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Ultrametric Algorithms and Automata

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Book cover Unconventional Computation and Natural Computation (UCNC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9252))

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Abstract

We introduce a notion of ultrametric automata and Turing machines using p-adic numbers to describe random branching of the process of computation. These automata have properties similar to the properties of probabilistic automata but complexity of probabilistic automata and complexity of ultrametric automata can differ very much.

The research was supported by Grant No. 271/2012 from the Latvian Council of Science and by the project ERAF Nr.2DP/2.1.1.1/13/APIA/VIAA/027. Partially supported by Latvian State Research programme NexIT project No.1 “Technologies of ontologies, semantic web and security”.

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References

  1. Ambainis, A.: Polynomial degree vs. quantum query complexity. J. Comput. Syst. Sci. 72(2), 220–238 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Radu, V.: Application. In: Radu, V. (ed.) Stochastic Modeling of Thermal Fatigue Crack Growth. ACM, vol. 1, pp. 63–70. Springer, Heidelberg (2015)

    Google Scholar 

  3. Freivalds, R.: Complexity of probabilistic versus deterministic automata. In: Bārzdinš, J., Bjørner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 565–613. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  4. Freivalds, R.: Non-constructive methods for finite probabilistic automata. Int. J. Found. Comput. Sci. 19(3), 565–580 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Freivalds, R.: Ultrametric automata and Turing machines. In: Voronkov, A. (ed.) Turing-100. EPiC Series, vol. 10, pp. 98–112. EasyChair (2012)

    Google Scholar 

  6. Freivalds, R.: Ultrametric finite automata and turing machines. In: Béal, M.-P., Carton, O. (eds.) DLT 2013. LNCS, vol. 7907, pp. 1–11. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  7. Gouvea, F.Q.: p-adic Numbers: An Introduction. Universitext, 2nd edn. Springer, Heidelberg (1983)

    Google Scholar 

  8. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th ACM Symposium on Theory of Computing, pp. 212–219 (1996)

    Google Scholar 

  9. Hirvensalo, M.: Quantum Computing. Springer, Heidelberg (2001)

    Book  MATH  Google Scholar 

  10. Khrennikov, A.Yu.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers, Dordrecht (1997)

    Google Scholar 

  11. Koblitz, N.: p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol. 58, 2nd edn. Springer, Heidelberg (1984)

    Book  Google Scholar 

  12. Kozyrev, S.V.: Ultrametric analysis and interbasin kinetics. In: Proceedings of the 2nd International Conference on p-Adic Mathematical Physics, American Institute Conference Proceedings, vol. 826, pp. 121–128 (2006)

    Google Scholar 

  13. Lunts, A.G.: A method of analysis of finite automata. Sov. Phys. Dokl. 10, 102–105 (1965)

    Google Scholar 

  14. Madore, D.A.: A first introduction to \(p\)-adic numbers. http://www.madore.org/david/math/padics.pdf

  15. Nisan, N., Wigderson, A.: On rank vs. communication complexity. Combinatorica 15(4), 557–565 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ostrowski, A.: Über einige Lösungen der funktionalgleichung \(\varphi (x)\varphi (y) = \varphi (xy)\). Acta Math. 41(1), 271–284 (1916)

    Article  MathSciNet  Google Scholar 

  17. Turakainen, P.: Generalized automata and stochastic languages. Proc. Am. Math. Soc. 21(2), 303–309 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  18. Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p-Adic Analysis and Mathematical Physics. World Scientific, Singapore (1995)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics and Computer Science, University of Latvia, Raiņa bulvāris 29, Riga, 1459, Latvia

    Rūsiņš Freivalds

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  1. Rūsiņš Freivalds
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Correspondence to Rūsiņš Freivalds .

Editor information

Editors and Affiliations

  1. University of Auckland, Auckland, New Zealand

    Cristian S. Calude

  2. University of Auckland, Auckland, New Zealand

    Michael J. Dinneen

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Freivalds, R. (2015). Ultrametric Algorithms and Automata. In: Calude, C., Dinneen, M. (eds) Unconventional Computation and Natural Computation. UCNC 2015. Lecture Notes in Computer Science(), vol 9252. Springer, Cham. https://doi.org/10.1007/978-3-319-21819-9_2

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  • DOI: https://doi.org/10.1007/978-3-319-21819-9_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21818-2

  • Online ISBN: 978-3-319-21819-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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