P Systems with Active Membranes Characterize PSPACE

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Now we construct the above mentioned function State(h i1 i 2 ...i n, n) which computesthe state of a membrane h i1 i 2 ...i nafter n computational steps of Π.(1) If n = 0 then return the state of membrane h in the initial configurationand exit.(2) Calculate recursively State(h i1 ...i n−1, n − 1) and store it in the variable S.(3) If S = nil or S = dissolved then return S and exit./* If membrane h i1 ...i n−1did not exist after (n−1) steps, then neither aftern steps exists membrane h i1 i 2 ...i nwhich could only evolve from h i1 i 2 ...i n−1during step n. */(4) Initialize the variable S ′ which will contain a final state of the membraneafter n-th computational step: set S ′ .M to ∅ and S ′ .p to S.p.(5) Initialize auxiliary variables X, X ′ : set X.M and X ′ .M to ∅ and set X.pand X ′ .p to 0.(6) /* Now we calculate how the membranes embedded in h i1 ...i n−1influenceits content at n-th step. */For each membrane g contained directly in h i1 ...i n−1calculate recursivelyState(g, n − 1) and store the result in X. Then:(a) Try to apply rules of type (a) with parameters g, X, X ′ , X ′ (see bellow).(b) Try to apply rules of type (b) with parameters g, X, X ′ , S. If any rulewas applied, skip steps (c) and (d).(c) Try to apply rules of type (c) with parameters g, X, X ′ , S ′ . If any rulewas applied, skip step (d).(d) Try to apply rules of type (d) with parameters g, X, X ′ , S ′ .(7) /* We calculate state of the parent membrane of h. */If h is not the skin membrane, then:– set g = Parent(h i1 ...i n−1, n − 1);– calculate recursively State(g, n − 1) and store the result to X.(8) /* Now we simulate evolution of membrane h i1 ...i n−1at step n. */(a) Try to apply rules of type (a) with parameters h, S, S ′ , X.(b) Try to apply rules of type (b) with parameters h, S, S ′ , X. If any rulewas applied, go to step 9.(c) Try to apply rules of type (c) with parameters h, S, S ′ , X. If any rulewas applied, go to step 9.(d) Try to apply rules of type (d) with parameters h, S, S ′ , X. If any rulewas applied, go to step 9.(e) If h i1 ...i n−1is an elementary membrane, try to apply rules of type (e) asfollows:– if i n = 1, then parameters are h, S, S ′ , X;– if i n = 2, then parameters are h, S, X, S ′ .(9) If i n = 2 and a rule of type (e) was not applied, set S ′ to nil./* If i n = 2, then membrane h i1 i 2 ...i ncould only be created by an applicationof an (e)-type rule during n-th step. */(10) If S ′ ≠ nil and S ′ ≠ dissolved then add the remaining content of S.M toS ′ .M.8
(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
Page 1 and 2: P Systems with Active MembranesChar
Page 3 and 4: ✬12✬✓5✒✗6✖✫✫✩ 1
Page 5 and 6: labelled h 0 contains other membran
Page 7: f✬1✩f 11✬✩✛✘✛✘✛
Page 11 and 12: at the third step membrane b 11 was
Page 13: not applied, then h i11 ...i 1n ,..
Page 17 and 18: Eventual simulation of this variant

P Systems with Active Membranes Characterize PSPACE

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