Scientific and Technical Information Processing

, Volume 44, Issue 5, pp 314–328 | Cite as

Computability via Cellular Automata

  • S. V. Gavrilov
  • I. V. Matyushkin
  • A. L. Stempkovsky


This review addresses the issues of computations using cellular automata (CA). It is shown that the generality of the connectionism paradigm allows some methods applicable to neural networks to be transferred into the domain of CA. Some special issues of computability are discussed based on the examples of the density classification task, the firing-squad synchronization problem, and the queen-bee problem, as well as sorting algorithms and Atrubin’s parallel multiplication algorithm.


cellular automata computability signal sorting parallel multiplication Atrubin’s algorithm Turing machine time-constructability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Al-Rabadi, A.N., Conservative reversible elementary cellular automata and their quantum computations, in Cellular Automata–Innovative Modelling for Science and Engineering. Scholar
  2. 2.
    Tumanov, V.E., Basics of Designing Relational Databases. Scholar
  3. 3.
    Gergel’, V.P. and Strongin, R.G., Osnovy parallel’nykh vychislenii dlya mnogoprotsessornykh vychislitel’nykh sistem. Uchebnoe posobie (Basics of Parallel Computing for Multiprocessor Computing Systems), Nizhnii Novgorod: Izd. NNGU im. N. I. Lobachevskogo, 2003.Google Scholar
  4. 4.
    Alekseev, V.E. and Talanov, V.A., Grafy. Modeli vychislenii. Struktury dannykh: Uchebnik (Graphs. Models of Calculations. Data Structures), Nizhnii Novgorod: Izd. NNGU, 2005.Google Scholar
  5. 5.
    Stefanova, T.S., Selection of the content of training for non-classical computing models, Izv. RGPU im. A. I. Gertsena, 2008, no. 58, pp. 440–451. http://cyberleninka. ru/article/n/otbor-soderzhaniya-obucheniya-neklassicheskim- vychislitelnym-modelyam. Cited April 15, 2015.Google Scholar
  6. 6.
    Anisov, A.M., Classical computability and signs of indeterminism, Logich. Res., 2007, no. 14, pp. 5–26.zbMATHGoogle Scholar
  7. 7.
    Savage, J.E., VLSI models of computation, in Models of Computation: Exploring the Power of Computing, 2008, pp. 575–603.Google Scholar
  8. 8.
    Hewitt, C., Bishop, P., and Steiger, R., A universal modular ACTOR formalism for artificial intelligence, Proceedings of the 3rd International Joint Conference on Artificial intelligence (IJCAI'73), San Francisco, 2003, pp. 235–245.Google Scholar
  9. 9.
    Medler, D.A., A brief history of connectionism, Neural Comput. Surv., 1998, vol. 1, no. 2, pp. 18–72.Google Scholar
  10. 10.
    Parallel Distributed Processing, vol. 1: Foundations, Rumelhart, D.E. McClelland, J.L., and the PDP Research Group, Eds., MIT Press, Cambridge, MA, 1986.Google Scholar
  11. 11.
    Wolfram, S., A New Kind of Science, New York: Wolfram Media, 2002.zbMATHGoogle Scholar
  12. 12.
    Langton, C., Computation at the edge of chaos, Phys. D, 1990, vol. 42, pp. 12–37.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kari, J., Cellular Automata: Tutorial. http://users.utu. fi/jkari/ca/CAintro.pdf.Google Scholar
  14. 14.
    Kudryavtsev, V., Podkolzin, A., and Bolotov, A., Osnovy teorii odnorodnykh struktur (Fundamentals of the Theory of Homogeneous Structures), Moscow: Moskva, 1990.zbMATHGoogle Scholar
  15. 15.
    Alad’ev, V.Z., Klassicheskie odnorodnye struktury. Kletochnye avtomaty (Classical Homogeneous Structures. Cellular Machines), FultusBooks, 2009.Google Scholar
  16. 16.
    Kari, J., Theory of cellular automata: A survey, Theor. Comput. Sci., 2005, vol. 334, pp. 3–33.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Life on the plane of Lobachevsky. http://habrahabr. ru/post/168421/.Google Scholar
  18. 18.
    Margenstern, M., Small Universal Cellular Automata in Hyperbolic Spaces. A Collection of Jewels, Springer, 2013.CrossRefzbMATHGoogle Scholar
  19. 19.
    Marques-Pita, M., and Rocha, L.M., Conceptual structure in cellular automata: The density classification task, Artificial Life XI: Eleventh International Conference on the Simulation and Synthesis of Living Systems, MIT Press, 2008 (in press).Google Scholar
  20. 20.
    Juillé, H. and Pollack, J.B., Coevolving the 'ideal' trainer: Application to the discovery of cellular automata rules, Genetic Programming 1998: Proceedings of the Third Annual Conference, San Francisco, 1998.Google Scholar
  21. 21.
    Stone, C. and Bull, L., Solving the density classification task using cellular automaton 184 with memory, Complex Syst., 2009, vol. 18, pp. 329–334. Scholar
  22. 22.
    Alonso-Sanz, R. and Bull, L., Elementary coupled cellular automata with memory, in Automata 2008: Theory and Applications of Cellular Automata, Adamatzky, A., Alonso-Sanz, R., Lawniczak, A., Martinez, G.J., Morita, K., and Worsch, T., Eds., Frome, UK: Luniver Press, 2008, pp. 72–99.Google Scholar
  23. 23.
    Gacs, P., Kurdyumov, L., and Levin, L., One-dimensional uniform arrays that wash out finite islands, Probl. Peredachi Inf., 1978, vol. 14, pp. 92–98.Google Scholar
  24. 24.
    Land, M. and Belew, R.K., No perfect two-state cellular automata for density classification exists, Phys. Rev. Lett., 1995, vol. 74, no. 25, p. 5148.CrossRefGoogle Scholar
  25. 25.
    Mitchell, M., Crutchfield, J.P., and Hraber, P.T., Evolving cellular automata to perform computations: Mechanisms and Impediments, Phys. D, 1994, vol. 75, pp. 361–391.CrossRefzbMATHGoogle Scholar
  26. 26.
    Yamashita, K., et al., The firing squad synchronization problems for number patterns on a seven-segment display and segment arrays, IEICE Trans. Inf. Syst., 2010, vol. 93, no. 12, pp. 3276–3283.CrossRefGoogle Scholar
  27. 27.
    Umeo H., Nishide K., Kubo K. A simple optimumtime FSSP algorithm for multi-dimensional cellular automata, arXiv preprint arXiv:1208.2761, 2012.Google Scholar
  28. 28.
    Ponikarov, M.V., Using games of cellular automates for synchronization in distributed systems, Biz. Inf., 2008, no. 3 (5), pp. 31–36.Google Scholar
  29. 29.
    Mazoyer, J., On optimal solutions to the firing squad synchronization problem, Theor. Comput. Sci., 1996, vol. 168, no. 2, pp. 367–404.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Vivien, H., An Introduction to Cellular Automata. pdfGoogle Scholar
  31. 31.
    Mazoyer, J. and Terrier, V., Signals in one-dimensional cellular automata, Theor. Comput. Sci., 1999, vol. 217, pp. 53–80.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Iwamoto, C., et al., Constructible functions in cellular automata and their applications to hierarchy results, Theor. Comput. Sci., 2002, vol. 270, no. 1, pp. 797–809.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Fisher, P.C., Generation on primes by one dimentional real time iterative array, J. ACM, 1965, vol. 12, pp. 388–394.CrossRefGoogle Scholar
  34. 34.
    Mahata, K. and Das, S., Gateway node in WSN as the queen bee in a honey bee colony, IEEE 2012 International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012, pp. 270–273.Google Scholar
  35. 35.
    Banda, R.N.D.P., Anonymous leader election in oneand two-dimensional cellular automata, Student Dissertation Thesis, Comenius University in Bratislava, 2014. Scholar
  36. 36.
    Beckers, A. and Worsch, T., A perimeter-time CA for the queen bee problem, Parallel Comput., 2001, vol. 27, no. 5, pp. 555–569.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Nichitiu, C., Mazoyer, J., and Rémila, E., Algorithms for leader election by cellular automata, J. Algorithms, 2001, vol. 41, no. 2, pp. 302–329.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Gornev, E.S., Matyushkin, I.V., and Teplov, G.S., Analysis of the concepts of non-classical computing and the paradigm of connectionism, Elektron. Tekh., Ser. 3: Mikroelektron., 2015, no. 2 (158), pp. 45–66.Google Scholar
  39. 39.
    Heen, O., Efficient constant speed-up for one dimensional cellular automata calculators, Parallel Comput., 1997, vol. 23, pp. 1663–1671MathSciNetCrossRefGoogle Scholar
  40. 40.
    Atrubin, A.J., A one-dimensional real-time iterative multiplier, in IEEE Transactions on Electronic Computers EC-14, 1965, pp. 394–399.Google Scholar
  41. 41.
    Even, S. and Litman, A., A systematic design and explanation of the Atrubin multiplier, in Sequences II. Methods in Communication, Security, and Computer Science, Springer New York, 1993, pp. 189–202.Google Scholar
  42. 42.
    Bandman, O.L., Invariants of cellular-automatic models of reaction-diffusion processes, PDM, 2012, no. 3, pp. 108–120.Google Scholar
  43. 43.
    Weimar, J.R., Cellular automata for reaction-diffusion systems, Parallel Comput., 1997, vol. 23, no. 11, pp. 1699–1715.MathSciNetCrossRefGoogle Scholar
  44. 44.
    Toffolli, T., Cellular automata as an alternative to (rather than approximation of) differential equations in modeling physics, Phys. D (Amsterdam), 1984, vol. 10, pp. 117–127.MathSciNetCrossRefGoogle Scholar
  45. 45.
    Vanag, V.K., Investigation of spatially distributed dynamical systems by the methods of a probabilistic cellular automaton, Usp. Fiz. Nauk, 1999, vol. 169, no. 5, pp. 481–505.CrossRefGoogle Scholar
  46. 46.
    Flocchini, P. and Cezar, V., Radial view of continuous cellular automata, Fundam. Inf., 2001, vol. 21, pp. 1001–1018.zbMATHGoogle Scholar
  47. 47.
    Bidlo, M., Vasicek, Z., and Slany, K., Sorting network development using cellular automata, Evolvable Systems: From Biology to Hardware. 9th International Conference, ICES 2010, York, UK, September 6–8, 2010, Proceedings, LNCS 6274, London, 2010, pp. 85–96.CrossRefGoogle Scholar
  48. 48.
    Lafe, O., Data compression and encryption using cellular automata transforms, IEEE International Joint Symposia on Intelligence and Systems 1996, 1996, pp. 234–241.CrossRefGoogle Scholar
  49. 49.
    Zheng, T., Discrete orthogonal transforms using cellular automata, NKS2004 (New Kind of Science Develop), 2004. Scholar
  50. 50.
    Shaw, B. and Wong, L.C., Algorithmic Self-Assembly of Circuits, 2002.Google Scholar
  51. 51.
    Evsyutin, O.O. and Shelupanov, A.A., Basic approaches to the use of the mathematical apparatus of the theory of cellular automata for solving information encoding problems, Dokl. TUSURa, 2007, no. 2 (32), pp. 60–65.Google Scholar
  52. 52.
    Speller, T.H., A study within the nanosystem architecture domain: Self-assembly of graphene, NKS2007 Conference, 2007.Google Scholar
  53. 53.
    Speller, T.H., Jr., Whitney, D., and Crawley, E., Using shape grammar to derive cellular automata rule patterns, Complex Syst., 2007, vol. 17, pp. 79–102.MathSciNetzbMATHGoogle Scholar
  54. 54.
    White, R., Engelen, G., and Uljee, I., The use of constrained cellular automata for high-resolution modelling of urban land-use dynamics, Environ. Plann. B: Plann. Des., 1997, vol. 24, no. 3, pp. 323–343.CrossRefGoogle Scholar
  55. 55.
    Krasnikov, G.Y., Matyushkin, I.V., and Korobov, S.V., Visualization of Cellular Automata in Nanotechnology, Model. Artif. Intell., 2014, vol. 3, no. 3, pp. 98–114.CrossRefGoogle Scholar
  56. 56.
    Buchholz, T., Klein, A., and Kutrib, M., Real-time language recognition by alternating cellular automata, Proceedings of the International Conference IFIP on Theoretical Computer Science, Exploring New Frontiers of Theoretical Informatics, Springer-Verlag, 2000, pp. 213–225.Google Scholar
  57. 57.
    Terrier, V., Characterization of real time iterative array by alternating device, Theor. Comput. Sci., 2003, vol. 290, no. 3, 2075–2084.MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Gordon, D., On the computational power of totalistic cellular automata, Math. Syst. Theory, 1987, vol. 20, no. 1, pp. 43–52.MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Jiirgen, A. and Culik, K., A simple universal cellular automaton and its one-way and totalistic version, Complex Syst., 1987, vol. 1, pp. 1–16.MathSciNetzbMATHGoogle Scholar
  60. 60.
    Iwamoto, C., et al., Constructible functions in cellular automata and their applications to hierarchy results, Theor. Comput. Sci., 2002, vol. 270, no. 1, pp. 797–809.MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Stempkovsky, A.L., Vlasov, P.A., and Kozin, G.V., Algorithmic environment for VLSI design on cellular automata, Proceedings of a Joint Symposium: Information Processing and Software, Systems Design Automation, Academy of Sciences of the USSR, Siemens AG, FRG, Moscow, June 5/6, 1990, 1990, pp. 308–312.Google Scholar
  62. 62.
    Stempkovskii, A.L., Osipov, L.B., and Seleznev, S.Z., Investigation of the implementation of the neural network by the SIP-technology for the construction of fault-tolerant homogeneous architectures, Inf. Tekhnol. Vychisl. Sist., 1995, no. 9, p. 58.Google Scholar
  63. 63.
    Stempkovskii, A.L., Osipov, L.B., and Seleznev, S.Z., Problems of implementation of fault-tolerant architectures of neurochips on the technology of systems with integration on the plate, Inf. Tekhnol., 1997, no. 5, p. 15.Google Scholar
  64. 64.
    Stempkovskii, A.L., Fault-tolerant architectures of microelectronic computing systems, Inf. Technol. Comput. Syst., 2001, nos. 2–3, p. 40.Google Scholar
  65. 65.
    Amoroso, S. and Cooper, G., The Garden-of-Eden theorem for finite configurations, Proceedings of the American Mathematical Society, 1970, vol. 26, no. 1, pp. 158–164.MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Jakubowski, M.H., Computing with solitons in bulk media, PhD Thesis, Princeton: University, 1999.Google Scholar
  67. 67.
    Rennard, J.-P., Implementation of logical functions in the game of life, in Collision-Based Computing, Adamatzky, A., Ed., Springer, 2002, pp. 491–512.CrossRefGoogle Scholar
  68. 68.
    Nepeiver, A.N., Reverse computations: A review of world experience, Materialy konferentsii Parallel’nyye vychisleniya i zadachi upravleniya (Proceedings of the Conference Parallel Computing and Control Problems), Moscow, 2012. Scholar

Copyright information

© Allerton Press, Inc. 2017

Authors and Affiliations

  • S. V. Gavrilov
    • 1
  • I. V. Matyushkin
    • 2
  • A. L. Stempkovsky
    • 1
  1. 1.Institute for Design Problems in MicroelectronicsRussian Academy of SciencesMoscowRussia
  2. 2.Research Institute for Molecular ElectronicsMoscowRussia

Personalised recommendations