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Pushdown Automata and Constant Height: Decidability and Bounds

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Descriptional Complexity of Formal Systems (DCFS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11612))

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Abstract

It cannot be decided whether a pushdown automaton accepts using constant pushdown height, with respect to the input length, or not. Furthermore, in the case of acceptance in constant height, the height cannot be bounded by any recursive function in the size of the description of the machine. In contrast, in the restricted case of pushdown automata over a one-letter input alphabet, i.e., unary pushdown automata, the above property becomes decidable. Moreover, if the height is bounded by a constant in the input length, then it is at most exponential with respect to the size of the description of the pushdown automaton. This bound cannot be reduced. Finally, if a unary pushdown automaton uses nonconstant height to accept, then the height should grow at least as the logarithm of the input length. This bound is optimal.

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Notes

  1. 1.

    In some papers pdas are presented in different forms. As pointed out in [2], it is possible to turn the definition of pdas into these equivalent forms, with a polynomial increase in size and by preserving the property of being constant height.

  2. 2.

    We point out that for unambiguous pdas, the property is decidable [10].

  3. 3.

    Notice that here H(n) is a function of the size of the pda and not of the input.

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Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini & Luca Prigioniero

Authors
  1. Giovanni Pighizzini
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  2. Luca Prigioniero
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Correspondence to Luca Prigioniero .

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Editors and Affiliations

  1. Slovak Academy of Sciences, Košice, Slovakia

    Michal Hospodár

  2. Slovak Academy of Sciences, Košice, Slovakia

    Galina Jirásková

  3. Saint Mary's University, Halifax, NS, Canada

    Stavros Konstantinidis

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Pighizzini, G., Prigioniero, L. (2019). Pushdown Automata and Constant Height: Decidability and Bounds. In: Hospodár, M., Jirásková, G., Konstantinidis, S. (eds) Descriptional Complexity of Formal Systems. DCFS 2019. Lecture Notes in Computer Science(), vol 11612. Springer, Cham. https://doi.org/10.1007/978-3-030-23247-4_20

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  • DOI: https://doi.org/10.1007/978-3-030-23247-4_20

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-23246-7

  • Online ISBN: 978-3-030-23247-4

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