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7.7. Mathematical Model of the Dynamic Semantics
The global vectors x and y with all inputs and outputs of a data flow model are defined as
follows:
⎡ ⎤
x 1,1
.
⎡ ⎤
x 1,m1
x 1
x 2,1
x 2
x = ⎢ ⎥
⎣ . ⎦ = .
x 2,m2
(7.16)
x nx
.
x nx,1
⎢ ⎥
⎣ . ⎦
y =
⎡
⎢
⎣
y 1
y 2
.
y
ny
⎡
⎤
⎥
⎦ = ⎢
⎣
x nx,m nx
y 1,1
.
y 1,p1
y 2,1
.
y 2,p2
.
y ny,1
.
⎤
⎥
⎦
(7.17)
y ny,p ny
Since there may not be any unused output in a data flow the number of outputs must always
be smaller or equal the number inputs. Correspondingly, the following mathematical constraint
must be fulfilled:
n
∑ y
∑n x
≤
(7.18)
u=1
p u
} {{ }
p
v=1
m v
} {{ }
m
Additionally, the following two constraints have to be fulfilled for C because its elements can
only be 0 or 1 and only one input can be connected with each output:
∀i :
c i,j ∈ {0; 1} (7.19)
p∑
c i,j
!
= 1 (7.20)
j=1
Example For clarification, the simple example for a data flow in Figure 7.15 is taken. It only
consists of a oSum object and three oVariableStorage objects. The object “0” is only used to
ensure that all inputs of the oSum object are used, but it does not influence the calculation.
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