On the computational complexity of membrane systems

We show how techniques in machine-based complexity can be used to analyze the complexity of membrane computing systems. We focus on catalytic syslems, communicating P systems, and systems with only symport/antiport rules, but our techniques are applicable to other P systems that are universal. We define space and time complexity measures and show hierarchies of complexity classes similar to well-known results concerning Turing machines and counter machines. We also show that the deterministic communicating P system simulating a deterministic counter machine in (Sosik (2002)) (Pre-Proc. of Workshop on Membrane Computing (WMC-CdeA2002), Curtea de Arges, Romania, 2002, pp. 371-382), (Sosik and Matysek (2002)) (Unconventional Models of Computation 2002, Lecture Notes in Computer Science, vol. 2509, Springer, Berlin, 2002, pp. 264-275.) can be constructed to have a fixed number of membranes, answering positively an open question in Sosik (2002), Sosik and Matysek (2002). We prove that reachability of extended configurations for symport/antiport systems (as well as for catalytic systems and communicating P systems) can be decided in nondeterministic log n space and, hence, in deterministic log2 n space or in polynomial time, improving the main result in Paun et al. (2002) (On the reachability problem for P systems with symport/antiport, 2002, submitted for publication.), We propose two equivalent systems that define languages (instead of multisets of objects): the first is a catalytic system language generator and the other is a communicating P system acceptor (or a symport/antiport system acceptor). These devices are universal and therefore can also be analyzed with respect to space and time complexity. Finally, we give a characterization of semilinear languages in terms of a restricted form of catalytic system language generator.

[1]  Oscar H. Ibarra.  A Note Concerning Nondeterministic Tape Complexities , 1972, JACM.

[2]  Carlos Martín-Vide,et al.  P Systems with Symport/Antiport Rules: The Traces of Objects , 2002, Grammars.

[3]  Hendrik Jan Hoogeboom,et al.  Simulating Counter Automata by P Systems with Symport/Antiport , 2002, WMC-CdeA.

[4]  Patrick C. Fischer,et al.  Turing Machines with Restricted Memory Access , 1966, Inf. Control..

[5]  Gheorghe Paun,et al.  Membrane Computing , 2002, Natural Computing Series.

[6]  KariLila,et al.  Computationally universal P systems without priorities , 2005 .

[7]  Mario J. Pérez-Jiménez,et al.  A Linear-Time Solution to the Knapsack Problem Using P Systems with Active Membranes , 2003, Workshop on Membrane Computing.

[8]  Rudolf Freund,et al.  A Short Note on Analysing P Systems with Antiport Rules , 2002, Bull. EATCS.

[9]  Gheorghe Paun,et al.  Membrane Computing with External Output , 2000, Fundam. Informaticae.

[10]  M. Minsky.  Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines , 1961 .

[11]  Gérard Berry,et al.  The chemical abstract machine , 1989, POPL '90.

[12]  Giancarlo Mauri,et al.  Solving NP-Complete Problems Using P Systems with Active Membranes , 2000, UMC.

[13]  Agustín Riscos Núñez,et al.  Solving the Subset-Sum problem by P systems with active membranes , 2005 .

[14]  Petr Sosík,et al.  Membrane Computing: When Communication Is Enough , 2002, UMC.

[15]  Rudolf Freund,et al.  Membrane Systems with Symport/Antiport Rules: Universality Results , 2002, WMC-CdeA.

[16]  Richard Edwin Stearns,et al.  Hierarchies of memory limited computations , 1965, SWCT.

[17]  Gheorghe Paun,et al.  A guide to membrane computing , 2002, Theor. Comput. Sci..

[18]  Rudolf Freund,et al.  P Systems without Priorities Are Computationally Universal , 2002, WMC-CdeA.

[19]  RozenbergGrzegorz,et al.  Membrane Systems with Coupled Transport , 2002 .

[20]  Gheorghe Paun,et al.  Computing with Membranes , 2000, J. Comput. Syst. Sci..

[21]  Erzsébet Csuhaj-Varjú,et al.  P Automata or Purely Communicating Accepting P Systems , 2002, WMC-CdeA.

[22]  Oscar H. Ibarra,et al.  Reversal-Bounded Multicounter Machines and Their Decision Problems , 1978, JACM.

[23]  Rudolf Freund,et al.  Computationally universal P systems without priorities: two catalysts are sufficient , 2005, Theor. Comput. Sci..

[24]  Oscar H. Ibarra,et al.  Characterizations of Catalytic Membrane Computing Systems , 2003, MFCS.

[25]  Walter J. Savitch.  A note on multihead automata and context-sensitive languages , 2004, Acta Informatica.

[26]  Andrei Paun,et al.  Computing by Communication in Networks of Membranes , 2002, Int. J. Found. Comput. Sci..

[27]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[28]  Tero Harju,et al.  Some Decision Problems Concerning Semilinearity and Commutation , 2002, J. Comput. Syst. Sci..

[29]  Andrei Paun,et al.  On the Power of P Systems with Symport Rules , 2002, J. Univers. Comput. Sci..

[30]  Arnold L. Rosenberg,et al.  Counter machines and counter languages , 1968, Mathematical systems theory.

[31]  Ronald V. Book,et al.  Tally Languages and Complexity Classes , 1974, Inf. Control..

[32]  Christel Baier,et al.  On Topological Hierarchies of Temporal Properties , 1997, Fundam. Informaticae.

[33]  Petr Sos ´ ik.  Membrane Computing: When Communication Is Enough , 2002 .

[34]  Andrei Paun,et al.  ReMembrane Systems with Coupled Transport: Universality and Normal Forms , 2002, Fundam. Informaticae.