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Operational Complexity: NFA-to-DFA Trade-Off

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Descriptional Complexity of Formal Systems (DCFS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13918))

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Abstract

We examine operational complexity assuming that the arguments are given as nondeterministic finite automata and the resulting language is represented by a deterministic finite automaton. We show that the known upper bounds for Boolean operations and concatenation are met by ternary languages, and we prove that they are asymptotically tight in the binary case. For the cut and square operations, we get tight upper bounds \(2^{m-1}(2^n+1)\) and \(\frac{3}{4}2^{2n}\), respectively. Our witnesses are described over a four-letter alphabet for cut, and a ten-letter alphabet for square. We also show that the tight upper bound on the syntactic complexity of a language given by an n-state NFA is \(2^{n^2}\). For the square root operation, we provide a lower bound \(2^{n^2-n}\) and an upper bound \(2^{n^2}\).

M. Hospodár—This publication was supported by the Operational Programme Integrated Infrastructure (OPII) for the project 313011BWH2: “InoCHF - Research and development in the field of innovative technologies in the management of patients with CHF”, co-financed by the European Regional Development Fund.

M. Hospodár and G. Jirásková—Supported by the Slovak Grant Agency for Science (VEGA) under contract 2/0096/23 “Automata and Formal Languages: Descriptional and Computational Complexity”.

J. Jirásek and J. Šebej—Supported by the Slovak Grant Agency for Science (VEGA) under contract 1/0177/21 “Descriptional and Computational Complexity of Automata and Algorithms”.

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Acknowledgment

We would like to thank Ivana Krajňáková for her wide and long-term contribution to some parts of this research.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01, Košice, Slovakia

    Michal Hospodár & Galina Jirásková

  2. Department of Computer Science, P. J. Šafárik University, Jesenná 5, 040 01, Košice, Slovakia

    Jozef Jirásek & Juraj Šebej

Authors
  1. Michal Hospodár
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  2. Jozef Jirásek
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  3. Galina Jirásková
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  4. Juraj Šebej
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Corresponding author

Correspondence to Galina Jirásková .

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Editors and Affiliations

  1. University of Potsdam, Potsdam, Germany

    Henning Bordihn

  2. Santa Clara University, Santa Clara, CA, USA

    Nicholas Tran

  3. University of Debrecen, Debrecen, Hungary

    György Vaszil

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Hospodár, M., Jirásek, J., Jirásková, G., Šebej, J. (2023). Operational Complexity: NFA-to-DFA Trade-Off. In: Bordihn, H., Tran, N., Vaszil, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2023. Lecture Notes in Computer Science, vol 13918. Springer, Cham. https://doi.org/10.1007/978-3-031-34326-1_6

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  • DOI: https://doi.org/10.1007/978-3-031-34326-1_6

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-031-34326-1

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