Biomedical Engineering Reference
In-Depth Information
Table 4.1
Possible values for
coefficients of zhegalkin
function
Coefficient
Set of Possible Values
Notation
a
0
0, 1
A
0
−
a
j
1, 0, 1
A
1
−
−
a
jk
2,
1, 0, 1 ,2
A
2
a
jkl
−
4,
−
3,
−
2, 0, 1, 2, 3, 4
A
3
.
.
.
2
(
n
−
1)
,
...
,
1, 0, 1,
...
,2
(
n
−
1)
a
123
...n
−
−
A
n
4.2.2
Zhegalkin Polynomial Function
To model the dynamics of continuous gene expression values and provide a continu-
ous representation of a Boolean function, our algorithm utilizes modified Zhegalkin
polynomial functions [
13
]. The following is a description of a Zhegalkin polynomial
function. The modifications we implement to express Boolean-like gene expressions
are described in Sect.
4.2.3
Definition IV.1:
A
Zhegalkin polynomial function
with
n
variables is given by
Eq.
4.1
.
n
n
k
−
1
f
(
x
1
,
...
,
x
n
)
=
a
0
+
a
j
x
j
+
a
jk
x
j
x
k
j
=
1
k
=
2
j
=
1
n
l
−
1
k
−
1
+
a
jkl
x
j
x
k
x
l
+
+
...
a
1
...n
x
1
...x
n
(4.1)
l
=
3
k
=
2
j
=
1
The Zhegalkin function is a linear function consisting of coefficients and products
of the input variables. The first term,
a
0
is a constant. The second term
j
=
1
a
j
x
j
is weighted sum of all possible single inputs. The third term
k
=
2
k
−
1
j
=
1
a
jk
x
j
x
k
is
weighted sum of all possible combinations of two inputs, and so on. The last term is
a weighted product of all inputs.
The coefficients or weights
a
0
,
a
1
,
...
,
a
1
...n
of a Zhegalkin function are called
Zhegalkin coefficients. In general, any Boolean function can be converted to a Zhe-
galkin function by selecting the appropriate Zhegalkin coefficients. In [
13
], it was
determined that the possible range of values for the Zhegalkin coefficients to define
any Boolean function
n
variables are listed in Table
4.1
.
To demonstrate how Zhegalkin function can represent a Boolean function, we
show two simple examples. Consider two Boolean functions:
f
B
=
x
1
x
2
g
B
=
x
1
x
2
+
x
1
x
2
