Skip to main content
SpringerLink
Log in

Visit-Bounded Stack Automata

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

Abstract

An automaton is k-visit-bounded if during any computation its work tape head visits each tape cell at most k times. In this paper we consider stack automata which are k-visit-bounded for some integer k. This restriction resets the visits when popping (unlike similarly defined Turing machine restrictions) which we show allows the model to accept a proper superset of context-free languages and also a proper superset of languages of visit-bounded Turing machines. We study two variants of visit-bounded stack automata: one where only instructions that move the stack head downwards increase the number of visits of the destination cell, and another where any transition increases the number of visits. We prove that the two types of automata recognize the same languages. We then show that all languages recognized by visit-bounded stack automata are effectively semilinear, and hence are letter-equivalent to regular languages, which can be used to show other properties.

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

Similar content being viewed by others

Notes

  1. A preliminary version of this paper appeared in the Proceedings of the 26th International Conference on Developments in Language Theory (DLT 2022), Lecture Notes in Computer Science 13257, 189-200, 2022. This version is substantially extended from the previous version to include the complete proof of Theorem 2, and full details of all algorithms used in the proof of Theorem 4, which are central to the work.

References

  1. Harrison, M.: Introduction to Formal Language Theory. Addison-Wesley, Reading, Ma (1978)

    MATH  Google Scholar 

  2. Parikh, R.: On context-free languages. J. ACM 13(4), 570–581 (1966)

    Article  MATH  Google Scholar 

  3. Ibarra, O., McQuillan, I.: Semilinearity of families of languages. International. J. Found. Comput. Sci. 31(8), 1179–1198 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Harju, T., Ibarra, O., Karhumäki, J., Salomaa, A.: Some decision problems concerning semilinearity and commutation. J. Comput. Syst. Sci. 65(2), 278–294 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ibarra, O.H., McQuillan, I.: On families of full trios containing counter machine languages. Theor. Comput. Sci. 799, 71–93 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ibarra, O.H., McQuillan, I., Ravikumar, B.: On counting functions and slenderness of languages. Theor. Comput. Sci. 777, 356–378 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Joshi, A.K.: Tree adjoining grammars: How much context-sensitivity is required to provide reasonable structural descriptions?, pp. 206–250. Cambridge University Press (1985)

  8. Ginsburg, S., Greibach, S., Harrison, M.: Stack automata and compiling. J. ACM 14(1), 172–201 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ginsburg, S., Greibach, S., Harrison, M.: One-way stack automata. J. ACM 14(2), 389–418 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Greibach, S.: Checking automata and one-way stack languages. J. Comput. Syst. Sci. 3(2), 196–217 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  11. Greibach, S.A.: One way finite visit automata. Theor. Comput. Sci. 6, 175–221 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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

  13. Harrison, M.A., Schkolnick, M.: A grammatical characterization of one-way nondeterministic stack languages. J. ACM 18(2), 148–172 (1971)

  14. Ibarra, O., Jirásek, J., McQuillan, I., Prigioniero, L.: Space complexity of stack automata models. Int. J. Found. Comput. Sci. 32(06), 801–823 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Supported, in part, by a grant from Natural Sciences and Engineering Research Council of Canada 2022-05092 (Ian McQuillan).

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Saskatchewan, Saskatoon, SK S7N 5A9, Saskatchewan, Canada

    Jozef Jirásek & Ian McQuillan

Authors
  1. Jozef Jirásek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ian McQuillan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jozef Jirásek.

Ethics declarations

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Detailed Algorithm Listings

Appendix A: Detailed Algorithm Listings

This section contains the description of two algorithms: one to advance the current transition counter of a history card and verify that all encountered transitions can be properly linked, and the other to match two history cards together. Since there are only finitely many possible history cards, the pushdown automaton P from the proof of Theorem 4 does not need to actually perform these procedures, but their result for every possible input card can be encoded in its transition function.

figure b

Advancing the completed transition counter of a history card to the next \(\texttt{push}\) or \(\texttt{pop}\) transition, or to the end of the computation. At the same time, it is verified that every transition in this card can be properly linked to some other transition.

figure c

Matching two history cards. Returns true if the transitions on the cards can be matched together, false if not.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jirásek, J., McQuillan, I. Visit-Bounded Stack Automata. Theory Comput Syst 67, 956–975 (2023). https://doi.org/10.1007/s00224-023-10124-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-023-10124-0

Keywords

Search

Navigation