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Computational Limitations of Affine Automata

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Unconventional Computation and Natural Computation (UCNC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11493))

Abstract

We present two new results on the computational limitations of affine automata. First, we show that the computation of bounded-error rational-valued affine automata is simulated in logarithmic space. Second, we give an impossibility result for algebraic-valued affine automata. As a result, we identify some unary languages (in logarithmic space) that are not recognized by algebraic-valued affine automata with cutpoints.

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Notes

  1. 1.

    It is known that \(\mathsf {L}\subsetneq \mathsf {PSPACE}\), so it is plausible that \(\mathsf {PSPACE}\)-complete languages are not in \(\mathsf {AfL}_{\mathbb {Q}}\).

  2. 2.

    Note that the new matrix obtained may not be affine, so it would be wrong to assume that all AfAs have to admit an equivalent one with only irrational eigenvalues. However, this does not affect this proof, since we do not require the new matrix to be affine, we only study the values that the fraction \(\frac{\left| P(\alpha M)^n\varvec{v}\right| }{\left| (\alpha M)^n\varvec{v}\right| } =\frac{\left| PM^n\varvec{v}\right| }{\left| M^n\varvec{v}\right| }\) take.

  3. 3.

    This is the only point we need the assumption that the matrix entries are algebraic.

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Acknowledgments

Yakaryılmaz was partially supported by Akadēmiskā personāla atjaunotne un kompetenču pilnveide Latvijaspilnveide Latvijas Universitātē līg Nr. 8.2.2.0/18/A/010 LU Nr. ESS2018/289 and ERC Advanced Grant MQC. Hirvensalo was partially supported by the Väisälä Foundation and Moutot by ANR project CoCoGro (ANR-16-CE40-0005).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Turku, 20014, Turku, Finland

    Mika Hirvensalo & Etienne Moutot

  2. LIP, ENS de Lyon – CNRS – UCBL – Université de Lyon, École Normale Supérieure de Lyon, Lyon, France

    Etienne Moutot

  3. Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Rīga, Latvia

    Abuzer Yakaryılmaz

Authors
  1. Mika Hirvensalo
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  2. Etienne Moutot
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  3. Abuzer Yakaryılmaz
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Corresponding author

Correspondence to Mika Hirvensalo .

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Editors and Affiliations

  1. University of Saskatchewan, Saskatoon, SK, Canada

    Ian McQuillan

  2. University of Electro-Communications, Tokyo, Japan

    Shinnosuke Seki

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Hirvensalo, M., Moutot, E., Yakaryılmaz, A. (2019). Computational Limitations of Affine Automata. In: McQuillan, I., Seki, S. (eds) Unconventional Computation and Natural Computation. UCNC 2019. Lecture Notes in Computer Science(), vol 11493. Springer, Cham. https://doi.org/10.1007/978-3-030-19311-9_10

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  • DOI: https://doi.org/10.1007/978-3-030-19311-9_10

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  • Online ISBN: 978-3-030-19311-9

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