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Linear-Time Limited Automata

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Descriptional Complexity of Formal Systems (DCFS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10952))

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Abstract

The time complexity of 1-limited automata is investigated from a descriptional complexity view point. Though the model recognizes regular languages only, it may use quadratic time in the input length. We show that, with a polynomial increase in size and preserving determinism, each 1-limited automaton can be transformed into an halting linear-time equivalent one. We also obtain polynomial transformations into related models, including weight-reducing Hennie machines, and we show exponential gaps for converse transformations in the deterministic case.

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Notes

  1. 1.

    In nondeterministic linear-time devices each accepting computation has linear length.

  2. 2.

    2dfa+cgs also correspond to synchronous two-way deterministic finite verifiers [6].

  3. 3.

    \(\tau _{z'X}(p)\) is undefined if one of the two following cases of the computation starting in \(z' \cdot p\cdot X\) occurs: either, after a finite number of steps, no successive transition is defined (incompleteness of \(\mathcal {A}\)), or the computation eventually enters a deterministic loop (non-haltingness of \(\mathcal {A}\)).

  4. 4.

    We could do a finer construction, based on Sipser’s backward construction [14], which has linear cost (without counting the relative position and relative frontier components) instead of the expensive \(O(n^3) \) cost of the clocked simulation presented here. For an adaptation to finite automata, see [2].

  5. 5.

    We implicitly fix a bijection from .

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Acknowledgement

We are very indebted to Giovanni Pighizzini for suggesting the problem and for many stimulating conversations.

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Authors and Affiliations

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

    Bruno Guillon & Luca Prigioniero

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  1. Bruno Guillon
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  2. Luca Prigioniero
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Correspondence to Bruno Guillon .

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Editors and Affiliations

  1. Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia, Canada

    Stavros Konstantinidis

  2. Dipartimento di Informatica e Comunicazi, Universita degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

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Guillon, B., Prigioniero, L. (2018). Linear-Time Limited Automata. In: Konstantinidis, S., Pighizzini, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2018. Lecture Notes in Computer Science(), vol 10952. Springer, Cham. https://doi.org/10.1007/978-3-319-94631-3_11

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  • DOI: https://doi.org/10.1007/978-3-319-94631-3_11

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