P Systems with Active Membranes Characterize PSPACE

not applied, then h i11 ...i 1n ,...,i k1 ...i kndoes not exist. */Application of rules of type (f) is implemented as follows:(f) For each rule [ h0[ h1] + h 1. . . [ hk] + h k[ hk+1] − h k+1. . . [ hn] − h n] α 2h 0→ [ h0[ h1] α 3h 1. . . [ hk] α 3h k] α 5h 0[ h0[ hk+1] α 4h k+1. . . [ hn] α 4h n] α 6such that S.p = α 2 : if, before application of steps 6(a)–(d), there existedembedded membranes with polarizations + and - (this informationcan be stored during step 6), then set S ′ .p to α 5 and T.p to α 6 . Skip allother applicable rules.Again, the recursive function State is defined correctly because each recursivecall during the computation of State(h i11 ...i 1n ,...,i k1 ...i kn, n) is in one of the formsh 0,State(h i11 ...i 1n ,...,i (k−1)1 ...i (k−1)n, n), (3)State(h i11 ...i 1(n−1) ,...,i k ′ 1 ...i k ′ (n−1), n − 1), 0 ≤ k ′ ≤ d(µ), (4)where d(µ) is the depth of the initial membrane structure µ. These two typesof calls include also the recursive calls during search for parent and childrenmembranes. One can describe the resulting graph of recursive calls as a twodimensionallattice with nodes corresponding to pairs (n, k), for n ≥ 0 and 0 ≤k ≤ d(µ), where d(µ) is depth of the initial membrane structure. An orientededge from (n, k) to (n ′ , k ′ ) denotes a call of State(h i11 ...i 1n ′ ,...,i k ′ 1 ...i k ′ n ′, n ′ ) fromState(h i11 ...i 1n ,...,i k1 ...i kn, n).By 3 and 4, there are edges from each node (n, k) to (n, k − 1), if k > 0, andto (n − 1, k ′ ), if n > 0, for 0 ≤ k ′ ≤ d(µ). Clearly, the graph is acyclic, henceall the recursive calls will be answered after a finite number of steps.5 Main resultsIn the previous section we have presented a recursive function State whichreturns a state of any membrane of a P system Π after a given number ofcomputational steps. To employ this function to simulate the computationof a confluent P system with active membranes, we can represent the outerenvironment surrounding the P system as another region with no applicablerules. Formally, we embed the whole membrane structure µ into a newmembrane, say, h 0 . Now it remains to subsequently calculate State(h 0 , n), forn = 1, 2, 3, . . . , until the object yes appears within the content of the new regionh 0 or until the computation halts (this we can test by calling State(h, n)13