Skip to main content
SpringerLink
Log in
Book cover

International Conference on Descriptional Complexity of Formal Systems

DCFS 2017: Descriptional Complexity of Formal Systems pp 299–310Cite as

The Quotient Operation on Input-Driven Pushdown Automata

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10316))

Abstract

The quotient of a formal language K by another language L is the set of all strings obtained by taking a string from K that ends with a suffix from L, and removing that suffix. The quotient of a regular language by any language is always regular, whereas the context-free languages and many of their subfamilies, such as the linear and the deterministic languages, are not closed under the quotient operation. This paper establishes the closure of the family of input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the quotient operation. A construction of automata representing the result of the operation is given, and its state complexity with respect to nondeterministic IDPDA is shown to be \(m^2n + O(m)\), where m and n is the number of states in the automata recognizing K and L, respectively.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Alur, R., Madhusudan, P.: Visibly pushdown languages. In: ACM Symposium on Theory of Computing, STOC 2004, Chicago, USA, pp. 202–211, 13–16 June 2004

    Google Scholar 

  2. Alur, R., Madhusudan, P.: Adding nesting structure to words. J. ACM 56, 3 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Birget, J.-C.: Intersection and union of regular languages and state complexity. Inf. Process. Lett. 43, 185–190 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. von Braunmühl, B., Verbeek, R.: Input driven languages are recognized in \(\log n\) space. Ann. Discret. Math. 24, 1–20 (1985)

    MathSciNet  MATH  Google Scholar 

  5. Ginsburg, S., Greibach, S.A.: Deterministic context-free languages. Inf. Control 9(6), 620–648 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ginsburg, S., Spanier, E.H.: Quotients of context-free languages. J. ACM 10(4), 487–492 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  7. Han, Y.-S., Salomaa, K.: Nondeterministic state complexity of nested word automata. Theoret. Comput. Sci. 410, 2961–2971 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hartmanis, J.: Context-free languages and Turing machine computations. In: Proceedings of Symposia in Applied Mathematics, vol. 19, pp. 42–51. AMS (1967)

    Google Scholar 

  9. Latteux, M., Leguy, B., Ratoandromanana, B.: The family of one-counter languages is closed under quotient. Acta Inform. 22(5), 579–588 (1985)

    MathSciNet  MATH  Google Scholar 

  10. Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL-recognition. In: Bakker, J., Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 422–435. Springer, Heidelberg (1980). doi:10.1007/3-540-10003-2_89

    Chapter  Google Scholar 

  11. Okhotin, A.: Input-driven languages are linear conjunctive. Theoret. Comput. Sci. 618, 52–71 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Okhotin, A., Salomaa, K.: Complexity of input-driven pushdown automata. SIGACT News 45(2), 47–67 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Okhotin, A., Salomaa, K.: Descriptional complexity of unambiguous input-driven pushdown automata. Theoret. Comput. Sci. 566, 1–11 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Okhotin, A., Salomaa, K.: State complexity of operations on input-driven pushdown automata. J. Comput. Syst. Sci. 86, 207–228 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Okhotin, A., Salomaa, K.: Edit distance neighbourhoods of input-driven pushdown automata. In: Weil, P. (ed.) CSR 2017. LNCS, vol. 10304, pp. 260–272. Springer, Cham (2017). doi:10.1007/978-3-319-58747-9_23

    Chapter  Google Scholar 

  16. Piao, X., Salomaa, K.: Operational state complexity of nested word automata. Theoret. Comput. Sci. 410, 3290–3302 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Salomaa, K.: Limitations of lower bound methods for deterministic nested word automata. Inf. Comput. 209, 580–589 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wood, D.: A further note on top-down deterministic languages. Comput. J. 14(4), 396–403 (1971)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

The authors are grateful to the anonymous reviewers for many pertinent comments and suggestions; the implementation of some of them is deferred until the full version of this paper.

Author information

Authors and Affiliations

  1. St. Petersburg State University, 14th Line V.O., 29B, Saint Petersburg, 199178, Russia

    Alexander Okhotin

  2. School of Computing, Queen’s University, Kingston, ON, K7L 2N8, Canada

    Kai Salomaa

Authors
  1. Alexander Okhotin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kai Salomaa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Okhotin .

Editor information

Editors and Affiliations

  1. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  2. University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada

    Cezar Câmpeanu

Rights and permissions

Reprints and permissions

Copyright information

© 2017 IFIP International Federation for Information Processing

About this paper

Cite this paper

Okhotin, A., Salomaa, K. (2017). The Quotient Operation on Input-Driven Pushdown Automata. In: Pighizzini, G., Câmpeanu, C. (eds) Descriptional Complexity of Formal Systems. DCFS 2017. Lecture Notes in Computer Science(), vol 10316. Springer, Cham. https://doi.org/10.1007/978-3-319-60252-3_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-60252-3_24

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-60251-6

  • Online ISBN: 978-3-319-60252-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation