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M
=(
S, R, H, Φ,
L
,
C
,
K
)
where:
S, H, Φ
are as in Definition 4;
L
is set of membrane labels that are identified with the membranes their-selves.
C
is a membrane structure describing the initial configuration.
R
is a set of localized reactions associated to membranes
k, l
∈L
with the forms:
meta
[
k
α
[
k
β
::
ϕ
the classical MP reaction rewritten in the multicompartimental con-
text: in membrane
k
at step
i
,
→
ϕ
(
X
[
i
])
units of multiset
α
are trans-
formed into
ϕ
(
X
[
i
])
units of multiset
β
.
in
α
[
k
→
[
k
β
::
ϕ
if
α
is in the immediate outside region of membrane
k
,
ϕ
(
X
[
i
])
units
of
α
are introduced in
k
and transformed into
ϕ
(
X
[
i
])
units of
β
.
out
[
k
α
β
[
k
::
ϕ
if
α
is contained in the membrane
k
, the reaction expels
→
ϕ
(
X
[
i
])
units
of
α
from
k
transforming it into
ϕ
(
X
[
i
])
units of
β
.
trans
[
k
α
[
l
β
::
ϕ
if
α
is contained in the membrane
k
and in the immediate out-
side region of
k
there is the sibling membrane
l
, the reaction trans-
forms
→
ϕ
(
X
[
i
])
units of
α
into
ϕ
(
X
[
i
])
units of
β
transferring it from
k
to
l
.
extra-in
[
k
→
[
k
α
::
ϕ
the reaction introduces
ϕ
(
X
[
i
])
units of
α
into membrane
k
from the
environment external to system
M
.
extra-out
[
k
α
[
k
::
ϕ
the reaction expels
→
ϕ
(
X
[
i
])
units of
α
from membrane
k
into the
environment external to system
M
.
K
is a labelling of substances and membranes marking
-asinput
In
and output
Out
substances or membranes:
In
,
Out
⊆
S
×L
-as
Start
and
Halt
only substances:
Start
,
Halt
⊆
S
.
The dynamics of
M
is written according to (2) following the semantics of the
rules in
R
.
The multicompartimental Arithmetical MP System defines a membrane model
with flux regulators in which the model of calculus is deterministic and therefore
effective for the implementations.
Example 1. (Abacus)
Let's define a 3-digit abacus with carry in base 10. Two
input numbers
N
= 728 and
M
= 322 are encoded, using the substance
u
inside the input membranes as [7
u
]
N
1
[2
u
]
N
2
[8
u
]
N
3
and [3
u
]
M
1
[2
u
]
M
2
[2
u
]
M
3
.An
insight is given in Fig. 5, where substances and membranes are denoted by circles
and rounded boxes respectively. Substance
u
(not shown for simplicity) from
input membranes is summed by transferring it digit-by-digit into the output
membranes. The output membrane
R
1
encodes the digit of thousands possibly
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