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Visit-Bounded Stack Automata

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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.

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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.

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Acknowledgements

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

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Authors and Affiliations

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

    Jozef Jirásek & Ian McQuillan

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  1. Jozef Jirásek
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  2. Ian McQuillan
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Correspondence to Jozef Jirásek.

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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.

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

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