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Nonuniform Families of Polynomial-Size Quantum Finite Automata and Quantum Logarithmic-Space Computation with Polynomial-Size Advice

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Language and Automata Theory and Applications (LATA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11417))

Abstract

The state complexity of a finite(-state) automaton intuitively measures the size of the description of the automaton. Sakoda and Sipser [STOC 1972, pp. 275–286] were concerned with nonuniform families of finite automata and they discussed the behaviors of nonuniform complexity classes defined by families of such finite automata having polynomial-size state complexity. In a similar fashion, we introduce nonuniform state complexity classes using families of quantum finite automata. Our primarily concern is one-way quantum finite automata empowered by garbage tapes. We show inclusion and separation relationships among nonuniform state complexity classes of various one-way finite automata, including deterministic, nondeterministic, probabilistic, and quantum finite automata of polynomial size. For two-way quantum finite automata equipped with garbage tapes, we discover a close relationship between the nonuniform state complexity of such a polynomial-size quantum finite automata family and the parameterized complexity class induced by quantum logarithmic-space computation assisted by polynomial-size advice.

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Notes

  1. 1.

    We use this term “1-way” in a strict sense that a tape head always moves to the right and is not allowed to stay still on the same cell. This term is called “real time” in certain literature.

  2. 2.

    In [4], the polynomial-time \(2\mathrm {BP}\) was considered under the name of \(\mathrm {2P}_2\) and the polynomial-time \(2\mathrm {P}\) was studied under the name of \(\mathrm {2P}\) but it is restricted to so-called “regular” language families. Here, we do not demand such a condition.

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Authors and Affiliations

  1. Faculty of Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui, 910-8507, Japan

    Tomoyuki Yamakami

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  1. Tomoyuki Yamakami
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Correspondence to Tomoyuki Yamakami .

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Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  2. Saint Petersburg State University, St. Petersburg, Russia

    Alexander Okhotin

  3. Ariel University, Ariel, Israel

    Dana Shapira

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Yamakami, T. (2019). Nonuniform Families of Polynomial-Size Quantum Finite Automata and Quantum Logarithmic-Space Computation with Polynomial-Size Advice. In: Martín-Vide, C., Okhotin, A., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2019. Lecture Notes in Computer Science(), vol 11417. Springer, Cham. https://doi.org/10.1007/978-3-030-13435-8_10

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  • DOI: https://doi.org/10.1007/978-3-030-13435-8_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-13434-1

  • Online ISBN: 978-3-030-13435-8

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