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Ta b l e 3 . 2 5
An MP grammar computing the square root
√
n
of a natural number
n
.Thevalue
n
is the initial value of substance
A
, while the (rounded) result can be found in
Y
when
Z
=
1,
and at the end, the difference between the initial value and result is put in
A
.
r
1
:
A
→
D
ϕ
1
=
2
B
+
C
r
2
:0
→
B
ϕ
2
=
C
r
3
:
D
→
0
ϕ
3
=
D
r
4
:
B
→
Y
ϕ
4
=
B
+
D
+
Y
r
5
:
C
→
W
ϕ
5
=
C
+
D
r
6
:
Y
→
Z
ϕ
6
=
W
r
7
:
W
→
0
ϕ
7
=
W
A
[
0
]=
n
B
[
0
]=
0
C
[
0
]=
1
D
[
]=
0
1
Y
[
0
]=
0
W
[
]=
0
0
Z
[
0
]=
0
3.8.2
Revisiting MP Grammars
We conclude this chapter with a general concept of MP grammar, which extends
the initial metabolic inspiration toward a general dynamical paradigm which can be
applied to any system of variables.
First, let us recall that a dynamical system is given by a set of real
variables
changing in time and a set of
invariants
, that is, conditions or constraints that are
satisfied by the variables during their change.
Now, let us assume a generalized notion of reaction among variables such as
X
Y
, meaning that
X
decreases by a certain amount, while
Y
increases by the
same amount (with respect to some measurement unit). The decreasing/increasing
amount, called flux of the rule, is determined by a regulator (regulation map) de-
pending on some variables of the system.
An MP rule is a
regulated reaction among variables
, that is, a pair constituted by
a reaction and a regulator. A reaction is constituted by left variables which decrease,
while some corresponding right variables increase (each variable is equipped with a
multiplicity). A
regulator
is a function providing the flux of the rule, in dependence
of the values of some regulation variables, called
tuners
of the rule (possibly includ-
ing parameters). Every left variable decreases of a quantity which is the product of its
multiplicity with the flux of the rule, and every right variable increases analogously.
An MP grammar is a set of MP rules. Given an MP grammar, when we start from
an initial state (the values of its variables at an initial time) by applying all the rules
of the grammar, we obtain the next state and so on, for all the subsequent steps. An
MP grammar becomes an MP system when some numerical values are fixed for the
physical interpretation of the time series: the time interval between two consecutive
→
