Information Technology Reference
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x
1
Neighborhood of x
0
x
0
x
2
2
nd
order clique
Figure
10.12.
NAND gate depicted as an MRF.
neighborhood, since the energy state of this node depends on its neighboring
nodes. The edges in Figure 10.12 depict the conditional probabilities with respect
to the other input x
1
and the output x
2
(nodes in the same clique). The operation of
the gate is designated by the logic compatibility function
j
(x
0
,x
1
,x
2
) shown as a
truth table in Table 10.2.
j¼
1 when (x
2
¼ð
x
0
^
x
1
Þ
0
(valid logic operations). Such a
function takes all valid and invalid logic functionality scenarios into account so as
to represent a Gibb's energy-based transformation of the NAND logic.
Entropy has been defined in many different yet equivalent ways. In this
discussion, we define entropy as a measure of the disorder of a system. It is
considered to have high values when the system under consideration is very
disordered. The concept of entropy originated from classical thermodynamics, but
has found widespread application in dynamical systems theory, communication
theory, information theory, etc. Considerable work has been done on statistical
thermodynamics [67], which became the inspiration for adopting the word entropy
in information theory. Let us consider a random variable X, which must take on
TABLE 10 . 2 . Logic Compatibility Function of a NAND Gate
I
x
0
x
1
x
2
F
0
0
0
1
1
1
0
0
0
0
2
0
1
1
1
3
0
1
0
0
4
1
0
1
1
5
1
0
0
0
6
1
1
0
1
7
1
1
1
0
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