Abstract
Brzozowski introduced the notion of derivatives for regular expressions. They can be used for a very simple regular expression matching algorithm. Sulzmann and Lu cleverly extended this algorithm in order to deal with POSIX matching, which is the underlying disambiguation 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. their specification; regrettably, it appears to us to have unfillable gaps. In the first part of this paper we give our inductive definition of what a POSIX value is and show (i) that such a value is unique (for given regular expression and string being matched) and (ii) that Sulzmann and Lu’s algorithm always generates such a value (provided that the regular expression matches the string). We also prove the correctness of an optimised version of the POSIX matching algorithm. Our definitions and proof are much simpler than those by Sulzmann and Lu and can be easily formalised in Isabelle/HOL. In the second part we analyse the correctness argument by Sulzmann and Lu and explain why the gaps in this argument cannot be filled easily.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
An extended version of [11] is available at the website of its first author; this extended version already includes remarks in the appendix that their informal proof contains gaps, and possible fixes are not fully worked out.
- 2.
Sulzmann and Lu state this clause as \(inj\ c\ c\ {(}{)}\) \(\,\mathop {=}\limits ^{\text{ def }}\,\) \(Char\ c\), but our deviation is harmless.
- 3.
All deviations we introduced are harmless.
References
Ausaf, F., Dyckhoff, R., Urban, C.: POSIX Lexing with Derivatives of Regular Expressions. Archive of Formal Proofs (2016). http://www.isa-afp.org/entries/Posix-Lexing.shtml, Formal proof development
Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11(4), 481–494 (1964)
Coquand, T., Siles, V.: A decision procedure for regular expression equivalence in type theory. In: Jouannaud, J.-P., Shao, Z. (eds.) CPP 2011. LNCS, vol. 7086, pp. 119–134. Springer, Heidelberg (2011)
Frisch, A., Cardelli, L.: Greedy regular expression matching. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 618–629. Springer, Heidelberg (2004)
Grathwohl, N.B.B., Henglein, F., Rasmussen, U.T.: A Crash-Course in Regular Expression Parsing and Regular Expressions as Types. Technical report, University of Copenhagen (2014)
Krauss, A., Nipkow, T.: Proof pearl: regular expression equivalence and relation algebra. J. Autom. Reasoning 49, 95–106 (2012)
Kuklewicz, C.: Regex Posix. https://wiki.haskell.org/Regex_Posix
Nipkow, T.: Verified lexical analysis. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 1–15. Springer, Heidelberg (1998)
Owens, S., Slind, K.: Adapting functional programs to higher order logic. High. Order Symbolic Comput. 21(4), 377–409 (2008)
Pierce, B.C., Casinghino, C., Gaboardi, M., Greenberg, M., Hriţcu, C., Sjöberg, V., Yorgey, B.: Software Foundations. Electronic Textbook (2015). http://www.cis.upenn.edu/~bcpierce/sf
Sulzmann, M., Lu, K.Z.M.: POSIX regular expression parsing with derivatives. In: Codish, M., Sumii, E. (eds.) FLOPS 2014. LNCS, vol. 8475, pp. 203–220. Springer, Heidelberg (2014)
Sulzmann, M., van Steenhoven, P.: A flexible and efficient ML lexer tool based on extended regular expression submatching. In: Cohen, A. (ed.) CC 2014 (ETAPS). LNCS, vol. 8409, pp. 174–191. Springer, Heidelberg (2014)
Vansummeren, S.: Type inference for unique pattern matching. ACM Trans. Program. Lang. Syst. 28(3), 389–428 (2006)
Acknowledgements
We are very grateful to Martin Sulzmann for his comments on our work and moreover for patiently explaining to us the details in [11]. We also received very helpful comments from James Cheney and anonymous referees.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Ausaf, F., Dyckhoff, R., Urban, C. (2016). POSIX Lexing with Derivatives of Regular Expressions (Proof Pearl). In: Blanchette, J., Merz, S. (eds) Interactive Theorem Proving. ITP 2016. Lecture Notes in Computer Science(), vol 9807. Springer, Cham. https://doi.org/10.1007/978-3-319-43144-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-43144-4_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-43143-7
Online ISBN: 978-3-319-43144-4
eBook Packages: Computer ScienceComputer Science (R0)