Membrane Computing: Basics and Frontiers

  • Gheorghe Păun


Membrane computing is a branch of natural computing inspired by the structure and the functioning of the living cell, as well as by the cooperation of cells in tissues, colonies of cells, and neural nets. This chapter briefly introduces the basic notions and (types of) results of this research area, also discussing open problems and research topics. Several central classes of computing models (called P systems) are considered: cell-like P systems with symbol objects processed by means of multiset rewriting rules, symport/antiport P systems, P systems with active membranes, spiking neural P systems, and numerical P systems.


Membrane Structure Turing Machine Evolution Rule Elementary Membrane Division Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    C.S. Calude, Gh. Păun, G. Rozenberg, A. Salomaa (eds.) Multiset Processing. Mathematical, Computer Science, and Molecular Computing Points of View. LNCS, vol. 2235 (Springer, Berlin, 2001) The first international meeting devoted to membrane computing was organized already in the summer of 2000, in Curtea de Argeş, Romania, and it was concerned both with the developments in the emerging research area of membrane computing and with the mathematical and computer science investigations of multisets. This LNCS volume is the proceedings of the workshop, edited after the meeting.Google Scholar
  2. 2.
    G. Ciobanu, Gh. Păun, M.J. Pérez-Jiménez (eds.) Applications of Membrane Computing (Springer, Berlin, 2006) The volume presents several classes of applications (in biology and biomedicine, computer science, linguistics), as well as the software available at the time of editing the book, and a selective bibliography of membrane computing. Here are the sections of the chapter Computer Science Applications: Static sorting P systems; Membrane-based devices used in computer graphics; An analysis of a public key protocol with membranes; Membrane algorithms: approximate algorithms for NP-complete optimization problems; and Computationally hard problems addressed through P systems.Google Scholar
  3. 3.
    P. Frisco, M. Gheorghe, M.J. Pérez-Jiménez (eds.) Applications of Membrane Computing in Systems and Synthetic Biology. (Springer, Berlin, 2014) Different from volume [2], this time only applications in biology and biomedicine are concerned, at the level of year 2013, with a detailed biological motivation, in most cases reporting research done in interdisciplinary teams, including both biologists and computer scientists.Google Scholar
  4. 4.
    R. Freund, Particular results for variants of P systems with one catalyst in one membrane, in Proc. Fourth Brainstorming Week on Membrane Computing, vol. II (Fénix Editora, Sevilla, 2006), pp. 41–50Google Scholar
  5. 5.
    R. Freund, Purely catalytic P systems: Two catalysts can be sufficient for computational completeness, in Proc. 14th Intern. Conf. on Membrane Computing (Chişinău, Moldova, 2013), pp. 153–166Google Scholar
  6. 6.
    R. Freund, O.H. Ibarra, A. Păun, P. Sosík, H.-C. Yen, Catalytic P systems. Chapter 4 of [30]Google Scholar
  7. 7.
    R. Freund, L. Kari, M. Oswald, P. Sosík, Computationally universal P systems without priorities: two catalysts are sufficient. Theor. Comput. Sci. 330, 251–266 (2005) After a series of previous papers, the first one, by P. Sosík, where the universality of catalytic P systems was proved for systems with 8, then 6, catalysts, this paper established the best result in this respect: two catalysts suffice.Google Scholar
  8. 8.
    R. Freund, Gh. Păun, Universal P systems: One catalyst can be sufficient, in Proc. 11th Brainstorming Week on Membrane Computing (Fénix Editora, Sevilla, 2013), pp. 81–96Google Scholar
  9. 9.
    M. Gheorghe, Gh. Păun, M.J. Pérez-Jiménez, G. Rozenberg, Frontiers of membrane computing: Open problems and research topics. Int. J. Found. Comput. Sci. 24(5), 547–623 (2013) (first version in Proc. Tenth Brainstorming Week on Membrane Computing, vol. I (Sevilla, 2012), pp. 171–249, January 30–February 3) This paper circulated in the membrane computing community in the brainstorming version under the title of “mega-paper.” In this form, it contains 26 sections, written by separate authors, covering most of the branches of this research area and presenting open problems and research topics of current interest. The titles of these 26 sections are worth recalling: A glimpse to membrane computing; Some general issues; The power of small numbers; Polymorphic P systems; P colonies and dP automata; Spiking neural P systems; Control words associated with P systems; Speeding up P automata; Space complexity and the power of elementary membrane division; The P-conjecture and hierarchies; Seeking sharper frontiers of efficiency in tissue P systems; Time-free solutions to hard computational problems; Fypercomputations; Numerical P systems; P systems formal verification and testing; Causality, semantics, behavior; Kernel P systems; Bridging P and R; P systems and evolutionary computing interactions; Metabolic P systems; Unraveling oscillating structures by means of P systems; Simulating cells using P systems; P systems for computational systems and synthetic biology; Biologically plausible applications of spiking neural P systems for an explanation of brain cognitive functions; Computer vision; and Open problems on simulation of membrane computing models.Google Scholar
  10. 10.
    O.H. Ibarra, Z. Dang, O. Egecioglu, Catalytic P systems, semilinear sets, and vector addition systems. Theor. Comput. Sci. 312, 379–399 (2004)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    O.H. Ibarra, Z. Dang, O. Egecioglu, G. Saxena, Characterizations of catalytic membrane computing systems, in 28th Intern. Symp. Math. Found. Computer Sci., ed. by B. Rovan, P. Vojtás. LNCS, vol. 2747 (Springer, 2003), pp. 480–489Google Scholar
  12. 12.
    M. Ionescu, Gh. Păun, T. Yokomori, Spiking neural P systems. Fundamenta Informaticae 71(2–3), 279–308 (2006) This is the paper where the spiking neural P systems were introduced, and two basic results of this area were proved: universality in the general case, and semilinearity of the computed sets of numbers in the bounded case. Similar results were later obtained for many classes of SN P systems.Google Scholar
  13. 13.
    M. Ionescu, Gh. Păun, M.J. Pérez-Jiménez, T. Yokomori, Spiking neural dP systems. Fundamenta Informaticae 11(4), 423–436 (2011)Google Scholar
  14. 14.
    S.N. Krishna, A. Păun, Results on catalytic and evolution-communication P systems. New Generat. Comput. 22, 377–394 (2004)CrossRefMATHGoogle Scholar
  15. 15.
    K. Krithivasan, Gh. Păun, A. Ramanujan, On controlled P systems. Fundamenta Informaticae 131(3–4), 451–464 (2014)MATHMathSciNetGoogle Scholar
  16. 16.
    A. Leporati, A.E. Porreca, C. Zandron, G. Mauri, Improving universality results on parallel enzymatic numerical P systems, in Proc. 11th Brainstorming Week on Membrane Computing (Fénix Editora, Sevilla, 2013), pp. 177–200Google Scholar
  17. 17.
    A. Leporati, A.E. Porreca, C. Zandron, G. Mauri, Enzymatic numerical P systems using elementary arithmetic operations, in Proc. 14th Intern. Conf. on Membrane Computing (Chişinău, Moldova, 2013), pp. 225–240Google Scholar
  18. 18.
    W. Maass, C. Bishop (eds.) Pulsed Neural Networks (MIT Press, Cambridge, 1999)Google Scholar
  19. 19.
    M. Mutyam, K. Krithivasan, P systems with membrane creation: Universality and efficiency, in Proc. MCU 2001 ed. by M. Margenstern, Y. Rogozhin. LNCS, vol. 2055 (Springer, Berlin, 2001), pp. 276–287Google Scholar
  20. 20.
    A.B. Pavel, C.I. Vasile, I. Dumitrache, Robot localization implemented with enzymatic numerical P systems, in Proc. Conf. Living Machines 2012, LNCS, vol. 7375 (Springer, 2012), pp. 204–215Google Scholar
  21. 21.
    A. Păun, Gh. Păun, The power of communication: P systems with symport/antiport. New Generat. Comput. 20, 295–305 (2002) The symport/antiport P systems were introduced here, and their universality was proved for rules of various complexities/sizes. These results were improved in a large number of papers, until reaching universality for minimal symport and antiport rules.Google Scholar
  22. 22.
    Gh. Păun, Computing with membranes. J. Comput. Syst. Sci. 61(1), 108–143 (2000) (and Turku Center for Computer Science-TUCS Report 208, November 1998, This is the paper where membrane computing was initiated. The cell-like P systems are introduced, both with symbol objects and string objects, and for both cases the universality was proved (using the characterization of Turing computable sets of numbers as the length sets of languages generated by context-free matrix grammars with appearance checking; later, more direct and simple proofs were obtained, starting from register machines). In the case of strings, both rewriting and splicing rules were investigated.
  23. 23.
    Gh. Păun, Computing with membranes—A variant. Int. J. Found. Comput. Sci. 11(1), 167–182 (2000)CrossRefGoogle Scholar
  24. 24.
    Gh. Păun, P systems with active membranes: attacking NP-complete problems. J. Autom. Lang. Combinat. 6, 75–90 (2001) Membrane division was introduced here, in the general framework of P systems with active membranes (the membranes are explicit parts of the object evolution rules), and a polynomial semi-uniform solution to SAT is provided. Later, uniform solutions were obtained (also for other NP-complete problems).Google Scholar
  25. 25.
    Gh. Păun, Membrane Computing. An Introduction (Springer, Berlin, 2002) This is the first survey of membrane computing, systematizing the notions and the results at only a few years after the initiation of this research area. After an informal introduction (“Membrane computing—what it is and what it is not”) and a chapter providing the biological and the computability prerequisites for the rest of the book, one presents the cell-like P systems with symbol objects and multiset rewriting rules, the systems with symport/antiport rules, the P systems with string objects, and then the tissue-like P systems; their computing power is investigated; then one passes to the computing efficiency (“Trading space for time”), considering P systems with membrane division, membrane creation, string replication, and precomputed resources. Two more chapters present “further technical results” and “(attempts to get) back to reality.” The book ends with a list of open problems and of universality results.Google Scholar
  26. 26.
    Gh. Păun, Towards “fypercomputations” (in membrane computing), in Languages Alive. Essays Dedicated to Jurgen Dassow on the Occasion of His 65 Birthdayed. by H. Bordihn, M. Kutrib, B. Truthe. LNCS, vol. 7300 (Springer, Berlin, 2012), pp. 207–221 The term “fypercomputation” (coming from “fast computation” and reminding of “hypercomputation” = a computation going beyond the “Turing barrier”) was coined to name situations when a computing device can solve NP-complete problems in polynomial time, hence when a significant efficiency speedup is obtained.Google Scholar
  27. 27.
    Gh. Păun, Some open problems about catalytic, numerical and spiking neural P systems, in Proc. 14th Intern. Conf. on Membrane Computing (Chişinău, Moldova, 2013), pp. 25–34Google Scholar
  28. 28.
    Gh. Păun, R. Păun, Membrane computing and economics: Numerical P systems. Fundamenta Informaticae 73, 213–227 (2006)MATHMathSciNetGoogle Scholar
  29. 29.
    Gh. Păun, M.J. Pérez-Jiménez, Solving problems in a distributed way in membrane computing: dP systems. Int. J. Comput. Commun. Cont. 5(2), 238–252 (2010)Google Scholar
  30. 30.
    Gh. Păun, G. Rozenberg, A. Salomaa (eds.) Handbook of Membrane Computing (Oxford University Press, 2010) The basics of membrane computing are given in the book [25] (translated in Chinese in 2013), but the domain has fast evolved beyond the contents of the volume; new classes of P systems were introduced; new results and applications were reported. This made both necessary and possible the editing of the present handbook, a comprehensive survey of membrane computing at the level of 2009. Its contents are a suggestive hint to the landscape of membrane computing: 1. An introduction to and an overview of membrane computing (Gh. Păun, G. Rozenberg); 2. Cell biology for membrane computing (D. Besozzi, I.I. Ardelean); 3. Computability elements for membrane computing (Gh. Păun, G. Rozenberg, A. Salomaa); 4. Catalytic P systems (R. Freund, O.H. Ibarra, A. Păun, P. Sosík, H.-C. Yen); 5. Communication P systems (R. Freund, A. Alhazov, Y. Rogozhin, S. Verlan); 6. P automata (E. Csuhaj-Varjú, M. Oswald, G. Vaszil); 7. P systems with string objects (C. Ferretti, G. Mauri, C. Zandron); 8. Splicing P systems (S. Verlan, P. Frisco); 9. Tissue and population P systems (F. Bernardini, M. Gheorghe); 10. Conformon P systems (P. Frisco); 11. Active membranes (Gh. Păun); 12. Complexity – Membrane division, membrane creation (M.J. Pérez-Jiménez, A. Riscos-Núñez, Á. Romero-Jiménez, D. Woods); 13. Spiking neural P systems (O.H. Ibarra, A. Leporati, A. Păun, S. Woodworth); 14. P systems with objects on membranes (M. Cavaliere, S.N. Krishna, A. Păun, Gh. Păun); 15. Petri nets and membrane computing (J. Kleijn, M. Koutny); 16. Semantics of P systems (G. Ciobanu); 17. Software for P systems (D. Díaz-Pernil, C. Graciani, M.A. Gutiérrez-Naranjo, I. Pérez-Hurtado, M.J. Pérez-Jiménez); 18. Probabilistic/stochastic models (P. Cazzaniga, M. Gheorghe, N. Krasnogor, G. Mauri, D. Pescini, F.J. Romero-Campero); 19. Fundamentals of metabolic P systems (V. Manca); 20. Metabolic P dynamics (V. Manca); 21. Membrane algorithms (T.Y. Nishida, T. Shiotani, Y. Takahashi); 22. Membrane computing and computer science (R. Ceterchi, D. Sburlan); 23. Other developments; 23.1. P Colonies (A. Kelemenová); 23.2. Time in membrane computing (M. Cavaliere, D. Sburlan); 23.3. Membrane computing and self-assembly (M. Gheorghe, N. Krasnogor); 23.4. Membrane computing and X-machines (P. Kefalas, I. Stamatopoulou, M. Gheorghe, G. Eleftherakis); 23.5. Q-UREM P systems (A. Leporati); 23.6. Membrane computing and economics (Gh. Păun, R.A. Păun); 23.7 Mobile membranes and mobile ambients (B. Aman, G. Ciobanu); 23.8. Other topics (Gh. Păun, G. Rozenberg)Google Scholar
  31. 31.
    Gh. Păun, S. Yu, On synchronization in P systems. Fundamenta Informaticae 38(4), 397–410 (1999)MATHMathSciNetGoogle Scholar
  32. 32.
    M.J. Pérez-Jiménez, A. Riscos-Núñez, A. Romero-Jiménez, Complexity— Membrane division and membrane creation. Chapter 12 of [30]Google Scholar
  33. 33.
    P. Sosík, A catalytic P system with two catalysts generating a non-semilinear set. Romanian J. Inf. Sci. Technology 16(1), 3–9 (2013)Google Scholar
  34. 34.
    C.I. Vasile, A.B. Pavel, J. Kelemen, Implementing obstacle avoidance and follower behaviors on Koala robots using numerical P systems, in Tenth Brainstorming Week on Membrane Computing, vol. II (Sevilla, 2012), pp. 215–227Google Scholar
  35. 35.
    C. Zandron, C. Ferretti, G. Mauri, Solving NP-complete problems using P systems with active membranes, in Proc. Unconventional Models of Computation ed. by I. Antoniou et al. (Springer, 2000), pp. 289–301 Among other results, one proves here the so-called “Milano theorem,” saying that P systems without membrane division cannot solve NP-complete problems in polynomial time (unless if P = NP)Google Scholar
  36. 36.
    The P Systems Website,

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institute of Mathematics of the Romanian AcademyBucureştiRomania

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