diff -r 73127f5db18f -r bea94f1e6771 Journal/document/root.tex --- a/Journal/document/root.tex Fri Apr 20 11:27:49 2012 +0000 +++ b/Journal/document/root.tex Fri Apr 20 11:45:06 2012 +0000 @@ -1,4 +1,4 @@ -\documentclass{ita} +\documentclass[acmtocl]{acmtrans2m} \usepackage{isabelle} \usepackage{isabellesym} \usepackage{amsmath} @@ -9,13 +9,12 @@ \usetikzlibrary{shapes,shapes.arrows,snakes,positioning} \usepgflibrary{shapes.misc} % LATEX and plain TEX and pure pgf \usetikzlibrary{matrix} -\usepackage{pdfsetup} +%%\usepackage{pdfsetup} \usepackage{ot1patch} \usepackage{times} -%%\usepackage{proof} -%%\usepackage{mathabx} \usepackage{stmaryrd} \usepackage{mathpartir} +\usepackage{longtable} \urlstyle{rm} \isabellestyle{it} @@ -35,39 +34,52 @@ \newcommand{\isasymbigplus}{\ensuremath{\bigplus}} \newcommand{\bigplus}{\mbox{\Large\bf$+$}} -\begin{document} + +\newtheorem{thrm}{Theorem}[section] +\newtheorem{lmm}{Lemma}[section] +\newtheorem{dfntn}{Definition}[section] +\newtheorem{prpstn}{Proposition}[section] + \title{A Formalisation of the Myhill-Nerode Theorem\\ based on Regular - Expressions} -\thanks{This is a revised and expanded version of \cite{WuZhangUrban11}.} -\author{Chunhan Wu}\address{PLA University of Science and Technology, China} -\author{Xingyuan Zhang}\sameaddress{1} -\author{Christian Urban}\address{King's College London, United Kingdom}\secondaddress{corresponding author} -\subjclass{68Q45} -\keywords{Myhill-Nerode theorem, regular expressions, Isabelle theorem prover} + Expressions$^\star$} +%%\thanks{This is a revised and expanded version of \cite{WuZhangUrban11}.} +\author{Chunhan Wu\\PLA University of Science and Technology Nanjing, China \and +Xingyuan Zhang\\PLA University of Science and Technology Nanjing, China \and +Christian Urban\\ King's College London, United Kingdom} \begin{abstract} There are numerous textbooks on regular languages. Nearly all of them -introduce the subject by describing finite automata and only mentioning on the -side a connection with regular expressions. Unfortunately, automata are difficult -to formalise in HOL-based theorem provers. The reason is that -they need to be represented as graphs, matrices or functions, none of which -are inductive datatypes. Also convenient operations for disjoint unions of -graphs, matrices and functions are not easily formalisiable in HOL. In contrast, regular -expressions can be defined conveniently as a datatype and a corresponding -reasoning infrastructure comes for free. We show in this paper that a central -result from formal language theory---the Myhill-Nerode Theorem---can be -recreated using only regular expressions. From this theorem many closure -properties of regular languages follow. +introduce the subject by describing finite automata and only +mentioning on the side a connection with regular +expressions. Unfortunately, automata are not so straightforward to +formalise in theorem provers. The reason is that a natural +representation for automata are graphs, matrices or functions, none of +which are inductive datatypes. Also convenient operations for disjoint +unions of graphs, matrices and functions are not easily formalisiable +in Isabelle/HOL, the theorem prover we are using. In contrast, regular +expressions can be defined conveniently as a datatype and a +corresponding reasoning infrastructure comes for free. We show in this +paper that a central result from formal language theory---the +Myhill-Nerode Theorem---can be recreated using only regular +expressions. From this theorem many closure properties of regular +languages follow. \end{abstract} +\category{F.4.1}{Mathematical Logic and Formal Languages}{Mechanical Theorem Proving} +\category{F.4.3}{Mathematical Logic and Formal Languages}{Formal Languages} +\terms{Interactive theorem proving, regular languages} +\keywords{Myhill-Nerode theorem, regular expressions, Isabelle theorem prover} +\begin{document} +\begin{bottomstuff} +Corresponding Author: Christian Urban, Department of Informatics, King's College London, +Strand, London WC2R 2LS, UK. Email: christian.urban@kcl.ac.uk.\\ +$^\star$This is a revised and expanded version of \cite{WuZhangUrban11}. +\end{bottomstuff} \maketitle + \input{session} -%%\mbox{}\\[-10mm] -\bibliographystyle{plain} -\bibliography{root} - \end{document} %%% Local Variables: