Undecidability and Hierarchy Results for Parallel Communicating Finite Automata

  • Henning Bordihn
  • Martin Kutrib
  • Andreas Malcher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6224)


Parallel communicating finite automata (PCFAs) are systems of several finite state automata which process a common input string in a parallel way and are able to communicate by sending their states upon request. We consider deterministic and nondeterministic variants and distinguish four working modes. It is known that these systems in the most general mode are as powerful as one-way multi-head finite automata. It is additionally known that the number of heads corresponds to the number of automata in PCFAs in a constructive way. Thus, undecidability results as well as results on the hierarchies induced by the number of heads carry over from multi-head finite automata to PCFAs in the most general mode. Here, we complement these undecidability and hierarchy results also for the remaining working modes. In particular, we show that classical decidability questions are not semi-decidable for any type of PCFAs under consideration. Moreover, it is proven that the number of automata in the system induces infinite hierarchies for deterministic and nondeterministic PCFAs in three working modes.


Turing Machine Finite Automaton Input String Input Symbol State Automaton 
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  1. 1.
    Bordihn, H., Kutrib, M., Malcher, A.: On the computational capacity of parallel communicating finite automata. In: Ito, M., Toyama, M. (eds.) DLT 2008. LNCS, vol. 5257, pp. 146–157. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Brand, D., Zafiropulo, P.: On communicating finite-state machines. J. ACM 30, 323–342 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Buda, A.: Multiprocessor automata. Inform. Process. Lett. 25, 257–261 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Choudhary, A., Krithivasan, K., Mitrana, V.: Returning and non-returning parallel communicating finite automata are equivalent. RAIRO Inform. Théor. 41, 137–145 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Csuhaj-Varjú, E., Dassow, J., Kelemen, J., Păun, G.: Grammar Systems: A Grammatical Approach to Distribution and Cooperation. Gordon and Breach, Yverdon (1984)Google Scholar
  6. 6.
    Csuhaj-Varjú, E., Martín-Vide, C., Mitrana, V., Vaszil, G.: Parallel communicating pushdown automata systems. Int. J. Found. Comput. Sci. 11, 633–650 (2000)zbMATHCrossRefGoogle Scholar
  7. 7.
    Ďuriš, P., Jurdziński, T., Kutyłowski, M., Loryś, K.: Power of cooperation and multihead finite systems. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 896–907. Springer, Heidelberg (1998)Google Scholar
  8. 8.
    Harrison, M.A., Ibarra, O.H.: Multi-tape and multi-head pushdown automata. Inform. Control 13, 433–470 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Language, and Computation. Addison-Wesley, Reading (1979)Google Scholar
  10. 10.
    Ibarra, O.H.: On two-way multihead automata. J. Comput. System Sci. 7, 28–36 (1973)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Ibarra, O.H., Karhumäki, J., Okhotin, A.: On stateless multihead automata: Hierarchies and the emptiness problem. Theoret. Comp. Sci. 411, 581–593 (2010)zbMATHCrossRefGoogle Scholar
  12. 12.
    Klemm, R.: Systems of communicating finite state machines as a distributed alternative to finite state machines. Phd thesis, Pennsylvania State University (1996)Google Scholar
  13. 13.
    Kutrib, M.: Cellular automata – a computational point of view. In: New Developments in Formal Languages and Applications, pp. 183–227. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Kutrib, M.: Cellular automata and language theory. In: Encyclopedia of Complexity and System Science, pp. 800–823. Springer, Heidelberg (2009)Google Scholar
  15. 15.
    Malcher, A.: Descriptional complexity of cellular automata and decidability questions. J. Autom., Lang. Comb. 7, 549–560 (2002)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Martín-Vide, C., Mitrana, V.: Some undecidable problems for parallel communicating finite automata systems. Inform. Process. Lett. 77, 239–245 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Martín-Vide, C., Mateescu, A., Mitrana, V.: Parallel finite automata systems communicating by states. Int. J. Found. Comput. Sci. 13, 733–749 (2002)zbMATHCrossRefGoogle Scholar
  18. 18.
    Rosenberg, A.L.: On multi-head finite automata. IBM J. Res. Dev. 10, 388–394 (1966)zbMATHCrossRefGoogle Scholar
  19. 19.
    Yao, A.C., Rivest, R.L.: k + 1 heads are better than k. J. ACM 25, 337–340 (1978)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Henning Bordihn
    • 1
  • Martin Kutrib
    • 2
  • Andreas Malcher
    • 2
  1. 1.Institut für InformatikUniversität PotsdamPotsdamGermany
  2. 2.Institut für InformatikUniversität GiessenGiessenGermany

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