functional programming (Abramksy, Howe) 
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\end{itemize} 
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\end{itemize} 
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Supported by \ldots ?, \ldots, Isabelle/ZF 
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\end{slide} 
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\begin{slide} 
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\heading{The KnasterTarksi Fixedpoint Theorem} 
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$h$ a monotone function 
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$D$ a set such that $h(D)\sbs D$ 
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The {\bf least} fixedpoint $\lfp(D,h)$ yields {\bf inductive} definitions 
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The {\bf greatest} fixedpoint $\gfp(D,h)$ yields {\bf coinductive} definitions 
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{\it A general approach\/}: 
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\begin{itemize} 
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\item handles all provably monotone definitions 
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\item works for set theory, higherorder logic, \ldots 
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\end{itemize} 
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\end{slide} 
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\begin{slide} 
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\heading{An Implementation in Isabelle/ZF}\centering 
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\begin{itemize} 
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\item {\bf Input} 
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\begin{itemize} 
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\item description of {\it introduction rules\/} \& tree's {\it 
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constructors\/} 
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\item {\it theorems\/} implying that the definition is {\it monotonic\/} 
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\end{itemize} 
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\item {\bf Output} 
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\begin{itemize} 
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\item (co)induction rules 
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\item case analysis rule and {\it rule inversion\/} tools, \ldots 
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\end{itemize} 
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\end{itemize} 
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\vfill 
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{\it flexible, secure, \ldots but fast\/} 
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\end{slide} 
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\begin{slide} 
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\heading{Working Examples} 
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\begin{itemize} 
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\item lists 
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\item terms recursive over lists: $\term(A) = A\times\lst(\term(A))$ 
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\item primitive recursive functions 
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\item lazy lists 
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\item bisimulations for lazy lists 
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\item combinator reductions; ChurchRosser Theorem 
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\item mutually recursive trees \& forests 
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\end{itemize} 
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\end{slide} 
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\begin{slide} 
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\heading{Other Work Using Fixedpoints} 
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{\bf The HOL system}: 
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\begin{itemize} 
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\item Melham's induction package: special case of Fixedpoint Theorem 
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\item Andersen \& Petersen's induction package 
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\item (no HOL datatype package uses fixedpoints) 
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\end{itemize} 
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{\bf Coq and LEGO}: 
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\begin{itemize} 
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\item (Co)induction {\it almost\/} expressible in base logic (CoC) 
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\item \ldots{} inductive definitions are builtin 
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\end{itemize} 
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\end{slide} 
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\begin{slide} 
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\heading{Limitations \& Future Developments}\centering 
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\begin{itemize} 
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\item {\bf infinitebranching trees} 
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\begin{itemize} 
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\item justification requires proof 
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\item would be easier to {\it build them in\/}! 
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\end{itemize} 
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\item {\bf recursive function definitions} 
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\begin{itemize} 
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\item use {\it wellfounded\/} recursion 
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\item distinct from datatype definitions 
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\end{itemize} 
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\item {\bf port to Isabelle/HOL} 
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\end{itemize} 
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\end{slide} 
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\end{document} 
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{\it flat\/} ordered pairs used to define infinite lists, \ldots 
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\begin{slide} 
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\heading{}\centering 
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\end{slide} 
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195 