On Parallel Addition and Multiplication via Symmetric Ternary Numeral System

  • Iurii V. StroganovEmail author
  • Liliya Volkova
  • Igor V. Rudakov
Conference paper
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 199)


This article is concerned with ternary logic application. Usage of ternary numeral system is recommended, particularly of symmetric ternary numeral system, as implementing arithmetic operations in ternary allows reducing roundoff errors, accumulated during finite-precision computation. A shift is suggested towards ternary computational machines. Ternary computations basis is given; addition and multiplication algorithms are discussed in classic and adapted versions, the latter is suggested as to develop a parallel implementation. Particular effects are highlighted which allow computing these operations in parallel mode, several examples illustrate the algorithms suggested. The resulting time and acceleration gain is discussed basing on data aggregated by means of an implementation in Haskell. Basing on experimental data, multithreaded implementation is recommended in order to accelerate addition and multiplication operations modelling. This research justifies the prospect of application of ternary co-processors for more precise computation.


Ternary computation Ternary logic Symmetric ternary numeral system Arithmetic operations implementation Parallel algorithms 


  1. 1.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, pp. 43–44 (2002)Google Scholar
  2. 2.
    Ralston, A., Rabinowitz, P.: A First Course in Numerical Analysis, Dover Books on Mathematics, 2nd edn. Courier Dover Publications, Mineola (2012)Google Scholar
  3. 3.
    Aksoy, P., DeNardis, L.: Information Technology in Theory. Cengage Learning, Boston (2007)Google Scholar
  4. 4.
    Anoprienko, A.I., Ivanitsa, S.V., Ivanitsa, S.V.: Features of representation of real numbers in post-binary formats (in Russian). Math. Mach. Syst. 1(3), 49–60 (2012)Google Scholar
  5. 5.
    Isupov, K.S.: Modular-positional format and software package for high-precision bit-parallel calculations in floating-point format (in Russian). Bull. South Ural. State Univ. Ser.: Comput. Math. Comput. Sci. 2, 65–79 (2012)Google Scholar
  6. 6.
    Lavrinenko, A.N., Chervyakov, N.I.: Study of non-modular operations in the system of residual classes (in Russian). Sci. Sheets Belgorod State Univ. Ser.: Econ. Inform. 21, 110–122 (2012)Google Scholar
  7. 7.
    Isupov, K.S.: On an algorithm for number comparison in the residue number system (in Russian). Bull. Astrakhan State Tech. Univ. Ser.: Manag. Comput. Eng. Inform. 3, 40–49 (2014)Google Scholar
  8. 8.
    Denisenko, B.: New physical effects in nanometer MOSFETs (in Russian). Compon. Technol. 12, 158–162 (2009)Google Scholar
  9. 9.
    Polyakov, V.I., Skorubsky, V.I.: The use of multivalued logic in the design of functional circuits (in Russian). Proc. High. Educ. Inst. Ser.: Instrum. 57(4), 57–60 (2014)Google Scholar
  10. 10.
    Budyakov, P.S., Chernov, N.I., Yugai, V. Ya., Yugai, N. N.: Logic functions representation and synthesis of k-valued digital circuits in linear algebra. In: 24th Telecommunications Forum (TELFOR 2016), pp. 1–4. IEEE (2016)Google Scholar
  11. 11.
    Hayes, B.: Third base. Am. Sci. 89(6), 490–494 (2001)CrossRefGoogle Scholar
  12. 12.
    Kushnerov, A.: Ternary digital technology. Retrospective and contemporary state (in Russian). Ben-Gurion University, Beersheba, pp. 1–5 (2005). Accessed 20 June 2015
  13. 13.
    Bobreshov, A.M., Koshelev, A.G., Zolotukhin, E.V.: Multichannel organic light emitting RGB diode, as ternary logic element. In: Proceedings of Voronezh State University, Voronezh (2016)Google Scholar
  14. 14.
    Voevodin, V.V., Kim, G. D.: A mathematic’s view on machine operations (in Russian). In: Computational Methods and Programs, vol. 26. MSU, Russia (1977)Google Scholar
  15. 15.
    Stroganov, I.V., Rudakov, I.V.: Ternary virtual machine for calculating. Int. J. Adv. Stud., vol. 4–3. Publishing House Science and Innovation Center, Ltd., Saint-Louis (2017)Google Scholar
  16. 16.
    Brusentsov, N.P., Maslov, S.P., Rozin, V.P., Tishulina, A.M.: Small digital Computer “Setun”. MSU publishing house, Moscow (1965). (in Russian)zbMATHGoogle Scholar
  17. 17.
    Knuth, D.: The Art of Computer Programming, vol. 2: Seminumerical Algorithms, chapter 4.1. Addison-Wesley Professional, Boston (2011)Google Scholar
  18. 18.
    Marlow, S.: Haskell 2010. Language Report, Accessed 21 Oct 2018
  19. 19.
    Marlow, S.: Parallel and Concurrent Programming in Haskell. O’Reilly Media Inc, Sebastopol (2013)Google Scholar
  20. 20.
    Mena, A.S.: Beginning Haskell: A Project-Based Approach. Apress, New-York (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.BMSTUMoscowRussia

Personalised recommendations