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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\documentclass[runningheads]{llncs}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\documentclass[10pt, conference, compsocconf]{IEEEtran}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{isabelle}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{isabellesym}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{times}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{amssymb}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{amsmath}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{stmaryrd}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{mathpartir}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{pdfsetup}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{tikz}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{pgf}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\usepackage{color}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%% for testing
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\usepackage{endnotes}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\let\footnote=\endnote
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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% urls in roman style, theory text in math-similar italics
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\urlstyle{rm}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\isabellestyle{it}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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% gray boxes
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\definecolor{mygrey}{rgb}{.80,.80,.80}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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% mathpatir
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\mprset{sep=0.9em}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\mprset{center=false}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\mprset{flushleft=true}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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% for uniform font size
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%\renewcommand{\isastyle}{\isastyleminor}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\def\dn{\,\stackrel{\mbox{\scriptsize def}}{=}\,}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\renewcommand{\isasymequiv}{$\dn$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\renewcommand{\isasymemptyset}{$\varnothing$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\renewcommand{\isacharunderscore}{\mbox{$\_$}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\renewcommand{\isasymiota}{}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\newcommand{\isasymulcorner}{$\ulcorner$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\newcommand{\isasymurcorner}{$\urcorner$}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{document}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\title{Mechanising Turing Machines and Computability Theory in Isabelle/HOL}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\author{Jian Xu\inst{1} \and Xingyuan Zhang\inst{1} \and Christian Urban\inst{2}}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\institute{PLA University of Science and Technology, China \and King's College London, UK}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\maketitle
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\begin{abstract}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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We present a formalised theory of computability in the theorem prover
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Isabelle/HOL.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%This theorem prover is based on classical logic which
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%precludes \emph{direct} reasoning about computability: every boolean
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%predicate is either true or false because of the law of excluded
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%middle. The only way to reason about computability in a classical
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%theorem prover is to formalise a concrete model of computation.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Following the textbook by Boolos et al, we formalise Turing machines
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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and relate them to abacus machines and recursive functions. We ``tie
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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the know'' between these three computational models by formalising a
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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universal function and obtaining from it a universal Turing machine by
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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our verified translation from recursive functions to abacus programs
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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and from abacus programs to Turing machine programs. Hoare-style reasoning techniques allow us
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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to reason about concrete Turing machine and abacus programs.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%Our theory can be
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%used to formalise other computability results.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%We give one example about the computational equivalence of
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%single-sided Turing machines.
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%{\it we give one example about the undecidability of Wang's tiling problem, whose proof uses
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%the notion of a universal Turing machine.}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{abstract}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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% generated text of all theories
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\input{session}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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% optional bibliography
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bibliographystyle{abbrv}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\bibliography{root}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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\end{document}
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%% Local Variables:
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%% mode: latex
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%% TeX-master: t
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Christian Urban <christian dot urban at kcl dot ac dot uk>
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%%% End:
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