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On the Dynamics of PB Systems with Volatile Membranes Giorgio Delzanno* and Laurent Van Begin** * Università di Genova, Italy ** Universitè Libre de Bruxelles, Belgium WMC8, Thessaloniki - 27 June 2007

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Contents of the Talk PB systems vs Petri nets Extensions with dissolution and creation Qualitative analysis –reachability –boundedness Decidability and undecidability results Conclusions

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1 2 3 MembranesInternal Rules 11 Boundary Rules 2 2 33 PB Systems [Bernardini-Manca WMC 2002]

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Petri (P/T) nets [Petri62] place transition token

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From PB Systems to Petri nets [Dal Zilio-Formenti WMC 2003]

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Membranes Places and Tokens 1 2 3 1 1 1 2 2 2 3 3 3

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PB Rules 11 Transitions 33 2 1 3 2

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Computational Properties PB configuration = Petri net marking The asynchronous evolution of a PB system with symbol objects is simulated step by step by a firing sequence of the Petri net Properties like reachability and boundedness are reduced to the corresponding decision problems for Petri nets Reachability: is conf. C1 reachable from C0? Boundedness: is a PB system finite-state?

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Decidability Results For a PB system with symbol objects and asynchronous semantics, reachability and boundedness are both decidable [Dal Zilio-Formenti WMC2003] Follows from results on Petri nets [Mayr,...]

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Can we extend these results? There is a natural connection between extensions of PB systems with volatile membranes (e.g. dissolution rules) and Petri nets with transfer arcs Unfortunately property like reachability are undecidable in presence of transfer, reset, or inhibitor (emptiness test) arcs For this reason, Dal Zilio and Formenti do not investigate further in extensions of PB systems But, do we really need extensions of Petri nets?

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Extensions of PB systems We consider here the following extensions Dissolution rules [ i u [ i v. Creation rules a [ i u ] where i is a membrane name a is an object u,v are multisets of objects dissolve!

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Theorem 1 For PB systems with dissolution rules, reachability is still decidable

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Proof part I From the initial configuration C0, we can extract an upper bound K on the number of applications of dissolution rules needed to reach the target configuration C1 We use this to extend the DalZilio-Formenti construction with special flags present/dissolved for each membrane in the initial configuration and two operating modes: normal and dissolving K= number of membranes in C0

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2 present2 3 present1 1 dissolved2 3 33 Boundary rule normalmode Transitions 1 2 3

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Proof: part II We model dissolution of a membrane by moving to a special operating mode dissolving In dissolving mode we transfer tokens (one by one) to the current immediate ancestor membrane The current immediate ancestor is determined by checking the status of the present/dissolved flags

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2 2 2 present2 dissolving2 normalmode 2 dissolving2 present1 Dissolution 1

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dissolving2normalmode Proof: part III The transfer is non-deterministically terminated. We then go back to the normal mode

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In the marking M1 that encodes the target configuration C1 we require that all places associated to objects of dissolved membranes are empty In other words we only keep good simulations in which transfers have never been interrupted Thus, M1 is reachable from M0 iff C1 is reachable form C0 Notice that the Petri net is not equivalent to the PBD system Proof: Final remarks

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Theorem 2 For PB systems with creation, reachability is still decidable Proof: The target configuration gives us an upper bound on the number of applications of creation rules Again, we use it for a reduction of PBC reachability to Petri net reachability

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Theorem 3 For PB systems with creation and deletion, reachability is undecidable Proof: We can reduce reachability of counter machines to this problem. Notice that the state-space we have to explore to reach the target configuration is unbounded in width (parallelism) and depth (nesting).

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Theorem 4 Reachability becomes decidable with dissolution and a restricted form of creation in which names of membranes are part of the current configuration and cannot be reused after dissolution Proof: The target configuration give us an upper bound on the number of membrane structures we have to explore. We use it for a reduction to Petri net reachability in which places are labeled with membrane structures

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Other Results Boundedness is decidable for PB systems with dissolution and restricted creation Boundedness is undecidable for PB systems with creation

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Conclusions We have investigated the applicability of decision procedures for Petri nets to extensions of PB systems Positive results for dissolution rules Creation is more problematic The results can be extended to movement operations

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