13th International Conference on Membrane Computing - MTA Sztaki

Mobile membranes with objects on surface as colored Petri nets
The evolution of the LDL degradation could be represented graphically as
in Figure 2. By M 1 , . . . , M 24 are denoted the possible configurations of the system,
and on each arrow from a M i to a M j is placed the number of the rule
which is applied in order to evolve from M i to M j . To denote that an object
recep changes its position and interacts with different objects, we use different
notations (namely, recep, recep 1 , . . . , recep 5 ) in the evolution of the system.
Remark 1. The number of the applied rules to reach the configuration M 24 starting
from the configuration M 1 is always 13.
4 Colored Petri Nets
Colored Petri nets (CPN) represent a graphical language used to describe systems
in which communication, synchronization and resource sharing play an
important role [9]. The CPN model contains places (drawn as ellipses or circles),
transitions (drawn as rectangular boxes), a number of directed arcs connecting
places and transitions, and finally some textual inscriptions located near the
places, transitions and arcs.
The places are used to represent the state of the modeled system, and this
state is given by the number of tokens of all the places. Such a state is called
a marking of the CPN model. By convention, we write the names of the places
inside the ellipses. The names have no formal meaning, but they have a practical
importance for the readability of a CPN model, just like the use of mnemonic
names in traditional programming.
The arc expressions on the input arcs of a transition determine when the
transition is enabled, i.e., to be activated by a certain marking. A transition is
enabled whenever it is possible to find a binding of the variables that appear in
the surrounding arc expressions of the transition such that the arc expression
of each input arc evaluates to a multiset of tokens that is present in the corresponding
input place. When a transition occurs with a given binding, it removes
from each input place the multiset of tokens to which the corresponding input
arc expression evaluates. Analogously, it adds to each output place the multiset
of tokens to which the corresponding output arc expression evaluates.
The colored Petri nets have also a mathematical representation with a well
defined syntax and semantics. This formal representation is the framework for
the study of different behavioral properties. We denote by EXP R the set of
expressions provided by the inscription language (which is ML in the case of
CPN Tools), and by T ype[e] we denote the type of an expression e ∈ EXP R,
i.e., the type of the values obtained when evaluating e. The set of free variables
in an expression e is denoted V ar[e], and the type of a variable x is denoted
T ype[x]. We denote the set of variables by X; the set of expressions e ∈ EXP R
for which V ar[e] ⊆ X is denoted by EXP R X . The set of all multisets over S is
denoted by S MS .
The following definition differs from that presented in [9] just because simultaneous
parallel arcs from the same place to the same transition are not allowed
(i.e., it is enough to have only one arc).
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