Basic mathematical definition Functional decomposition
an example of sparsely connected dependency structure. direction of causal flow upward.
for multivariate function
y
=
f
(
x
1
,
x
2
,
…
,
x
n
)
{\displaystyle y=f(x_{1},x_{2},\dots ,x_{n})}
, functional decomposition refers process of identifying set of functions
{
g
1
,
g
2
,
…
g
m
}
{\displaystyle \{g_{1},g_{2},\dots g_{m}\}}
such that
f
(
x
1
,
x
2
,
…
,
x
n
)
=
ϕ
(
g
1
(
x
1
,
x
2
,
…
,
x
n
)
,
g
2
(
x
1
,
x
2
,
…
,
x
n
)
,
…
g
m
(
x
1
,
x
2
,
…
,
x
n
)
)
{\displaystyle f(x_{1},x_{2},\dots ,x_{n})=\phi (g_{1}(x_{1},x_{2},\dots ,x_{n}),g_{2}(x_{1},x_{2},\dots ,x_{n}),\dots g_{m}(x_{1},x_{2},\dots ,x_{n}))}
where
ϕ
{\displaystyle \phi }
other function. thus, function
f
{\displaystyle f}
decomposed functions
{
g
1
,
g
2
,
…
g
m
}
{\displaystyle \{g_{1},g_{2},\dots g_{m}\}}
. process intrinsically hierarchical in sense can (and do) seek further decompose functions
g
i
{\displaystyle g_{i}}
collection of constituent functions
{
h
1
,
h
2
,
…
h
p
}
{\displaystyle \{h_{1},h_{2},\dots h_{p}\}}
such that
g
i
(
x
1
,
x
2
,
…
,
x
n
)
=
γ
(
h
1
(
x
1
,
x
2
,
…
,
x
n
)
,
h
2
(
x
1
,
x
2
,
…
,
x
n
)
,
…
h
p
(
x
1
,
x
2
,
…
,
x
n
)
)
{\displaystyle g_{i}(x_{1},x_{2},\dots ,x_{n})=\gamma (h_{1}(x_{1},x_{2},\dots ,x_{n}),h_{2}(x_{1},x_{2},\dots ,x_{n}),\dots h_{p}(x_{1},x_{2},\dots ,x_{n}))}
where
γ
{\displaystyle \gamma }
other function. decompositions of kind interesting , important wide variety of reasons. in general, functional decompositions worthwhile when there sparseness in dependency structure; is, when constituent functions found depend on approximately disjoint sets of variables. thus, example, if can obtain decomposition of
x
1
=
f
(
x
2
,
x
3
,
…
,
x
6
)
{\displaystyle x_{1}=f(x_{2},x_{3},\dots ,x_{6})}
hierarchical composition of functions
{
g
1
,
g
2
,
g
3
}
{\displaystyle \{g_{1},g_{2},g_{3}\}}
such
x
1
=
g
1
(
x
2
)
{\displaystyle x_{1}=g_{1}(x_{2})}
,
x
2
=
g
2
(
x
3
,
x
4
,
x
5
)
{\displaystyle x_{2}=g_{2}(x_{3},x_{4},x_{5})}
,
x
5
=
g
3
(
x
6
)
{\displaystyle x_{5}=g_{3}(x_{6})}
, shown in figure @ right, considered highly valuable decomposition.
example: arithmetic
a basic example of functional decomposition expressing 4 binary arithmetic operations of addition, subtraction, multiplication, , division in terms of 2 binary operations of addition
a
+
b
{\displaystyle a+b}
, multiplication
a
×
b
,
{\displaystyle a\times b,}
, 2 unary operations of additive inversion
−
a
{\displaystyle -a}
, multiplicative inversion
1
/
a
.
{\displaystyle 1/a.}
subtraction can realized composition of addition , additive inversion,
a
−
b
=
a
+
(
−
b
)
,
{\displaystyle a-b=a+(-b),}
, division can realized composition of multiplication , multiplicative inverse,
a
÷
b
=
a
×
(
1
/
b
)
.
{\displaystyle a\div b=a\times (1/b).}
simplifies analysis of subtraction , division, , makes easier axiomatize these operations in notion of field, there 2 binary , 2 unary operations, rather 4 binary operations.
example: decomposition of continuous functions
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