P Systems with Elementary Active Membranes: Beyond NP and coNP

  • Antonio E. Porreca
  • Alberto Leporati
  • Giancarlo Mauri
  • Claudio Zandron
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6501)

Abstract

We prove that a uniform family of P systems with active membranes, where division rules only operate on elementary membranes and dissolution rules are avoided, can be used to solve the following PP-complete decision problem in polynomial time: given a Boolean formula of m variables in 3CNF, do at least \(\sqrt{2^m}\) among the 2m possible truth assignments satisfy it? As a consequence, the inclusion \(\mathbf{PP} \subseteq \mathbf{PMC}_{\mathcal{AM}(\mathrm{-d,-n})}\) holds: this provides an improved lower bound on the class of languages decidable by this kind of P systems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alhazov, A., Burtseva, L., Cojocaru, S., Rogozhin, Y.: Solving PP-complete and #P-complete problems by P systems with active membranes. In: Corne, D.W., Frisco, P., Păun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2008. LNCS, vol. 5391, pp. 108–117. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Alhazov, A., Martín-Vide, C., Pan, L.: Solving a PSPACE-complete problem by recognizing P systems with restricted active membranes. Fundamenta Informaticae 58(2), 67–77 (2003)MathSciNetMATHGoogle Scholar
  3. 3.
    Bailey, D.D., Dalmau, V., Kolaitis, P.G.: Phase transitions of PP-complete satisfiability problems. Discrete Applied Mathematics 155(12), 1627–1639 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Balcázar, J.L., Díaz, J., Gabarró, J.: Structural Complexity I, 2nd edn. Springer, Heidelberg (1995)CrossRefMATHGoogle Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York (1979)MATHGoogle Scholar
  6. 6.
    Gill, J.T.: Computational complexity of probabilistic Turing machines. In: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, pp. 91–95 (1974)Google Scholar
  7. 7.
    Lagoudakis, M.G., LaBean, T.H.: 2D DNA self-assembly for satisfiability. In: Winfree, E., Gifford, D.K. (eds.) DNA Based Computers V. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 54, pp. 139–152 (1999)Google Scholar
  8. 8.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1993)MATHGoogle Scholar
  9. 9.
    Pérez-Jiménez, M.J., Romero Jiménez, A., Sancho Caparrini, F.: Complexity classes in models of cellular computing with membranes. Natural Computing 2(3), 265–285 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Porreca, A.E., Leporati, A., Mauri, G., Zandron, C.: P systems with active membranes: Trading time for space. Natural Computing (in press), doi:10.1007/s11047-010-9189-xGoogle Scholar
  11. 11.
    Păun, G.: P systems with active membranes: Attacking NP-complete problems. Journal of Automata, Languages and Combinatorics 6(1), 75–90 (2001)MathSciNetMATHGoogle Scholar
  12. 12.
    Sosík, P.: The computational power of cell division in P systems: Beating down parallel computers? Natural Computing 2(3), 287–298 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Sosík, P., Rodríguez-Patón, A.: Membrane computing and complexity theory: A characterization of PSPACE. Journal of Computer and System Sciences 73(1), 137–152 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zandron, C., Ferretti, C., Mauri, G.: Solving NP-complete problems using P systems with active membranes. In: Antoniou, I., Calude, C., Dinneen, M.J. (eds.) Unconventional Models of Computation, UMC’2K: Proceedings of the Second International Conference, pp. 289–301. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

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

Personalised recommendations