Advertisement

Ultrametric Vs. Quantum Query Algorithms

  • Rūsiņš Freivalds
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8890)

Abstract

Ultrametric algorithms are similar to probabilistic algorithms but they describe the degree of indeterminism by p-adic numbers instead of real numbers. This paper introduces the notion of ultrametric query algorithms and shows an example of advantages of ultrametric query algorithms over deterministic, probabilistic and quantum query algorithms.

Keywords

Nature-inspired models of computation ultrametric algorithms probabilistic algorithms quantum algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ablayev, F.M., Freivalds, R.: Why sometimes probabilistic algorithms can be more effective. In: Wiedermann, J., Gruska, J., Rovan, B. (eds.) MFCS 1986. LNCS, vol. 233, pp. 1–14. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  2. 2.
    Ambainis, A.: Polynomial degree vs. quantum query complexity. Journal of Computer and System Sciences 72(2), 220–238 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: Proc. IEEE FOCS 1998, pp. 332–341 (1998)Google Scholar
  4. 4.
    Balodis, K., Beriņa, A., Cīpola, K., Dimitrijevs, M., Iraids, J., Jēriņš, K., Kacs, V., Kalājs, J., Krišlauks, R., Lukstiņš, K., Raumanis, R., Scegulnaja, I., Somova, N., Vanaga, A., Freivalds, R.: On the state complexity of ultrametric finite automata. In: Proceedings of SOFSEM, vol. 2, pp. 1–9 (2013)Google Scholar
  5. 5.
    Bērziņa, A., Freivalds, R.: On quantum query complexity of kushilevitz function. In: Proceedings of Baltic DB&IS 2004, vol. 2, pp. 57–65 (2004)Google Scholar
  6. 6.
    Buhrman, H., Wolf, R.D.: Complexity measures and decision tree complexity: a survey. Theoretical Computer Science 288(1), 21–43 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Moore, C., Quantum, J.C.: automata and quantum grammars. Theoretical Computer Science 237(1-2), 275–306 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dragovich, B., Dragovich, A.: A p-adic model of dna sequence and genetic code. p-Adic Numbers, Ultrametric Analysis, and Applications 1(1), 34–41 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Freivalds, R.: Recognition of languages with high probability on different classes of automata. Doklady Akademii Nauk SSSR 239(1), 60–62 (1978)MathSciNetGoogle Scholar
  10. 10.
    Freivalds, R.: Projections of languages recognizable by probabilistic and alternating finite multi-tape automata. Information Processing Letters 13(4-5), 195–198 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Freivalds, R.: On the growth of the number of states in result of the determinization of probabilistic finite automata. Avtomatika i Vichislitel’naya Tekhnika (3), 39–42 (1982)Google Scholar
  12. 12.
    Freivalds, R.: Complexity of probabilistic versus deterministic automata. In: Barzdins, J., Bjorner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 565–613. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  13. 13.
    Freivalds, R.: Languages recognizable by quantum finite automata. In: Farré, J., Litovsky, I., Schmitz, S. (eds.) CIAA 2005. LNCS, vol. 3845, pp. 1–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Freivalds, R.: Non-constructive methods for finite probabilistic automata. International Journal of Foundations of Computer Science 19, 565–580 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Freivalds, R., Zeugmann, T.: Active learning of recursive functions by ultrametric algorithms. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds.) SOFSEM 2014. LNCS, vol. 8327, pp. 246–257. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  17. 17.
    Gouvea, F.Q.: p-adic numbers: An introduction, universitext (1983)Google Scholar
  18. 18.
    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
  19. 19.
    Khrennikov, A.Y.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers (1997)Google Scholar
  20. 20.
    Koblitz, N.: P-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edn. Graduate Texts in Mathematics, vol. 58. Springer (1984)Google Scholar
  21. 21.
    Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: Proc. IEEE FOCS 1997, pp. 66–75 (1997)Google Scholar
  22. 22.
    Kozyrev, S.V.: Ultrametric analysis and interbasin kinetics. In: Proc. of the 2nd International Conference on p-Adic Mathematical Physics, vol. 826, pp. 121–128. American Institute Conference Proceedings (2006)Google Scholar
  23. 23.
    Krišlauks, R., Rukšāne, I., Balodis, K., Kucevalovs, I., Freivalds, R., Agele, I.N.: Ultrametric turing machines with limited reversal complexity. In: Proceedings of SOFSEM, vol. 2, pp. 87–94 (2013)Google Scholar
  24. 24.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press (2000)Google Scholar
  25. 25.
    Nisan, N., Wigderson, A.: On rank vs. communication complexity. Combinatorica 15(4), 557–565 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Ostrowski, A.: Über einige Lösungen der Funktionalgleichung ϕ(x)ϕ(y) = ϕ(xy). Acta Mathematica 41(1), 271–284 (1916)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Papadimitriou, C.H.: Computational complexity. John Wiley and Sons Ltd, Chichester (2003)Google Scholar
  28. 28.
    Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p-Adic Analysis and Mathematical Physics. World Scientific, Singapore (1995)Google Scholar
  29. 29.
    Zariņa, S., Freivalds, R.: Visualisation and and ultrametric analysis of koch fractals. In: Proc. 16th Japan Conference on Discrete and Computational Geometry and Graphs, pp. 84–85. Tokyo (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Rūsiņš Freivalds
    • 1
  1. 1.Institute of Mathematics and Computer ScienceUniversity of LatviaRigaLatvia

Personalised recommendations