P Systems with Active Membranes Characterize PSPACE

(11) Return S ′ and exit.The application of rules of the types (a)–(e) is implemented as follows:Parameters:h – label of the membrane processedS – original state of the membraneS ′ – final state of the membraneT – state of another membrane eventually acting at the operation(a) For each rule [ h a → v] α h in R such that S.p = α, remove all the occurrencesof a from S.M and add to S ′ .M the corresponding number of occurrencesof v (i.e. of multisets corresponding to v).(b) For each rule a[ h] α 1h → [ h b] α 2hin R such that S.p = α 1: if T.M containsa, then remove a from T.M, add b to S ′ .M, set S ′ .p to α 2 and skip allother applicable rules.(c) For each rule [ ha ] α 1h → [ h ] α 2h b in R such that S.p = α 1: if S.M containsa, then remove a from S.M, add b to T.M, set S ′ .p to α 2 and skip allother applicable rules.(d) For each rule [ ha ] α → b in R such that S.p = α: if S.M contains a, thenhremove a from S.M, add b to S ′ .M, add S.M ∪ S ′ .M to T.M, set S ′ todissolved and skip all other applicable rules.(e) For each rule [ ha ] α 1h → [ h b ] α 2h [ h c ] α 3hin R such that S.p = α 1 : if S.Mcontains a, then– remove a from S.M– add b to S ′ .M and set S ′ .p to α 2 ,– add c to T.M and set T.p to α 3 ,– skip all other applicable rules.Observe that, with the aid of the function State, we can uniquely determinethe parent and the children (in terms of the membrane structure tree) of agiven membrane h i1 i 2 ...i n, without actually storing the membrane structure ofΠ after n-th step. The function Parent can be implemented as follows:Parameters:h i1 i 2 ...i n– a membrane whose parent is searched forn – a number of step(a) Let g be the parent membrane of h in the initial membrane structure µ.Calculate State(g 1...1 , n).(b) If the state of g 1...1 was dissolved then calculate recursively Parent(g 1...1 , n)and return the result, else return g 1...1 .Similarly, at step 6 we needed to find all children membranes of h i1 ...i n−1. Ifg is a child of h in the initial configuration, then each membrane g j1 j 2 ...j n,9