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Variations of Checking Stack Automata: Obtaining Unexpected Decidability Properties

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Developments in Language Theory (DLT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10396))

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Abstract

We introduce a model of one-way language acceptors (a variant of a checking stack automaton) and show the following decidability properties:

  1. 1.

    The deterministic version has a decidable membership problem but has an undecidable emptiness problem.

  2. 2.

    The nondeterministic version has an undecidable membership problem and emptiness problem.

There are many models of accepting devices for which there is no difference with these problems between deterministic and nondeterministic versions, i.e., the membership problem for both versions are either decidable or undecidable, and the same holds for the emptiness problem. As far as we know, the model we introduce above is the first one-way model to exhibit properties (1) and (2). We define another family of one-way acceptors where the nondeterministic version has an undecidable emptiness problem, but the deterministic version has a decidable emptiness problem. We also know of no other model with this property in the literature. We also investigate decidability properties of other variations of checking stack automata (e.g., allowing multiple stacks, two-way input, etc.). Surprisingly, two-way deterministic machines with multiple checking stacks and multiple reversal-bounded counters are shown to have a decidable membership problem, a very general model with this property.

The research of O.H. Ibarra was supported, in part, by NSF Grant CCF-1117708.

The research of I. McQuillan was supported, in part, by Natural Sciences and Engineering Research Council of Canada Grant 2016-06172.

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Notes

  1. 1.

    A counter is reversal-bounded if there is a bound on the number of changes between increasing and decreasing.

References

  1. Baker, B.S., Book, R.V.: Reversal-bounded multipushdown machines. J. Comput. Syst. Sci. 8(3), 315–332 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Engelfriet, J.: The power of two-way deterministic checking stack automata. Inf. Comput. 80(2), 114–120 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ginsburg, S.: Algebraic and Automata-Theoretic Properties of Formal Languages. North-Holland Publishing Company, Amsterdam (1975)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Gurari, E.M., Ibarra, O.H.: Two-way counter machines and diophantine equations. J. ACM 29(3), 863–873 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

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

    MATH  Google Scholar 

  8. Ibarra, O., McQuillan, I.: On store languages of language acceptors (2017, submitted). A preprint appears in https://arxiv.org/abs/1702.07388

  9. Ibarra, O., McQuillan, I.: Variations of checking stack automata: Obtaining unexpected decidability properties (2017). Full version of current paper with all definitions and proofs. https://arxiv.org/abs/1705.09732

  10. Ibarra, O.H.: Reversal-bounded multicounter machines and their decision problems. J. ACM 25(1), 116–133 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ibarra, O.H., Jiang, T., Tran, N., Wang, H.: New decidability results concerning two-way counter machines. SIAM J. Comput. 23(1), 123–137 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of California, Santa Barbara, CA, 93106, USA

    Oscar H. Ibarra

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

    Ian McQuillan

Authors
  1. Oscar H. Ibarra
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  2. Ian McQuillan
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Corresponding author

Correspondence to Ian McQuillan .

Editor information

Editors and Affiliations

  1. Université de Liège, Liège, Belgium

    Émilie Charlier

  2. Université de Liège, Liège, Belgium

    Julien Leroy

  3. Université de Liège, Liège, Belgium

    Michel Rigo

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Ibarra, O.H., McQuillan, I. (2017). Variations of Checking Stack Automata: Obtaining Unexpected Decidability Properties. In: Charlier, É., Leroy, J., Rigo, M. (eds) Developments in Language Theory. DLT 2017. Lecture Notes in Computer Science(), vol 10396. Springer, Cham. https://doi.org/10.1007/978-3-319-62809-7_17

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  • DOI: https://doi.org/10.1007/978-3-319-62809-7_17

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-62808-0

  • Online ISBN: 978-3-319-62809-7

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