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Limited Automata: Properties, Complexity and Variants

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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

Limited automata are single-tape Turing machines with severe rewriting restrictions. They have been introduced in 1967 by Thomas Hibbard, who proved that they have the same computational power as pushdown automata. Hence, they provide an alternative characterization of the class of context-free languages in terms of recognizing devices. After that paper, these models have been almost forgotten for many years. Only recently limited automata were reconsidered in a series of papers, where several properties of them and of their variants have been investigated. In this work we present an overview of the most important results obtained in these researches. We also discuss some related models and possible lines for future investigations.

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Notes

  1. 1.

    Actually, the original definition of d-limited automata was given by Hibbard by considering some kinds of rewriting systems [9]. It is not difficult to reformulate it, as we did, in terms of Turing machines.

  2. 2.

    We could have moves that do not change the contents of a cell even in the first visit, as we seen in the above example of 1-limited automaton accepting \(K_n\). For a such a move, we can imagine that the cell is rewritten by the same symbol which is already in it.

  3. 3.

    The proof of this result has been given by Hennie, for the deterministic case [8]. Several improvements have been presented in the literature. See [24] for a survey.

  4. 4.

    The maximum number of visits to a cell up to the last rewriting, namely the measure corresponding to limited automata, is sometimes called dual return complexity [34].

  5. 5.

    A nonunary rewriting alphabet is necessary for strongly limited automata to recognize all context-free languages. For instance, to recognize the set of palindromes, a working alphabet of at least 3 symbols is required [27].

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Acknowledgment

I am very grateful to Luca Prigioniero for his valuable and helpful comments.

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Authors and Affiliations

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

    Giovanni Pighizzini

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  1. Giovanni Pighizzini
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Correspondence to Giovanni Pighizzini .

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  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. (2019). Limited Automata: Properties, Complexity and Variants. 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_4

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