13th International Conference on Membrane Computing - MTA Sztaki

A. Ţurcanu, F. Ipate
2 Simplifying Event-B Models of Cell-like
P Systems
In this section we present some general ideas about how to build a reduced and
“more non-deterministic” model of a cell-like P system which uses transformation
rules. The main idea of our modelling approach is that all the rules with
the same number of objects on the left side are similar and so we can associate
to all of them only one Event-B event.
The set of P system objects will be denoted SYMBOLS, and a subset of
the cartesian product SYMBOLS × N × SYMBOLS × N denoted PAIRS will
contain all the pairs of objects from the left and the right side of each rule
together with their multiplicities. If all the objects have the multiplicity 1, then
the multiplicities can be omitted. If two rules have the same left side then the
corresponding objects are renamed and added to SYMBOLS, and their values
are modified in the same time with the values of the initial objects.
Two functions objects, pobjects : SY MBOLS → N are used to represent
the existing objects and the objects produced between two steps of maximal
parallelism in each membrane, respectively. In a special event, called update,
each value of pobjects is added to the corresponding value in objects and the
values of pobjects are reset to 0.
We consider as an example a simple P system with one membrane and
three rules: s → ab, a → ac, b → c. In this case SYMBOLS= {s, a, b, c} and
PAIRS = {(s, a); (s, b); (a, a); (a, c); (b, c)} and we associate to all the rules the
following Event-B event:
Rule123
any
x
where
then
end
grd1 : x ∈ SY MBOLS & objects(x) > 0
grd2 : ∃y.y ∈ SY MBOLS&(x, y) ∈ PAIRS
act1 : objects : |(∀y.y ∈ SY MBOLS& y ≠ x ⇒
objects ′ (y) =objects(y)&objects ′ (x) :=objects(x) − 1)
act2 : pobjects : |((∀y.y ∈ SY MBOLS&(x, y) ∈ PAIRS
⇒ pobjects ′ (y) =pobjects(y) + 1&(∀y.y ∈ SY MBOLS&
(x, y) /∈ PAIRS ⇒ pobjects ′ (y) =pobjects(y))
This event can be used, in general, in any P system, to substitute all the rules
with one object on the left side. If the generic objects x and y have multiplicities
then the values of objects(x) is decremented with the multiplicity of x in the
first action and the value of pobjects(y) is incremented with its multiplicity in
the second one.
Communication rules are quite similar except that, in the second action,
they modify the values of the function pobject associated with another membrane.
These ideas can be extended for rules with many objects on the left side,
specifying that the number of event parameters will increase accordingly.
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