on the dynamics of pb systems with volatile membranes giorgio delzanno* and laurent van begin** *...
TRANSCRIPT
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
Contents of the Talk
• PB systems vs Petri nets• Extensions with dissolution and creation• Qualitative analysis
– reachability
– boundedness
• Decidability and undecidability results
• Conclusions
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?
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,...]
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?
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 objectu,v are multisets of objects
dissolve!
Proof part I
From the initial configuration C0, we can extract an upperbound K on the number of applications of dissolution rulesneeded to reach the target configuration C1
We use this to extend the DalZilio-Formenti constructionwith special flags present/dissolved for each membrane inthe initial configuration and two operating modes: normal and dissolving
K= number of membranes in C0
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
dissolving2 normalmode
Proof: part III
The transfer is non-deterministically terminated.We then go back to the normal mode
In the marking M1 that encodes the target configuration C1we require that all places associated to objects of dissolvedmembranes are empty
In other words we only keep good simulations in whichtransfers 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
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
Theorem 3
For PB systems with creation and deletion,reachability is undecidable
Proof: We can reduce reachability of counter machines to thisproblem.Notice that the state-space we have to explore to reach thetarget configuration is unbounded in width (parallelism) and depth (nesting).
Theorem 4
Reachability becomes decidable with dissolution anda restricted form of creation in which names ofmembranes are part of the current configuration and cannot be reused after dissolution
Proof: The target configuration give us an upper bound on thenumber 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
Other Results
• Boundedness is decidable for PB systems with dissolution and restricted creation
• Boundedness is undecidable for PB systems with creation