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Time and Space Complexity of P Systems — And Why They Matter

  • Alberto LeporatiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11399)

Abstract

Computational complexity theory allows one to investigate the amount of resources (usually, time and/or space) which are needed to solve a given computational problem. Indeed, since the appearance of P systems several computational complexity techniques have been applied to study their computational power and efficiency. In this paper, starting from some results which have been obtained in the last few years by the group of Membrane Computing at the University of Milan-Bicocca (also known as the “Milano Team”), sometimes in collaboration with colleagues from the Membrane Computing community, I will make some observations on what is the relevance (in my opinion) of time and space complexity theory for P systems. Speaking about the results, I will focus in particular on the ideas lying behind them, without delving into technical details. I will also comment on the importance of these results for applications, such as modelling complex systems and implementing decentralized applications. I will finally conclude with some (somewhat provocative) connections with other Computer Science subjects, related with Cryptography, Computer and Network Security, and Decentralized Applications.

References

  1. 1.
    Alhazov, A., Martín-Vide, C., Pan, L.: Solving a PSPACE-complete problem by recognizing P systems with restricted active membranes. Fundam. Informaticae 58(2), 67–77 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alhazov, A., Leporati, A., Mauri, G., Porreca, A.E., Zandron, C.: Space complexity equivalence of P systems with active membranes and Turing machines. Theor. Comput. Sci. 529, 69–81 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cormen, T.H., Leiserson, C.H., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge (1990)zbMATHGoogle Scholar
  4. 4.
    Freund, R., Leporati, A., Oswald, M., Zandron, C.: Sequential P systems with unit rules and energy assigned to membranes. In: Margenstern, M. (ed.) MCU 2004. LNCS, vol. 3354, pp. 200–210. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-31834-7_16CrossRefzbMATHGoogle Scholar
  5. 5.
    Furst, M., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17, 13–27 (1984)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 169–178. ACM (2009)Google Scholar
  7. 7.
    Ionescu, M., Păun, G., Yokomori, T.: Spiking neural P systems. Fundam. Informaticae 71(3), 279–308 (2006)Google Scholar
  8. 8.
    Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: How to run turing machines on encrypted data. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 536–553. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40084-1_30CrossRefGoogle Scholar
  9. 9.
    Konur, S., Gheorghe, M., Dragomir, C., Mierla, L., Ipate, F., Krasnogor, N.: Qualitative and quantitative analysis of systems and synthetic biology constructs using P systems. ACS Synth. Biol. 4(1), 83–92 (2015)CrossRefGoogle Scholar
  10. 10.
    Korec, I.: Small universal register machines. Theor. Comput. Sci. 168, 267–301 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Leporati, A., Ferretti, C., Mauri, G., Pérez-Jiménez, M.J., Zandron, C.: Complexity aspects of polarizationless membrane systems. Nat. Comput. 8, 703–717 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: Constant-space P systems with active membranes. Fundam. Informaticae 134(1–2), 111–128 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Leporati, A., Mauri, G., Porreca, A.E., Zandron, C.: A gap in the space hierarchy of P systems with active membranes. J. Automata Lang. Comb. 19(1–4), 173–184 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Leporati, A., Mauri, G., Zandron, C., Păun, G., Péréz-Jiménez, M.J.: Uniform solutions to SAT and Subset Sum by spiking neural P systems. Nat. Comput. 8, 681–702 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Leporati, A., Zandron, C., Ferretti, C., Mauri, G.: Solving numerical NP-complete problems with spiking neural P systems. In: Eleftherakis, G., Kefalas, P., Păun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2007. LNCS, vol. 4860, pp. 336–352. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-77312-2_21CrossRefGoogle Scholar
  16. 16.
    Leporati, A., Zandron, C., Ferretti, C., Mauri, G.: On the computational power of spiking neural P systems. Int. J. Unconventional Comput. 5, 459–473 (2009)Google Scholar
  17. 17.
    Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: A toolbox for simpler active membrane algorithms. Theor. Comput. Sci. 673, 42–57 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mauri, G., Leporati, A., Porreca, A.E., Zandron, C.: Recent complexity-theoretic results on P systems with active membranes. J. Logic Comput. 25(4), 1047–1071 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mix Barrington, D.A., Immerman, N., Straubing, H.: On uniformity within NC\({}^1\). J. Comput. Syst. Sci. 41(3), 274–306 (1990)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Păun, G.: Computing with membranes. J. Comput. Syst. Sci. 1(61), 108–143 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Păun, G.: P systems with active membranes: attacking NP-complete problems. J. Automata Lang. Comb. 6(1), 75–90 (2001)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Păun, G., Pérez-Jiménez, M.J.: Solving Problems in a distributed way in membrane computing: dP systems. Int. J. Comput. Commun. Control 5(2), 238–250 (2010)CrossRefGoogle Scholar
  23. 23.
    Păun, G., Rozenberg, G., Salomaa, A. (eds.): The Oxford Handbook of Membrane Computing. Oxford University Press, Oxford (2010)zbMATHGoogle Scholar
  24. 24.
    Pérez-Jiménez, M.J., Romero Jiménez, A., Sancho Caparrini, F.: Complexity classes in models of cellular computing with membranes. Nat. Comput. 2(3), 265–285 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Porreca, A.E., Leporati, A., Mauri, G., Zandron, C.: P systems with active membranes: trading time for space. Nat. Comput. 10(1), 167–182 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Porreca, A.E., Leporati, A., Mauri, G., Zandron, C.: P systems with elementary active membranes: beyond NP and coNP. In: Gheorghe, M., Hinze, T., Păun, G., Rozenberg, G., Salomaa, A. (eds.) CMC 2010. LNCS, vol. 6501, pp. 338–347. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-18123-8_26CrossRefzbMATHGoogle Scholar
  27. 27.
    Porreca, A.E., Leporati, A., Mauri, G., Zandron, C.: Elementary active membranes have the power of counting. Int. J. Nat. Comput. Res. 2(3), 35–48 (2011)CrossRefGoogle Scholar
  28. 28.
    Porreca, A.E., Leporati, A., Mauri, G., Zandron, C.: P systems with active membranes working in polynomial space. Int. J. Found. Comput. Sci. 2(1), 65–73 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Porreca, A.E., Leporati, A., Mauri, G., Zandron, C.: P systems simulating oracle computations. In: Gheorghe, M., Păun, G., Rozenberg, G., Salomaa, A., Verlan, S. (eds.) CMC 2011. LNCS, vol. 7184, pp. 346–358. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-28024-5_23CrossRefzbMATHGoogle Scholar
  30. 30.
    Porreca, A.E., Leporati, A., Mauri, G., Zandron, C.: Sublinear-space P systems with active membranes. In: Csuhaj-Varjú, E., Gheorghe, M., Rozenberg, G., Salomaa, A., Vaszil, G. (eds.) CMC 2012. LNCS, vol. 7762, pp. 342–357. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36751-9_23CrossRefzbMATHGoogle Scholar
  31. 31.
    Rong, H., Wu, T., Pan, L., Zhang, G.: Spiking neural P systems: theoretical results and applications. In: Bulletin of the International Membrane Computing Society (IMCS), June 2018. http://membranecomputing.net/IMCSBulletin/index.php?page=SNP-review
  32. 32.
    Sosík, P.: The computational power of cell division in P systems: beating down parallel computers? Nat. Comput. 2(3), 287–298 (2003)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sosík, P., Rodríguez-Patón, A.: Membrane computing and complexity theory: a characterization of PSPACE. J. Comput. Syst. Sci. 73(1), 137–152 (2007)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Stockmeyer, L.J.: The polynomial hierarchy. Theor. Comput. Sci. 3, 1–22 (1976)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20(5), 865–877 (1991)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Valsecchi, A., Porreca, A.E., Leporati, A., Mauri, G., Zandron, C.: An efficient simulation of polynomial-space turing machines by P systems with active membranes. In: Păun, G., Pérez-Jiménez, M.J., Riscos-Núñez, A., Rozenberg, G., Salomaa, A. (eds.) WMC 2009. LNCS, vol. 5957, pp. 461–478. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-11467-0_31CrossRefzbMATHGoogle Scholar
  37. 37.
    Wood, G.: Ethereum: a secure decentralised generalised transaction ledger. Ethereum Project Yellow Paper (2014). https://github.com/ethereum/yellowpaper
  38. 38.
    Zandron, C., Ferretti, C., Mauri, G.: Solving NP-complete problems using P systems with active membranes. In: Antoniou, I., Calude, C.S., Dinneen, M.J. (eds.) Unconventional Models of Computation, UMC’2K. Discrete Mathematics and Theoretical Computer Science, pp. 289–301. Springer, London (2001).  https://doi.org/10.1007/978-1-4471-0313-4_21CrossRefGoogle Scholar
  39. 39.
    Zandron, C., Leporati, A., Ferretti, C., Mauri, G., Pérez-Jiménez, M.J.: On the computational efficiency of polarizationless recognizer P systems with strong division and dissolution. Fundam. Informaticae 87(1), 79–91 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi di Milano-BicoccaMilanoItaly

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