P Systems with Active Membranes Characterize PSPACE

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T O(1) (n)-time bounded P system Π is T O(1) (n)-space bounded which concludesthe proof.Now, consider the relation (2) and the facts that the second machine class isclosed under polynomial time reduction, and that deterministic P systems withactive membranes are universal computers [1]. Therefore we can generalize (2)as follows:SPACE(T O(1) (n)) ⊆ FAM div-ne -TIME(T O(1) (n)) ⊆ FAM S div-ne-TIME(T O(1) (n))Together with Theorem 2 we can conclude that the parallel computation thesisholds for uniform families of confluent P systems with active membranes:Corollary 3FAM div-ne -TIME(T O(1) (n)) = FAM S div-ne-TIME(T O(1) (n)) = SPACE(T O(1) (n))Furthermore, as its special case for a polynomial time T (n) we obtain:Corollary 4PMC div-ne = PMC S div-ne = PSPACE.6 DiscussionThe results presented in this paper establish an upper bound on the powerof confluent P systems with active membranes. However, we do not knowthe upper bound of the power of their non-deterministic and non-confluentvariant. The presented proof cannot be simply extended to this case by usinga non-deterministic RAM machine. The reason is that during the simulationof a computation of Π, we did not store its configurations but re-calculatedthem again and again when needed. Therefore, if these calculations were donenon-deterministically, we could get different results in different calculationsof the same configuration. Consequently, the whole simulation would not beconsistent.Another variant of P systems with active membranes is presented in [5]. In thisvariant the non-elementary membrane division is not performed by rules oftype (f) above. Instead, rules of the form [ ha ] α 1h → [ h b ] α 2h [ h c ] α 3are used forhboth elementary and non-elementary membrane division. If in the membraneh there are other objects than a and any embedded membranes, then duringthe same step they may evolve in a usual way and then the results of thisevolution are copied into both copies of membrane h.16
Eventual simulation of this variant of P systems with active membranes wouldfollow almost exactly the way described in Section 4.1. The only differenceswould be that also non-elementary membranes can divide and that the structuralindexing of the type i 11 . . . i 1n , . . . , i k1 . . . i kn would be used. Therefore,we claim that Theorem 2 holds also in this case, and the class of problemssolvable in a polynomial time by this variant of P systems is bounded fromabove by PSPACE. However, there is no formal proof yet that these P systemscan also solve PSPACE-complete problems in polynomial time. Hence,the validity of parallel computation thesis (3) in this case remains open. Anotheropen problem is the (non-)validity of the results contained in this paperin the case of P systems with active membranes and minimal parallelism [5].AcknowledgementsResearch was partially supported by the Czech Science Foundation, grant No.201/06/0567, and by the Programa Ramón y Cajal, Ministerio de Ciencia yTecnología, Spain.References[1] A. Alhazov, R. Freund, A. Riscos-Núñez, One and two polarizations, membranecreation and objects complexity in P systems, in: G. Ciobanu, Gh. Păun(Eds.), Technical Report 05-11, Institute e-Austria, Timişoara, Romania, FirstInternational Workshop on Theory and Application of P Systems (TAPS), 2005,pp. 9–18.[2] A. Alhazov, C. Martin-Vide, L. Pan, Solving a PSPACE-complete problem byP systems with restricted active membranes, Fundamenta Informaticae, 58, 2(2003), 67–77.[3] J.L. Balcazar, J. Diaz, J. Gabarro, Structural Complexity I, Second Edition,Springer-Verlag, Berlin, 1995.[4] J.L. Balcazar, J. Diaz, J. Gabarro, Structural Complexity II, Springer-Verlag,Berlin, 1991.[5] G. Ciobanu, L. Pan, G. Păun, M.J. Pérez-Jiménez, P Systems with MinimalParallelism, submitted.[6] L.M. Goldschlager, A universal interconnection pattern for parallel computers,J. Assoc. Comput. Mach., 29 (1982), 1073–1086.[7] Gh. Păun, Computing with Membranes, J. Comput. System Sci., 61 (2000),108–143.17
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