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

1

2

3

Membranes Internal Rules

1 1Boundary Rules

2 2

3 3

PB Systems [Bernardini-Manca WMC 2002]

Petri (P/T) nets [Petri62]

place

transition

token

From PB Systems to Petri nets

[Dal Zilio-Formenti WMC 2003]

Membranes Places and Tokens1

2

3

1

1

1

2

2

2

3

3

3

PB Rules

1 1

Transitions

3 3

21

32

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!

Theorem 1

For PB systems with dissolution rules,

reachability is still decidable

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

2

present2

3

present1

1

dissolved2

3

3 3

Boundary rule

normalmode

normalmode

Transitions

1

23

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

2

2 2

present2

dissolving2

normalmode

2

dissolving2 present1

Dissolution

1

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

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