P Systems with Active Membranes Characterize PSPACE

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Here we complete the characterization of the computational power of P systemswith active membranes. We show that the class of problems solvable inpolynomial time by these P systems is exactly the class PSPACE. As a consequence,these P systems satisfy the so-called Parallel Computation Thesis [6]for a computer M :M -TIME(T O(1) (n)) = SPACE(T O(1) (n)). (1)Computers satisfying this thesis form the second machine class [13,4]. Amongits members there are standard models of parallel computers as SIMDAG(known also as SIMD PRAM), APM – a model of existing vector supercomputers[14], P-RAM – a model of MIMD PRAM computer, alternating Turingmachine and others [4]. An interesting member is the genetic Turing machine[11], a computational model of genetic crossing-over. It is also straightforwardthat the second machine class is closed under the polynomial-time reduction.To demonstrate that any f(n) time-bounded computation of a P system withactive membranes can be simulated in a space polynomial w.r.t. f(n), we adoptthe technique of reverse-time simulation. Instead of simulating a computationof a P system from its initial configuration onwards (which would require anexponential space for storing configurations), we create a recursive functionwhich returns the state of any membrane h after a given number of steps.The recursive calls evaluate contents of the membranes interacting with h ina reverse time order (towards the initial configuration). In such a manner wedo not need to store a state of any membrane, but instead we calculate itrecursively whenever it is needed.2 DefinitionsIn this section we give a brief description of P systems with active membranesdue to [9] or [8], where also more details can be found. A membrane structureis represented by a Venn diagram (or a rooted tree) and is identified by astring of correctly matching parentheses, with a unique external pair of parenthesescorresponding to the external membrane, called the skin. A membranewithout any other membrane inside is said to be elementary. The followingexample from [9] illustrates the situation: the membrane structure at Figure1 is identified by the stringµ = [ 1[ 2[ 5] 5[ 6] 6] 2[ 3] 3[ 4[ 7[ 8] 8] 7] 4] 1.In what follows, we interpret the membrane structure of Π as a rooted treeand refer occasionally to its elements – membranes as nodes in this tree. The2
✬12✬✓5✒✗6✖✫✫✩ 1✩ 4✬✩ ✪❅ 3✏★✥✬7✩ ✪ ❅❅✪★8✥ 2 3 4✑✂❆ ✂ ❆✔✂❆7✧✦✧✦ 5 6✕✫ ✪✪8✪Fig. 1. A membrane structure and its associated tree.membranes can be further marked with + or −, and this is interpreted as an“electrical charge”, or with 0, and this means “neutral charge”. We will write[ i] + , [ i i ]−, [ i i ]0 in the three cases, respectively.iThe membranes delimit regions, precisely identified by the membranes (theregion of a membrane is delimited by the membrane and all membranes placedimmediately inside it, if any such a membrane exists). In these regions we placeobjects, which are represented by symbols of an alphabet. Several copies of thesame object can be present in a region, so we work with multisets of objects. Amultiset m over an alphabet V can be represented by any string x ∈ V ∗ (by V ∗we denote the free monoid generated by V with respect to the concatenationand the identity λ) such that the number of occurrences of a symbol a ∈ V inx represents the multiplicity of the object a in the multiset m.A P system with active membranes is a constructwhere:Π = (V, H, µ, w 1 , . . . , w m , R),(i) m ≥ 1;(ii) V is an alphabet;(iii) H is a finite set of labels for membranes;(iv) µ is a membrane structure, consisting of m membranes, labelled (not necessarilyin a one-to-one manner) with elements of H; all membranes in µ aresupposed to be neutral;(v) w 1 , . . . , w m are strings over V , describing the multisets of objects placed inthe m regions of µ;(vi) R is a finite set of developmental rules, of the following forms:(a) [ ha → v] α h ,for h ∈ H, α ∈ {+, −, 0}, a ∈ V, v ∈ V ∗3
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Page 7 and 8: f✬1✩f 11✬✩✛✘✛✘✛
Page 9 and 10: (11) Return S ′ and exit.The appl
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Page 17 and 18: Eventual simulation of this variant

P Systems with Active Membranes Characterize PSPACE

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