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International Conference on Unconventional Computation and Natural Computation

UCNC 2016: Unconventional Computation and Natural Computation pp 116–129Cite as

Language Recognition Power and Succinctness of Affine Automata

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9726))

Abstract

In this work we study a non-linear generalization based on affine transformations of probabilistic and quantum automata proposed recently by Díaz-Caro and Yakaryılmaz [6] referred as affine automata. First, we present efficient simulations of probabilistic and quantum automata by means of affine automata which allows us to characterize the class of exclusive stochastic languages. Then, we initiate a study on the succintness of affine automata. In particular, we show that an infinite family of unary regular languages can be recognized by 2-state affine automata, whereas the number of states of any quantum and probabilistic automata cannot be bounded. Finally, we present the characterization of all (regular) unary languages recognized by two-state affine automata.

The omitted proofs can be found in [22].

A.Yakaryılmaz—Yakaryılmaz was partially supported by CAPES with grant 88881.030338/2013-01 and some parts of this work was done while he was visiting Universidad Nacional de Asunción in September 2015.

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Notes

  1. 1.

    This way of scanning an input tape is sometimes referred to as “strict realtime.”.

  2. 2.

    A superoperator or quantum operator is a positive-semidefinite operation that maps density matrices to density matrices [4, 19].

  3. 3.

    Pures states are vectors in a complex Hilbert space normalized with respect to the \(\ell _2\)-norm.

  4. 4.

    A promise problem \(\mathtt L=(L_{yes},L_{no})\) is solved by a machine M, or M solves \(\mathtt L\), if for all \( w \in \mathtt L_{yes}\), M accepts w, and for all \( w \in \mathtt L_{no}\), M rejects w.

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Acknowledgements

We thank the anonymous referees for their helpful comments.

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Authors and Affiliations

  1. Universidad Nacional de Asunción NIDTEC, Campus Universitario, San Lorenzo, C.P. 2619, Paraguay

    Marcos Villagra

  2. National Laboratory for Scientific Computing, Petrópolis, RJ, 25651-075, Brazil

    Abuzer Yakaryılmaz

Authors
  1. Marcos Villagra
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  2. Abuzer Yakaryılmaz
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Corresponding authors

Correspondence to Marcos Villagra or Abuzer Yakaryılmaz .

Editor information

Editors and Affiliations

  1. Manchester Metropolitan University, Manchester, United Kingdom

    Martyn Amos

  2. University of British Columbia, Vancouver, BC, Canada

    ANNE CONDON

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Villagra, M., Yakaryılmaz, A. (2016). Language Recognition Power and Succinctness of Affine Automata. In: Amos, M., CONDON, A. (eds) Unconventional Computation and Natural Computation. UCNC 2016. Lecture Notes in Computer Science(), vol 9726. Springer, Cham. https://doi.org/10.1007/978-3-319-41312-9_10

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  • DOI: https://doi.org/10.1007/978-3-319-41312-9_10

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