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3 Adding Membranes
In the examples above there was the implicit assumption that the
IN
values
were available at once and together with the
. Indeed biological networks
transform matter as soon as appropriate rules and substances are put together.
If we want to compose two computation processes together, a sort of buffering
layer is needed in-between preventing any partial values being fed into the sec-
ond system possibly corrupting its initial configuration and triggering altered
computations. We want to be sure that such a composition is possible at all.
Start
Theorem 1.
(Sequential Composition)
Given two MP Computable functions
f
and
g
, the composition
f
◦
g
is a MP Computable function.
A mechanism proving the theorem is shown in Fig. 4 where top and bottom sub-
stances belong to
f
and
g
respectively. Let's consider, without loss of generality,
only the output
x
of
f
being transformed into the input
x
for
g
.
t
h
...
x'
z'
1
1
x'
z'
t
h
1
x
z
0
0
s
Fig. 4.
The Sequential Composition mechanism
Anytime before
h
reaches 1, both the rules consuming
x
are
blocked
because
the total requested flux
x
+
t
exceeds the value of
x
.Once
h
reaches 1, substance
t
gets its value decreased to 0 and the
x
-availability constraint is satisfied:
Start
and
In
substances are synchronously provided to
g
.The
Start
and
Halt
substances of
f
◦
g
are the
Start
of
f
and the
Halt
of
g
, respectively.
3.1 Multicompartimental Setting
The functional composition, a very desirable property when complex algorithms
are to be designed, naturally brings the idea of having separate and composable
computing entities: the membranes.
A boundary rule [1] is
localized
in the membrane of its reactants. Let
k
,
l
be regions, a boundary rule has always the form of
localized
k
reactants
→
localized
l
products
. If the regions
k
and
l
are the same, the rule transforms matter,
otherwise there is a communication and a transformation of substances across
membranes.
Definition 6.
(Multicompartimental Arithmetical MP System)
AMulticom-
partimental Arithmetical MP System of type
(
n, m, k
)
is a construct:
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