13th International Conference on Membrane Computing - MTA Sztaki

B. Aman, G. Ciobanu
A state space is a directed graph where there is a node for each reachable
marking, and an arc for each occurring transition. The state space of a CPN
model can be computed fully automatically, and this makes it possible to automatically
analyze and verify several properties concerning the behavior of the
model: the minimum and maximum numbers of tokens in a place, reachability,
boundedness, etc. When working with Petri nets, some behavioral properties
(e.g., reachability, boundedness, liveness, fairness) are easier to study once a
state space is calculated; a good survey for known decidability issues for Petri
nets is given in [7].
We can define now similar properties for mobile membranes with objects
on surface. Given a mobile membrane with object on surface Π with initial
configuration M 0 , we say that a configuration M is reachable in Π if there
exist the sets of transitions U 1 , . . . , U n such that M 0 [U 1 〉 . . . [U n 〉M n = M. A
home configuration is a configuration which can be reached from any reachable
configuration. We say that a membrane system is bounded if the set of reachable
configurations is finite. A membrane system has the liveness property if each
rule can be applied again in another evolution step, and it is fair if no infinite
execution sequence contains some configurations which occur only finitely. By
considering a colored Petri net CP N Π obtained from a mobile membrane Π, we
have the following decidability result.
Proposition 1. If the reachability problem is decidable for CP N Π , then the
reachability problem is also decidable for Π.
Proof (Sketch). The initial marking of CP N Π is the same as the initial configuration
of Π according to the construction presented in Section 4, and each step of
the Petri nets corresponds to an evolution of the mobile membranes with objects
on surface (according to Theorem 1). Thus the reachability problem becomes decidable
for mobile membranes with objects on surface as soon it is decidable for
colored Petri nets.
In a similar way, we can prove several properties for mobile membranes with
objects on surface as soon they hold for their corresponding colored Petri nets.
Proposition 2.
• If CP N Π is bounded, then Π is bounded.
• If CP N Π has the liveness property, then Π has the liveness property.
• If CP N Π is fair, then Π is fair.
Since the properties of reachability, boundedness, liveness and fairness can be
derived automatically by using CPN Tools, these results are of great help when
studying similar properties for mobile membranes with objects on surface. For
instance, using the CPN Tools and the model for the LDL degradation pathway,
we can check whether we can reach the configuration in which the membrane
marked by apoB is inside the membrane marked by lyso, for instance.
Using CPN Tools for the LDL degradation pathway model, we obtain the
following output file:
138