Small Universal Non-deterministic Petri Nets with Inhibitor Arcs

  • Sergiu Ivanov
  • Elisabeth Pelz
  • Sergey Verlan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8614)

Abstract

This paper investigates the universality problem for Petri nets with inhibitor arcs. Four descriptional complexity parameters are considered: the number of places, transitions, inhibitor arcs, and the maximal degree of a transition. Each of these parameters is aimed to be minimized, a special attention being given to the number of places. Four constructions are presented having the following values of parameters (listed in the above order): (5, 877, 1022, 729), (5, 1024, 1316, 379), (4, 668, 778, 555), and (4, 780, 1002, 299). The decrease of the number of places with respect to previous work is primarily due to the consideration of non-deterministic computations in Petri nets. Using equivalencies between models our results can be translated to multiset rewriting with forbidding conditions, or to P systems with inhibitors.

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References

  1. 1.
    Alhazov, A., Verlan, S.: Minimization strategies for maximally parallel multiset rewriting systems. Theoretical Computer Science 412(17), 1581–1591 (2011)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Barzdin, I.M.: Ob odnom klasse machin Turinga (machiny Minskogo), russian. Algebra i Logika 1, 42–51 (1963)MathSciNetGoogle Scholar
  3. 3.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming, 2nd edn. Duxbury Press, Brooks/Cole Publishing Company (2002)Google Scholar
  4. 4.
    Ivanov, S., Pelz, E., Verlan, S.: Small universal Petri nets with inhibitor arcs. In: Computability in Europe (2014)Google Scholar
  5. 5.
    Koiran, P., Moore, C.: Closed-form analytic maps in one and two dimensions can simulate universal turing machines. Theor. Comput. Sci. 210(1), 217–223 (1999)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Korec, I.: Small universal register machines. Theoretical Computer Science 168(2), 267–301 (1996)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Malcev, A.I.: Algorithms and Recursive Functions. Wolters-Noordhoff Pub. Co., Groningen (1970)Google Scholar
  8. 8.
    Margenstern, M.: Frontier between decidability and undecidability: A survey. Theoretical Computer Science 231(2), 217–251 (2000)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Margenstern, M.: An algorithm for buiding inrinsically universal automata in hyperbolic spaces. In: Arabnia, H.R., Murgin, M. (eds.) FCS, pp. 3–9. CSREA Press (2006)Google Scholar
  10. 10.
    Minsky, M.: Size and structure of universal Turing machines using tag systems. In: Recursive Function Theory: Proceedings, Symposium in Pure Mathematics, Provelence, vol. 5, pp. 229–238 (1962)Google Scholar
  11. 11.
    Minsky, M.: Computations: Finite and Infinite Machines. Prentice Hall, Englewood Cliffts (1967)Google Scholar
  12. 12.
    Gurobi Optimization, Inc. Gurobi optimizer reference manual (2014)Google Scholar
  13. 13.
    Păun, G.: Membrane Computing. An Introduction. Springer, Berlin (2002)CrossRefMATHGoogle Scholar
  14. 14.
    Pelz, E.: Closure properties of deterministic Petri nets. In: Brandenburg, F.J., Wirsing, M., Vidal-Naquet, G. (eds.) STACS 1987. LNCS, vol. 247, pp. 371–382. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  15. 15.
    Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages, vol. 1–3. Springer (1997)Google Scholar
  16. 16.
    Schroeppel, R.: A two counter machine cannot calculate 2N. In: AI Memos. MIT AI Lab (1972)Google Scholar
  17. 17.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 42(2), 230–265 (1936)MathSciNetGoogle Scholar
  18. 18.
    Woods, D., Neary, T.: The complexity of small universal Turing machines: A survey. Theor. Comput. Sci. 410(4-5), 443–450 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Zaitsev, D.A.: Universal Petri net. Cybernetics and Systems Analysis 48(4), 498–511 (2012)CrossRefGoogle Scholar
  20. 20.
    Zaitsev, D.A.: A small universal Petri net. EPTCS 128, 190–202 (2013); In Proceedings of Machines, Computations and Universality (MCU 2013), arXiv:1309.1043Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Sergiu Ivanov
    • 1
  • Elisabeth Pelz
    • 1
  • Sergey Verlan
    • 1
  1. 1.Laboratoire d’Algorithmique, Complexité et LogiqueUniversité Paris EstCréteilFrance

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