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Solving a Special Case of the P Conjecture Using Dependency Graphs with Dissolution

  • Alberto Leporati
  • Luca Manzoni
  • Giancarlo Mauri
  • Antonio E. Porreca
  • Claudio Zandron
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10725)

Abstract

We solve affirmatively a new special case of the P conjecture by Gh. Păun, which states that P systems with active membranes without charges and without non-elementary membrane division cannot solve \(\mathbf {NP}\)-complete problems in polynomial time. The variant we consider is monodirectional, i.e., without send-in communication rules, shallow, i.e., with membrane structures consisting of only one level besides the external membrane, and deterministic, rather than more generally confluent. We describe a polynomial-time Turing machine simulation of this variant of P systems, exploiting a generalised version of dependency graphs for P systems which, unlike the original version introduced by Cordón-Franco et al., also takes membrane dissolution into account.

References

  1. 1.
    Alhazov, A., Freund, R.: On the efficiency of P systems with active membranes and two polarizations. In: Mauri, G., Păun, G., Pérez-Jiménez, M.J., Rozenberg, G., Salomaa, A. (eds.) WMC 2004. LNCS, vol. 3365, pp. 146–160. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-31837-8_8 CrossRefGoogle Scholar
  2. 2.
    Alhazov, A., Pérez-Jiménez, M.J.: Uniform solution of QSAT using polarizationless active membranes. In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 122–133. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74593-8_11 CrossRefGoogle Scholar
  3. 3.
    Cordón-Franco, A., Gutiérrez-Naranjo, M.A., Pérez-Jiménez, M.J., Riscos-Núñez, A.: Exploring computation trees associated with P systems. In: Mauri, G., Păun, G., Pérez-Jiménez, M.J., Rozenberg, G., Salomaa, A. (eds.) WMC 2004. LNCS, vol. 3365, pp. 278–286. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-31837-8_16 CrossRefGoogle Scholar
  4. 4.
    Gazdag, Z., Kolonits, G.: Remarks on the computational power of some restricted variants of P systems with active membranes. In: Leporati, A., Rozenberg, G., Salomaa, A., Zandron, C. (eds.) CMC 2016. LNCS, vol. 10105, pp. 209–232. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-54072-6_14 CrossRefGoogle Scholar
  5. 5.
    Gutiérrez-Naranjo, M.A., Pérez-Jiménez, M.J., Riscos-Núñez, A., Romero-Campero, F.J.: P systems with active membranes, without polarizations and without dissolution: a characterization of P. In: Calude, C.S., Dinneen, M.J., Păun, G., Pérez-Jímenez, M.J., Rozenberg, G. (eds.) UC 2005. LNCS, vol. 3699, pp. 105–116. Springer, Heidelberg (2005).  https://doi.org/10.1007/11560319_11 CrossRefGoogle Scholar
  6. 6.
    Gutiérrez–Naranjo, M.A., Pérez–Jiménez, M.J., Riscos–Núñez, A., Romero–Campero, F.J.: On the power of dissolution in P systems with active membranes. In: Freund, R., Păun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2005. LNCS, vol. 3850, pp. 224–240. Springer, Heidelberg (2006).  https://doi.org/10.1007/11603047_16 CrossRefGoogle Scholar
  7. 7.
    Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: Simulating elementary active membranes. In: Gheorghe, M., Rozenberg, G., Salomaa, A., Sosík, P., Zandron, C. (eds.) CMC 2014. LNCS, vol. 8961, pp. 284–299. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-14370-5_18 Google Scholar
  8. 8.
    Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: Monodirectional P systems. Nat. Comput. 15(4), 551–564 (2016).  https://doi.org/10.1007/s11047-016-9565-2 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Murphy, N., Woods, D.: Active membrane systems without charges and using only symmetric elementary division characterise P. In: Eleftherakis, G., Kefalas, P., Păun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2007. LNCS, vol. 4860, pp. 367–384. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-77312-2_23 CrossRefGoogle Scholar
  10. 10.
    Murphy, N., Woods, D.: The computational power of membrane systems under tight uniformity conditions. Nat. Comput. 10(1), 613–632 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Păun, G.: Further twenty six open problems in membrane computing. In: Gutíerrez-Naranjo, M.A., Riscos-Nuñez, A., Romero-Campero, F.J., Sburlan, D. (eds.) Proceedings of the Third Brainstorming Week on Membrane Computing, pp. 249–262. Fénix Editora (2005)Google Scholar
  12. 12.
    Pérez-Jiménez, M.J., Romero-Jiménez, A., Sancho-Caparrini, F.: Complexity classes in models of cellular computing with membranes. Nat. Comput. 2(3), 265–284 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sosík, P., Rodríguez-Patón, A.: Membrane computing and complexity theory: a characterization of PSPACE. J. Comput. Syst. Sci. 73(1), 137–152 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Woods, D., Murphy, N., Pérez-Jiménez, M.J., Riscos-Núñez, A.: Membrane dissolution and division in P. In: Calude, C.S., Costa, J.F., Dershowitz, N., Freire, E., Rozenberg, G. (eds.) UC 2009. LNCS, vol. 5715, pp. 262–276. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-03745-0_28 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Alberto Leporati
    • 1
  • Luca Manzoni
    • 1
  • Giancarlo Mauri
    • 1
  • Antonio E. Porreca
    • 1
  • Claudio Zandron
    • 1
  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi di Milano-BicoccaMilanoItaly

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