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applications of rules, and other values related to the quantity units (depending on
the physical nature of variables).
In mathematical terms an MP grammar is specified by: variables, reactions,
regulators, initial values . Variables that do not occur in reactions, but occur
in regulators are called parameters , while variables that are not parameters are
called (proper) variables . Given some initial values of its proper variables, an MP
grammar deterministically generates a time series for each of its proper variables.
If parameters are present in the grammar, then also the time series of the parameters
take part in the generation of dynamics. It is easy to realize, and it will be evident in
the following definition, that this notion of MP grammar defines a system of finite
difference recurrence equations, which represent the invariant of the dynamical sys-
tem generated from it. In the following generalization of MP grammar, the letters
MP may be related to Minus-Plus rules, which express the dynamics by means of
(regulated) decreasing-increasing of (proper) variables.
Definition 3.9 (Generalized MP Grammar). A (generalized) MP grammar is a
structure:
(
X
,
Y
,
R
,
X
[
0
])
where:
X is a set of real variables ;
Y is a (possibly empty) subset of variables, called parameters . The elements of
set difference X
Y are also called proper variables ;
R is a set of MP rules , that is, pairs of type ( reaction, regulator ):
α
β
;
ϕ
.
r
r
r
For any r
R , a reaction
α r β r is a pair of multisets over the set X
/
Y of proper
variables, and a regulator
ϕ r is a function from X -states to
R
(an X -state is a
function from X to
R
). Variables that are arguments of
ϕ r are the tuners of
ϕ r ;
X
[
0
]=(
x
[
0
] |
x
X
)
is an initial state constituted by the values of variables at
an initial time 0.
For any X -state s , and for any proper variable x
X
/
Y , the rules determine a
decrease-increase
of x , according to the following formula (each reaction de-
creases any left occurrence of x and increases any right occurrence of x ):
Δ x (
s
)
)= r R ( β r ( x ) α r ( x )) ϕ r ( s ) .
Δ
(
s
(3.49)
x
(
α r (
x
)= β r (
x
)=
0if x does not occur in the reaction of r ). Let us set, for every
i
N
:
X
[
i
]=(
x
[
i
] |
x
X
)
then, the (proper) variable variation (column) vector
Δ [
i
]
is given by (superscript T
denotes matrix transposition):
T
Δ [
i
]=( Δ x [
i
] |
x
X
/
Y
)
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