Hierarchies and Undecidability Results for Iterative Arrays with Sparse Communication

  • Andreas MalcherEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10875)


Iterative arrays with restricted internal inter-cell communication are investigated. A quantitative measure for the communication is defined by counting the number of uses of the links between cells and it is differentiated between the sum of all communications of an accepting computation and the maximum number of communications per cell occurring in accepting computations. The computational complexity of both classes of devices is investigated and put into relation. In addition, a strict hierarchy depending on the maximum number of communications per cell is established. Finally, it is shown that almost all commonly studied decidability questions are not semidecidable for iterative arrays with restricted communication and, moreover, it is not semidecidable as well whether a given iterative array belongs to a given class with restricted communication.



Thanks are given to Victor Roussanaly for several discussions on the topic while his internship at our institute in 2014.


  1. 1.
    Chang, J.H., Ibarra, O.H., Palis, M.A.: Parallel parsing on a one-way array of finite-state machines. IEEE Trans. Comput. C–36, 64–75 (1987)CrossRefGoogle Scholar
  2. 2.
    Cole, S.N.: Real-time computation by \(n\)-dimensional iterative arrays of finite-state machines. IEEE Trans. Comput. C–18(4), 349–365 (1969)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fischer, P.C.: Generation of primes by a one-dimensional real-time iterative array. J. ACM 12, 388–394 (1965)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ibarra, O.H., Palis, M.A.: Some results concerning linear iterative (systolic) arrays. J. Parallel Distrib. Comput. 2, 182–218 (1985)CrossRefGoogle Scholar
  5. 5.
    Ibarra, O.H., Palis, M.A.: Two-dimensional iterative arrays: Characterizations and applications. Theor. Comput. Sci. 57, 47–86 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kutrib, M.: Cellular automata - a computational point of view. In: Bel-Enguix, G., Jiménez-López, M.D., Martín-Vide, C. (eds.) New Developments in Formal Languages and Applications, Chap. 6. SCI, vol. 113, pp. 183–227. Springer, Heidelberg (2008). Scholar
  7. 7.
    Kutrib, M.: Cellular automata and language theory. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, pp. 800–823. Springer, New York (2009). Scholar
  8. 8.
    Kutrib, M., Malcher, A.: Computations and decidability of iterative arrays with restricted communication. Parallel Process. Lett. 19(2), 247–264 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kutrib, M., Malcher, A.: On one-way one-bit \({O}(1)\)-message cellular automata. Electr. Notes Theor. Comput. Sci. 252, 77–91 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kutrib, M., Malcher, A.: Cellular automata with sparse communication. Theor. Comput. Sci. 411(38–39), 3516–3526 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kutrib, M., Malcher, A.: One-way cellular automata, bounded languages, and minimal communication. J. Autom. Lang. Comb. 15(1/2), 135–153 (2010)zbMATHGoogle Scholar
  12. 12.
    Kutrib, M., Malcher, A.: Cellular automata with limited inter-cell bandwidth. Theor. Comput. Sci. 412(30), 3917–3931 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Malcher, A.: On the descriptional complexity of iterative arrays. IEICE Trans. Inf. Syst. E87–D, 721–725 (2004)Google Scholar
  14. 14.
    Mazoyer, J., Terrier, V.: Signals in one-dimensional cellular automata. Theor. Comput. Sci. 217(1), 53–80 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Smith III, A.R.: Real-time language recognition by one-dimensional cellular automata. J. Comput. Syst. Sci. 6(3), 233–253 (1972)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Umeo, H., Kamikawa, N.: A design of real-time non-regular sequence generation algorithms and their implementations on cellular automata with 1-bit inter-cell communications. Fund. Inform. 52, 257–275 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Umeo, H., Kamikawa, N.: Real-time generation of primes by a 1-bit-communication cellular automaton. Fund. Inform. 58, 421–435 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Worsch, T.: Linear time language recognition on cellular automata with restricted communication. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 417–426. Springer, Heidelberg (2000). Scholar

Copyright information

© IFIP International Federation for Information Processing 2018

Authors and Affiliations

  1. 1.Institut für InformatikUniversität GiessenGiessenGermany

Personalised recommendations