Advertisement

Fast Iterative Arrays with Restricted Inter-cell Communication: Constructions and Decidability

  • Martin Kutrib
  • Andreas Malcher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

Iterative arrays (IAs) are one-dimensional arrays of interconnected interacting finite automata with sequential input mode. We investigate IAs which work in real time and whose inter-cell communication is bounded by some constant number of bits not depending on the number of states. It is known [13] that such IAs can recognize rather complicated unary languages with a minimum amount of communication, namely one-bit communication, in real time. Some examples are the languages \(\{a^{2^n} \mid n \ge 1\}\), \(\{a^{n^2} \mid n \ge 1\}\), and {a p |p is prime}. Here, we consider non-unary languages and it turns out that the non-unary case is quite different. We present several real-time constructions for certain non-unary languages. For example, the languages {a n b n |n ≥1}, {a n (b n ) m |n,m ≥1}, and {a n ba m b(ba) n .m |n,m ≥1} are recognized in real time by 1-bit IAs. Moreover, it is shown that real-time 1-bit IAs can, in some sense, add and multiply integer numbers. Furthermore, closure properties and decidability questions of communication restricted IAs are investigated. Due to the constructions provided, non-closure results as well as undecidability results can be shown. It turns out that emptiness is still undecidable for 1-bit IAs despite their restricted communication. Thus, also the questions of finiteness, infiniteness, inclusion, and equivalence are undecidable.

Keywords

Cellular Automaton Communication Cell Regular Language Closure Property Input Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Buchholz, T., Klein, A., Kutrib, M.: Iterative arrays with a wee bit alternation. In: Ciobanu, G., Păun, G. (eds.) FCT 1999. LNCS, vol. 1684, pp. 173–184. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  2. 2.
    Buchholz, T., Klein, A., Kutrib, M.: Iterative arrays with small time bounds. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 243–252. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Buchholz, T., Klein, A., Kutrib, M.: Iterative arrays with limited nondeterministic communication cell. In: Ito, M., Imaoka, T. (eds.) Words, Languages and Combinatorics III, pp. 73–87. World Scientific Publishing, Singapore (2003)CrossRefGoogle Scholar
  4. 4.
    Chang, J.H., Ibarra, O.H., Palis, M.A.: Parallel parsing on a one-way array of finite-state machines. IEEE Transactions Computers C-36, 64–75 (1987)CrossRefGoogle Scholar
  5. 5.
    Cole, S.N.: Real-time computation by n-dimensional iterative arrays of finite-state machines. IEEE Transactions Computers C-18, 349–365 (1969)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Fischer, P.C.: Generation of primes by a one-dimensional real-time iterative array. Journal of the ACM 12, 388–394 (1965)MATHCrossRefGoogle Scholar
  7. 7.
    Ibarra, O.H.: Reversal-bounded multicounter machines and their decision problems. Journal of the ACM 25, 116–133 (1978)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Iwamoto, C., Hatsuyama, T., Morita, K., Imai, K.: On time-constructible functions in one-dimensional cellular automata. In: Ciobanu, G., Păun, G. (eds.) FCT 1999. LNCS, vol. 1684, pp. 316–326. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Kutrib, M., Malcher, A.: Fast cellular automata with restricted inter-cell communication: computational capacity. In: Proceedings of IFIP TCS 2006, Santiago de Chile (to appear)Google Scholar
  10. 10.
    Malcher, A.: On the descriptional complexity of iterative arrays. IEICE Transactions on Information Sciences E87-D, 721–725 (2004)Google Scholar
  11. 11.
    Seidel, S.R.: Language recognition and the synchronization of cellular automata. Technical Report 79-02, University of Iowa (1979)Google Scholar
  12. 12.
    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. Fundamenta Informaticae 52, 257–275 (2002)MATHMathSciNetGoogle Scholar
  13. 13.
    Umeo, H., Kamikawa, N.: Real-time generation of primes by a 1-bit-communication cellular automaton. Fundamenta Informaticae 58, 421–435 (2003)MATHMathSciNetGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Martin Kutrib
    • 1
  • Andreas Malcher
    • 2
  1. 1.Institut für InformatikUniversität GiessenGiessenGermany
  2. 2.Institut für InformatikJohann Wolfgang Goethe UniversitätFrankfurt am MainGermany

Personalised recommendations