<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN" "http://www.w3.org/TR/REC-html40/loose.dtd"> <HEAD><TITLE>ADU</TITLE><BASE HREF="http://nms.kcl.ac.uk/christian.urban/"></HEAD><BODY TEXT="#000000" BGCOLOR="#4169E1" LINK="#0000EF" VLINK="#51188E" ALINK="#FF0000"><TABLE WIDTH="100%" BGCOLOR="#4169E1" BORDER="0" FRAME="border" CELLPADDING="10" CELLSPACING="2" RULES="all"><!-- right column --><TR><TD BGCOLOR="#FFFFFF" WIDTH="75%"><H2>POSIX Lexing with Derivatives of Regular Expressions (Proof Pearl)</H2>Fahad Ausaf, Roy Dyckhoff, Christian Urban<p>Brzozowski introduced the notion of derivatives for regularexpressions. They can be used for a very simple regular expressionmatching algorithm. Sulzmann and Lu cleverly extended this algorithmin order to deal with POSIX matching, which is the underlyingdisambiguation strategy for regular expressions needed in lexers.Sulzmann and Lu have made available on-line what they call a``rigorous proof'' of the correctness of their algorithm w.r.t. theirspecification; regrettably, it appears to us to have unfillable gaps.In the first part of this paper we give our inductive definition ofwhat a POSIX value is and show (i) that such a value is unique (forgiven regular expression and string being matched) and (ii) thatSulzmann and Lu's algorithm always generates such a value (providedthat the regular expression matches the string). We also prove thecorrectness of an optimised version of the POSIX matchingalgorithm. Our definitions and proof are much simpler than those bySulzmann and Lu and can be easily formalised in Isabelle/HOL. In thesecond part we analyse the correctness argument by Sulzmann and Lu andexplain why the gaps in this argument cannot be filled easily.<H3>Theory Files for Isabelle 2016</H3><ul><li> <A HREF="http://talisker.inf.kcl.ac.uk/%7Eurbanc/cgi-bin/repos.cgi/lexing/raw-file/tip/thys/Lexer.thy">Lexer.thy</A><li> <A HREF="http://talisker.inf.kcl.ac.uk/%7Eurbanc/cgi-bin/repos.cgi/lexing/raw-file/tip/thys/Simplifying.thy">Simplifying.thy</A></ul><H3>Links</H3><ul><li> <A HREF="http://talisker.inf.kcl.ac.uk/%7Eurbanc/cgi-bin/repos.cgi/lexing/raw-file/tip/thys/paper.pdf">our paper</A><li> <A HREF="http://talisker.inf.kcl.ac.uk/%7Eurbanc/cgi-bin/repos.cgi/lexing/raw-file/tip/Literature/sulzmann14-new.pdf">the paper</A> by Sulzmann and Lu</ul></TABLE><P><!-- hhmts start --> Last modified: Wed May 18 15:01:19 BST 2016 <!-- hhmts end --><a href="http://validator.w3.org/check/referer">[Validate this page.]</a></BODY></HTML>