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Universality in Infinite Petri Nets

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Machines, Computations, and Universality (MCU 2015)

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Abstract

Finite classical Petri nets are non-Turing-complete. Two infinite Petri nets are constructed which simulate the linear cellular automaton Rule 110 via expanding traversals of the cell array. One net is obtained via direct simulation of the cellular automaton while the other net simulates a Turing machine, which simulates the cellular automaton. They use cell models of 21 and 14 nodes, respectively, and simulate the cellular automaton in polynomial time. Based on known results we conclude that these Petri nets are Turing-complete and run in polynomial time. We employ an induction proof technique that is applicable for the formal proof of Rule 110 ether and gliders properties further to the constructs presented by Matthew Cook.

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References

  1. Agerwala, T.: A complete model for representing the coordination of asynchronous processes, John Hopkins University, Hopkins Computer Science Program, Baltimore, MD, Research Report no. 32, July 1974

    Google Scholar 

  2. Berthomieu, B., Ribet, O.-P., Vernadat, F.: The tool TINA-construction of abstract state space for Petri nets and time Petri nets. Int. J. Prod. Res. 42(14), 2741–2756 (2004)

    Article  MATH  Google Scholar 

  3. Burkhard, H.-D.: On priorities of parallelism: Petri nets under the maximum firing strategy. In: Salwicki, A. (ed.) Logics of Programs and Their Applications. LNCS, vol. 148, pp. 86–97. Springer, Heidelberg (1983)

    Chapter  Google Scholar 

  4. Cook, M.: Universality in elementary cellular automata. Complex Syst. 15(1), 1–40 (2004). http://www.complex-systems.com/pdf/15-1-1.pdf

    Google Scholar 

  5. Cook, M.: A Concrete View of Rule 110 Computation. In: Neary, T., Woods, D., Seda, A.K., Murphy, N. (eds.) The Complexity of Simple Programs 2008, EPTCS 1, pp. 31–55 (2009). doi:10.4204/EPTCS.1.4

  6. Esparza, J.: Decidability and complexity of PN problems. In: Reisig, W., Rozenberg, G. (eds.) APN 1998. LNCS, vol. 1491, pp. 374–428. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  7. Fischer, P.C., Meyer, A.R., Rosenberg, A.L.: Counter machines and counter languages. Math. Syst. Theory 2(3), 265–283 (1968). http://dx.doi.org/10.1007/BF01694011

    Article  MathSciNet  Google Scholar 

  8. Ivanov, S., Pelz, E., Verlan, S.: Small Universal Petri Nets with Inhibitor Arcs. In: Computability in Europe, pp. 23–27, Budapest, Hungary, June 2014. ( http://arxiv.org/abs/arXiv:1312.4414)

  9. Korec, I.: Small universal register machines. Theoret. Comput. Sci. 168(2), 267–301 (1996)

    Article  MathSciNet  Google Scholar 

  10. Kotov, V.E.: Seti Petri. Nauka, Moscow (1984)

    Google Scholar 

  11. Neary, T.: Small universal Turing machines. PhD thesis, Department of Computer Science, National University of Ireland, Maynooth (2008)

    Google Scholar 

  12. Neary, T., Woods, D.: Small weakly universal turing machines. In: Kutyłowski, M., Charatonik, W., Gębala, M. (eds.) FCT 2009. LNCS, vol. 5699, pp. 262–273. Springer, Heidelberg (2009). http://www.ini.uzh.ch/\(\sim \)tneary/NearyWoods\_FCT2009.pdf

    Chapter  Google Scholar 

  13. Neary, T.: On the computational complexity of spiking neural P systems. Nat. Comput. 9(4), 831–851 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nielsen, M., Plotkin, G., Winskel, G.: Petri nets, event structures and domains, part i. Theoretical Computer Science 13, 85–108 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Peterson, J.: A note on colored petri nets. Inf. Proces. Lett. 11(1), 40–43 (1980)

    Article  MATH  Google Scholar 

  16. Peterson, J.: Petri Net Theory and the Modelling of Systems, Prentice-Hall (1981)

    Google Scholar 

  17. Petri, C.: Kommunikation mit Automaten. Bonn: Institut fur Instrumentelle Mathematik, Schriften des IIM, Nr. 2 (1962)

    Google Scholar 

  18. Smith, E.: Principles of high-level net theory. In: Reisig, W., Rozenberg, G. (eds.) APN 1998. LNCS, vol. 1491, pp. 174–210. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  19. Winkowski, J.: Reachability in contextual nets. In: Fundamenta Informaticae - Concurrency Specification and Programming Workshop (CSP 2001), vol. 51(1–2), pp. 235–250 (2002)

    Google Scholar 

  20. Zaitsev, D.A., Sleptsov, A.I.: State equations and equivalent transformations for timed petri nets. Cybern. Syst. Anal. 33(5), 659–672 (1997). doi:10.1007/BF02667189

    Article  MathSciNet  Google Scholar 

  21. Zaitsev, D.A.: A Small universal Petri net. In: Neary, T., Cook, M. (eds.) Proceedings Machines, Computations and Universality 2013 (MCU 2013), Zurich, Switzerland, September 9–11, Electronic Proceedings in Theoretical Computer Science 128, 190–202 (2013). doi:10.4204/EPTCS.128.22

  22. Zaitsev, D.A.: Small polynomial time universal petri nets, September 2013. arXiv:1309.7288

  23. Zaitsev D.A., Zaitsev I.D., Shmeleva T.R.: Infinite Petri Nets as Models of Grids. In: Khosrow-Pour, M. (ed.) Encyclopedia of Information Science and Technology, 3rd edn., vol. 10, Chap. 19, pp. 187–204. IGI-Global USA (2014)

    Google Scholar 

  24. Zaitsev, D.A.: Simulating cellular automata by infinite synchronous Petri nets. In: 21st Annual International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2015) Exploratory papers, vol. 24, pp. 91–100. TUCS Lecture Notes, Turku, Finland, 8–10 June 2015

    Google Scholar 

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Acknowledgement

The author would like to thank reviewers whose comments allowed the refinement of the presentation and Jacob Hendricks for his help in improving the readability of the paper.

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Authors and Affiliations

  1. Faculty of Computer Science, Vistula University, St. Stoklosy, 3, 02-787, Warsaw, Poland

    Dmitry A. Zaitsev

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  1. Dmitry A. Zaitsev
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Correspondence to Dmitry A. Zaitsev .

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Editors and Affiliations

  1. Université d'Orléans, Orléans, France

    Jerome Durand-Lose

  2. Eastern Mediterranean University, Famagusta, Cyprus

    Benedek Nagy

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Zaitsev, D.A. (2015). Universality in Infinite Petri Nets. In: Durand-Lose, J., Nagy, B. (eds) Machines, Computations, and Universality. MCU 2015. Lecture Notes in Computer Science(), vol 9288. Springer, Cham. https://doi.org/10.1007/978-3-319-23111-2_12

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  • DOI: https://doi.org/10.1007/978-3-319-23111-2_12

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