Finite and Infinite Computations and a Classification of Two-Dimensional Cellular Automata Using Infinite Computations

  • Louis D’AlottoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10421)


This paper proposes an application of the Infinite Unit Axiom and grossone, introduced by Yaroslav Sergeyev (see [19, 20, 21, 22, 23]), to the development and classification of two-dimensional cellular automata. This application establishes, by the application of grossone, a new and more precise nonarchimedean metric on the space of definition for two-dimensional cellular automata, whereby the accuracy of computations is increased. Using this new metric, open disks are defined and the number of points in each disk computed. The forward dynamics of a cellular automaton map are also studied by defined sets. It is also shown that using the Infinite Unit Axiom, the number of configurations that follow a given configuration, under the forward iterations of the cellular automaton map, can now be computed and hence a classification scheme developed based on this computation.


Cellular automata Infinite Unit Axiom Grossone Nonarchimedean metric Dynamical systems 


  1. 1.
    Baetens, J.M., Gravner, J.: Stability of cellular automata trajectories revisited: branching walks and Lyapunov profiles. J. Nonlinear Sci. 26, 1329–1367 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways for Your Mathematical Plays, vol. 4, 2nd edn. A. K. Peters, Wellesley (2004)zbMATHGoogle Scholar
  3. 3.
    Calidonna, C.R., Naddeo, A., Trunfio, G.A., Di Gregorio, S.: From classical infinite space-time CA to a hybrid CA model for natural sciences modeling. Appl. Math. Comput. 218(16), 8137–8150 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chopard, B., Droz, M.: Cellular Automata Modeling of Physical Systems. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  5. 5.
    D’Alotto, L.: Cellular automata using infinite computations. Appl. Math. Comput. 218(16), 8077–8082 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    D’Alotto, L.: A classification of one-dimensional cellular automata using infinite computations. Appl. Math. Comput. 255, 15–24 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D’Alotto, L., Pizzuti, C.: Characterization of one-dimensional cellular automata rules through topological network features. In: Numerical Computations Theory and Algorithms 2016, AIP Conference Proceedings, vol. 1776, pp. 090048-1–090048-4 (2016)Google Scholar
  8. 8.
    D’Ambrosio, D., Filippone, G., Marocco, D., Rongo, R., Spataro, W.: Efficient application of GPGPU for lava flow hazard mapping. J. Supercomput. 65(2), 630–644 (2013)CrossRefGoogle Scholar
  9. 9.
    De Cosmis, S., De Leone, R.: The use of grossone in mathematical programming and operations research. Appl. Math. Comput. 218(16), 8029–8038 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gilman, R.: Classes of linear automata. Ergod. Theor. Dyn. Syst. 7, 105–118 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hedlund, G.A.: Edomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theor. 3, 51–59 (1969)CrossRefGoogle Scholar
  12. 12.
    Iudin, D.I., Sergeyev, Y.D., Hayakawa, M.: Interpretation of percolation in terms of infinity computations. Appl. Math. Comput. 218(16), 8099–8111 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lolli, G.: Infinitesimals and infinities in the history of mathematics: a brief survey. Appl. Math. Comput. 218(16), 7979–7988 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lolli, G., Metamathematical Investigations on the Theory of Grossone, Preprint, Applied Mathematics and Computation. Elsevier (submitted and accepted for publication)Google Scholar
  15. 15.
    Mart’nez, G.J.: A note on elementary cellular automata classification. J. Cell. Automata 8, 233–259 (2013)MathSciNetGoogle Scholar
  16. 16.
    Margenstern, M.: Using grossone to count the number of elements of infinite sets and the connection with bijections. p-Adic Numbers Ultrametric Anal. Appl. 3(3), 196–204 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Margenstern, M.: An application of grossone to the study of a family of tilings of the hyperbolic plane. Appl. Math. Comput. 218(16), 8005–8018 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Narici, L., Beckenstein, E., Bachman, G.: Functional Analysis and Valuation Theory. Marcel Dekker Inc., New York (1971)zbMATHGoogle Scholar
  19. 19.
    Sergeyev, Y.D.: Arithmetic of Infinity. Edizioni Orizzonti Meridionali, Italy (2003)zbMATHGoogle Scholar
  20. 20.
    Sergeyev, Y.D.: Numerical Point of view on calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains. Nonlinear Anal. Ser. A Theor. Methods Appl. 71(12), e1688–e1707 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sergeyev, Y.D.: Numerical computations with infinite and infinitesimal numbers: theory and applications. In: Sorokin, A., Pardalos, P.M. (eds.) Dynamics of Information Systems: Algorithmic Approaches, pp. 1–66. Springer, New York (2013)Google Scholar
  22. 22.
    Sergeyev, Y.D.: A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica 19(4), 567–596 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sergeyev, Y.D.: Measuring fractals by infinite and infinitesimal numbers. Math. Methods Phys. Methods Simul. Sci. Technol. 1(1), 217–237 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Sergeyev, Y.D., Garro, A.: Observability of turing machines: a refinement of the theory of computation. Informatica 21(3), 425–454 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sergeyev, Y.D., Garro, A.: Single-tape and multi-tape turing machines through the lens of grossone methodology. J. Supercomput. 65(2), 645–663 (2013)CrossRefGoogle Scholar
  26. 26.
    Sirakoulis, G.C., Krafyllidis, I., Spataro, W.: A computational intelligent oxidation process model and its VLSI implementation. In: International Conference on Scientific Computing Proceedings, pp. 329–335 (2009)Google Scholar
  27. 27.
    Trunfio, G.A.: Predicting wildfire spreading through a hexagonal cellular automata model. In: Sloot, P.M.A., Chopard, B., Hoekstra, A.G. (eds.) ACRI 2004. LNCS, vol. 3305, pp. 385–394. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-30479-1_40 CrossRefGoogle Scholar
  28. 28.
    Trunfio, G.A., D’Ambrosio, D., Rongo, R., Spataro, W., Di Gregorio, S.: A new algorithm for simulating wilfire spread through cellular automata. ACM Trans. Model. Comput. Simul. 22, 1–26 (2011)CrossRefGoogle Scholar
  29. 29.
    Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55(3), 601–644 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wolfram, S.: A New Kind of Science. Wolfram Media Inc., Champaign (2002)zbMATHGoogle Scholar
  31. 31.
    Wolfram, S.: Universality and complexity in cellular automata. Phys. D 10, 1–35 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zhigljavsky, A.: Computing sums of conditionally convergent and divergent series using the concept of grossone. Appl. Math. Comput. 218(16), 8064–8076 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceYork College/City University of New YorkJamaica, New YorkUSA
  2. 2.The Doctoral Program in Computer ScienceCUNY Graduate CenterNew YorkUSA

Personalised recommendations