On the computational complexity of membrane systems

• Authors: Oscar H. Ibarra
• Venue: Theoretical Computer Science
• Year: 2004

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.