Skip to main content
SpringerLink
Log in

Extended Regular Expressions: Succinctness and Decidability

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

Most modern implementations of regular expression engines allow the use of variables (also called backreferences). The resulting extended regular expressions (which, in the literature, are also called practical regular expressions, rewbr, or regex) are able to express non-regular languages.

The present paper demonstrates that extended regular-expressions cannot be minimized effectively (neither with respect to length, nor number of variables), and that the tradeoff in size between extended and “classical” regular expressions is not bounded by any recursive function. In addition to this, we prove the undecidability of several decision problems (universality, regularity, and cofiniteness) for extended regular expressions. Furthermore, we show that all these results hold even if the extended regular expressions contain only a single variable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Abigail: Re: Random number in perl. Posting in the newsgroup comp.lang.perl.misc, October 1997. Message-ID slrn64sudh.qp.abigail@betelgeuse.wayne.fnx.com

  2. Aho, A.: Algorithms for finding patterns in strings. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. A, Chap. 5, pp. 255–300. Amsterdam, Elsevier (1990)

    Google Scholar 

  3. Aho, A., Hopcroft, J., Ullman, J.: The Design and Analysis of Computer Algorithms, Chap. 10.6, pp. 395–400. Addison-Wesley, Reading (1974)

    MATH  Google Scholar 

  4. Albert, J., Wegner, L.: Languages with homomorphic replacements. Theor. Comput. Sci. 16, 291–305 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bordihn, H., Dassow, J., Holzer, M.: Extending regular expressions with homomorphic replacements. RAIRO Theor. Inform. Appl. 44(2), 229–255 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bremer, J., Freydenberger, D.D.: Inclusion problems for patterns with a bounded number of variables. In: Proc. 14th International Conference on Developments in Language Theory, DLT 2010. LNCS, vol. 6224, pp. 100–111. Springer, Heidelberg (2010)

    Google Scholar 

  7. Câmpeanu, C., Santean, N.: On the intersection of regex languages with regular languages. Theor. Comput. Sci. 410(24–25), 2336–2344 (2009)

    Article  MATH  Google Scholar 

  8. Câmpeanu, C., Yu, S.: Pattern expressions and pattern automata. Inf. Process. Lett. 92(6), 267–274 (2004)

    Article  MATH  Google Scholar 

  9. Câmpeanu, C., Salomaa, K., Yu, S.: A formal study of practical regular expressions. Int. J. Found. Comput. Sci. 14, 1007–1018 (2003)

    Article  MATH  Google Scholar 

  10. Carle, B., Narendran, P.: On extended regular expressions. In: Proc. Language and Automata Theory and Applications, Third International Conference, LATA 2009. LNCS, vol. 5457, pp. 279–289. Springer, Heidelberg (2009)

    Google Scholar 

  11. Cassaigne, J.: Unavoidable patterns. In: Lothaire, M. (ed.) Algebraic Combinatorics on Words, Chap. 3, pp. 111–134. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  12. Currie, J.: Open problems in pattern avoidance. Am. Math. Mon. 100(8), 790–793 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cutland, N.: Computability. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  14. Della Penna, G., Intrigila, B., Tronci, E., Zilli, M.V.: Synchronized regular expressions. Acta Inform. 39(1), 31–70 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Freydenberger, D.D.: Extended regular expressions: Succinctness and decidability. In: 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), vol. 9, pp. 507–518. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl (2011)

    Google Scholar 

  16. Friedl, J.: Mastering Regular Expressions, 3rd edn. O’Reilly Media, Sebastopol (2006)

    Google Scholar 

  17. Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Inf. Process. Lett. 59(2), 75–77 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Goldstine, J., Kappes, M., Kintala, C.M.R., Leung, H., Malcher, A., Wotschke, D.: Descriptional complexity of machines with limited resources. J. Univers. Comput. Sci. 8(2), 193–234 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Hartmanis, J.: On Gödel speed-up and succinctness of language representations. Theor. Comput. Sci. 26(3), 335–342 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Holzer, M., Kutrib, M.: The complexity of regular(-like) expressions. In: Proc. 14th Conference on Developments in Language Theory, DLT 2010. LNCS, vol. 6224, pp. 16–30. Springer, Heidelberg (2010)

    Google Scholar 

  21. Holzer, M., Kutrib, M.: Descriptional complexity—an introductory survey. In: Martín-Vide, C. (ed.) Scientific Applications of Language Methods, pp. 1–58. Imperial College Press, London (2010)

    Chapter  Google Scholar 

  22. Hopcroft, J., Ullman, J.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)

    MATH  Google Scholar 

  23. Kleene, S.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J., Ashby, W.R. (eds.) Automata Studies, pp. 3–42. Princeton University Press, Princeton (1956)

    Google Scholar 

  24. Kutrib, M.: The phenomenon of non-recursive trade-offs. Int. J. Found. Comput. Sci. 16(5), 957–973 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Larsen, K.: Regular expressions with nested levels of back referencing form a hierarchy. Inf. Process. Lett. 65(4), 169–172 (1998)

    Article  Google Scholar 

  26. Meyer, A.R., Fischer, M.J.: Economy of description by automata, grammars, and formal systems. In: 12th Annual Symposium on Switching and Automata Theory, SWAT (FOCS), pp. 188–191. IEEE Computer Society, Washington (1971)

    Chapter  Google Scholar 

  27. Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall, Upper Saddle River (1967)

    MATH  Google Scholar 

  28. Odifreddi, P.: Classical Recursion Theory, vol. I. Elsevier, Amsterdam (1989)

    MATH  Google Scholar 

  29. Odifreddi, P.: Classical Recursion Theory, vol. II. Elsevier, Amsterdam (1999)

    MATH  Google Scholar 

  30. Reidenbach, D., Schmid, M.L.: A polynomial time match test for large classes of extended regular expressions. In: Proc. 15th International Conference on Implementation and Application of Automata, CIAA 2010. LNCS, vol. 6482, pp. 241–250. Springer, Heidelberg (2010)

    Google Scholar 

Download references

Acknowledgements

The author wishes to thank Nicole Schweikardt and the anonymous referees for the conference version [15] and the present version for their helpful remarks.

Author information

Authors and Affiliations

  1. Institute for Computer Science, Goethe-University, Postfach 11 19 32, 60054, Frankfurt am Main, Germany

    Dominik D. Freydenberger

Authors
  1. Dominik D. Freydenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dominik D. Freydenberger.

Additional information

A preliminary version of this article appeared as [15].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Freydenberger, D.D. Extended Regular Expressions: Succinctness and Decidability. Theory Comput Syst 53, 159–193 (2013). https://doi.org/10.1007/s00224-012-9389-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-012-9389-0

Keywords

Search

Navigation