Going Beyond Turing with P Automata: Partial Adult Halting and Regular Observer \(\omega \)-Languages

  • Rudolf Freund
  • Sergiu Ivanov
  • Ludwig Staiger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9252)


In this paper we investigate several variants of P automata having infinite runs on finite inputs. By imposing specific conditions on the infinite evolution of the systems, it is easy to find ways for going beyond Turing if we are watching the behavior of the systems on infinite runs. As specific variants we introduce a new halting variant for P automata which we call partial adult halting with the meaning that a specific predefined part of the P automaton does not change any more from some moment on during the infinite run. In a more general way, we can assign \(\omega \)-languages as observer languages to the infinite runs of a P automaton. Specific variants of regular \(\omega \)-languages then, for example, characterize the red-green P automata.


  1. 1.
    Alhazov, A., Aman, B., Freund, R.: P systems with anti-matter. In: [10], pp. 66–85Google Scholar
  2. 2.
    Aman, B., Csuhaj-Varjú, E., Freund, R.: Red–green P automata. In: [10], pp. 139–157Google Scholar
  3. 3.
    Budnik, P.: What Is and What Will Be. Mountain Math Software, Los Gatos (2006)Google Scholar
  4. 4.
    Calude, C.S., Păun, Gh.: Bio-steps beyond Turing. Biosystems 77, 175–194 (2004)Google Scholar
  5. 5.
    Calude, C.S., Staiger, L.: A note on accelerated Turing machines. Math. Struct. Comput. Sci. 20(6), 1011–1017 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Csuhaj-Varjú, E., Vaszil, Gy.: P automata or purely communicating accepting P systems. In: Păun, Gh., Rozenberg, G., Salomaa, A., Zandron, C. (eds.) Membrane Computing. Lecture Notes in Computer Science, vol. 2597, pp. 219–233. Springer, Heidelberg (2003)Google Scholar
  7. 7.
    Freund, R., Pérez-Hurtado, I., Riscos-Núñez, A., Verlan, S.: A formalization of membrane systems with dynamically evolving structures. Int. J. Comput. Math. 90(4), 801–815 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Freund, R., Oswald, M.: A short note on analysing P systems. Bull. EATCS 78, 231–236 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Freund, R., Oswald, M., Staiger, L.: \({\omega }\)-P automata with communication rules. In: Martín-Vide, C., Mauri, G., Păun, Gh., Rozenberg, G., Salomaa, A. (eds.) WMC 2003. LNCS, vol. 2933, pp. 203–217. Springer, Heidelberg (2004)Google Scholar
  10. 10.
    Pérez-Jiménez, M.J.: A bioinspired computing approach to model complex systems. In: Gheorghe, M., Rozenberg, G., Salomaa, A., Sosík, P., Zandron, C. (eds.) CMC 2014. LNCS, vol. 8961, pp. 20–34. Springer, Heidelberg (2014) Google Scholar
  11. 11.
    van Leeuwen, J., Wiedermann, J.: Computation as an unbounded process. Theor. Comput. Sci. 429, 202–212 (2012)CrossRefzbMATHGoogle Scholar
  12. 12.
    Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice Hall, Englewood Cliffs (1967) zbMATHGoogle Scholar
  13. 13.
    Păun, Gh.: Computing with membranes. J. Comput. Syst. Sci. 61(1), 108–143 (2000). (and Turku Center for Computer Science-TUCS Report 208, November 1998.
  14. 14.
    Păun, Gh.: Membrane Computing: An Introduction. Springer, Heidelberg (2002)Google Scholar
  15. 15.
    Păun, Gh., Rozenberg, G., Salomaa, A. (eds.): The Oxford Handbook of Membrane Computing. Oxford University Press, Oxford (2010)Google Scholar
  16. 16.
    Perrin, D., Pin, J.-É.: Infinite Words. Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  17. 17.
    Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages: 3 volumes. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  18. 18.
    Sosík, P., Valík, O.: On evolutionary lineages of membrane systems. In: Freund, R., Păun, Gh., Rozenberg, G., Salomaa, A. (eds.) WMC 2005. LNCS, vol. 3850, pp. 67–78. Springer, Heidelberg (2006)Google Scholar
  19. 19.
    Staiger, L.: Finite-state \(\omega \)-languages. J. Comput. Syst. Sci. 27(3), 434–448 (1983)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Staiger, L.: \(\omega \)-computations on Turing machines and the accepted languages. In: Lovász, L., Szemerédi, E. (eds.) Theory of Algorithms (Colloquia Mathematica Societatis Janos Bolyai), vol. 44, pp. 393–403. North Holland, Amsterdam (1986)Google Scholar
  21. 21.
    Staiger, L.: \(\omega \)-languages. In: [17], vol. 3, pp. 339–387Google Scholar
  22. 22.
    Staiger, L., Wagner, K.: Automatentheoretische und automatenfreie Charakterisierungen topologischer Klassen regulärer Folgenmengen. Elektron. Informationsverarb. Kybernetik 10(7), 379–392 (1974)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 133–192. North Holland, Amsterdam (1990)Google Scholar
  24. 24.
    Wagner, K.: On \(\omega \)-regular sets. Inf. Control 43(2), 123–177 (1979)CrossRefzbMATHGoogle Scholar
  25. 25.
    The P Systems Website.

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Technische Universität WienViennaAustria
  2. 2.Université Paris EstParisFrance
  3. 3.Martin-Luther-Universität Halle-WittenbergHalleGermany

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