Characterizations of Catalytic Membrane Computing Systems

  • Oscar H. Ibarra
  • Zhe Dang
  • Omer Egecioglu
  • Gaurav Saxena
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2747)


We look at 1-region membrane computing systems which only use rules of the form CaCv, where C is a catalyst, a is a noncatalyst, and v is a (possibly null) string of noncatalysts. There are no rules of the form av. Thus, we can think of these systems as “purely” catalytic. We consider two types: (1) when the initial configuration contains only one catalyst, and (2) when the initial configuration contains multiple (not necessarily distinct) catalysts. We show that systems of the first type are equivalent to communication-free Petri nets, which are also equivalent to commutative context-free grammars. They define precisely the semilinear sets. This partially answers an open question in [19]. Systems of the second type define exactly the recursively enumerable sets of tuples (i.e., Turing machine computable). We also study an extended model where the rules are of the form q: (pCaCv) (where q and p are states), i.e., the application of the rules is guided by a finite-state control. For this generalized model, type (1) as well as type (2) with some restriction correspond to vector addition systems.


membrane computing catalytic system semilinear set vector addition system reachability problem 


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  1. 1.
    Baker, H.G.: Rabin’s proof of the undecidability of the reachability set inclusion problem for vector addition systems. In: C.S.C. Memo 79, Project MAC, MIT (1973)Google Scholar
  2. 2.
    Berry, G., Boudol, G.: The chemical abstract machine. In: POPL 1990, pp. 81–94. ACM Press, New York (1990)CrossRefGoogle Scholar
  3. 3.
    Bottoni, P., Martin-Vide, C., Paun, G., Rozenberg, G.: Membrane systems with promoters/inhibitors. Acta Informatica 38(10), 695–720 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dassow, J., Paun, G.: On the power of membrane computing. Journal of Universal Computer Science 5(2), 33–49 (1999)MathSciNetGoogle Scholar
  5. 5.
    Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. In: Reichel, H. (ed.) FCT 1995. LNCS, vol. 965, pp. 221–232. Springer, Heidelberg (1995)Google Scholar
  6. 6.
    Freund, R., Oswald, M.: P Systems with activated/prohibited membrane channels. In: Păun, G., Rozenberg, G., Salomaa, A., Zandron, C. (eds.) WMC 2002. LNCS, vol. 2597, pp. 261–269. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Freund, R., Oswald, M., Sosik, P.: Reducing the number of catalysts needed in computationally universal P systems without priorities. In: the 5th Descriptional Complexity of Formal Systems Workshop (DFCS), July 12-14, Budapest, Hungary (2003) Google Scholar
  8. 8.
    Frisco, P., Jan Hoogeboom, H.: Simulating counter automata by P Systems with symport/antiport. In: Păun, G., Rozenberg, G., Salomaa, A., Zandron, C. (eds.) WMC 2002. LNCS, vol. 2597, pp. 288–301. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Hack, M.H.: The equality problem for vector addition systems is undecidable. In: C.S.C. Memo 121, Project MAC, MIT (1975)Google Scholar
  10. 10.
    Hopcroft, J., Pansiot, J.-J.: On the reachability problem for 5-dimensional vector addition systems. TCS 8(2), 135–159 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Huynh, D.T.: Commutative grammars: The complexity of uniform word problems. Information and Control 57, 21–39 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Martin-Vide, C., Paun, G.: Computing with membranes (P Systems): Universality results. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 82–101. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Mayr, E.: Persistence of vector replacement systems is decidable. Acta Informatica 15, 309–318 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Minsky, M.: Recursive unsolvability of Post’s problem of Tag and other topics in the theory of Turing machines. Ann. of Math. 74, 437–455 (1961)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Parikh, R.: On context-free languages. Journal of the ACM 13, 570–581 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Paun, G.: Computing with membranes. JCSS 61(1), 108–143 (2000)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Paun, G.: Computing with membranes (P Systems): A variant. International Journal of Foundations of Computer Science 11(1), 167–181 (2000)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Paun, G., Rozenberg, G.: A guide to membrane computing. TCS 287(1), 73–100 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sosik, P., Freund, R.: P Systems without priorities are computationally universal. In: Păun, G., Rozenberg, G., Salomaa, A., Zandron, C. (eds.) WMC 2002. LNCS, vol. 2597, pp. 400–409. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    van Leeuwen, J.: A partial solution to the reachability problem for vector addition systems. In: Andersson, S.I. (ed.) Analysis of Dynamical and Cognitive Systems. LNCS, vol. 888, pp. 303–309. Springer, Heidelberg (1995)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Oscar H. Ibarra
    • 1
  • Zhe Dang
    • 2
  • Omer Egecioglu
    • 1
  • Gaurav Saxena
    • 1
  1. 1.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA
  2. 2.School of Electrical Engineering and Computer ScienceWashington State UniversityPullmanUSA

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