Information Technology Reference
In-Depth Information
Moreover, some statement definitions may be divided into several statements
connected with conjunctions or disjunctions such that each sentence can be better
associated with a particular complexity measure. Let us recall Fig. 4.7 in Chap. 4
where we concluded that a “good random number generator” (i.e. our objective
statement “
A
”) is mainly characterized by the following two phrases:
(A1
) The clustering coefficient
Clus
shall be close to 0.5.
(A2)
The variance of the clustering coefficient
Var
shall be larger than 0 but not
so large, actually it should be situated within the region 1
where
v1
and
v2
are two bifurcation parameters that can be easily determined for a
given family of cellular systems (Sect. 4.5.1).
v
rv
2
Now one may consider the two statements associated with two sieves: The first
sieve rejects all CA cells not satisfying the “
A1
” statement. Among the CA cells
which passed, a second sieve is applied such that it will further reject all cells not
satisfying the “
A2
” statement. Let us consider the family “2s5” with 1,024 members.
As a first sieve one may choose
. Assuming a degree
P
Clus
1
Clus
0
1
of truth
1
T
the following is a list of 12 cell IDs that passed after this first
sieve: 198, 315, 323, 325, 346, 362, 373, 627, 650, 677, 714, 761.
0
.
999
v
1
v
2
As a second sieve one may choose
. Assuming
P
Var
1
Var
A
2
2
2
T
,
after applying this sieve to the above list results in the following list with only five
cells: 315, 325, 346, 362, 714.
Running each of the CAs in the list proves that a desired behavior, as specified
by statement “
A
” emerges in the corresponding CA.
The choice of an absolute value function is subjective, and one can freely
select
and
as indicated in Fig. 4.14, and a degree of truth
v
1
0
02
v
2
0
08
0
.
99
other
functions.
For
instance,
if
one
uses
the
function
>
@
2
P
instead, for the same degree of truth, obtains the
same list of genes. But the absolute value function is much easier to evaluate than
the exponential, so in practice sieves based on absolute values may be preferred.
The degrees of truth are also subjective choices of the designer, smaller values
resulting in larger lists with a wider range of behaviors. On the opposite choosing
a too small degree of truth leads to the risk of an empty list.
Instead of successively applying the sieves, one may apply the same strategy
as used in the definition of fuzzy expert systems and define a unique sieve, for
instance, in our case:
Var
exp
100
Var
0
05
2
>
@
P
, where the “min”
operator stands for “Fuzzy AND” (i.e. A is true if A1 AND A2 is true).
For the above example, using this single sieve with a degree of truth of 0.999
results in only two cells: 315 and 714. Relaxing the degree of truth to 0.998 gives
the following list: 178, 202, 315, 714 with two members (ID = 178 and 202) not
previously listed. Though, simulations of the corresponding CA indicates its
behavior in accord to the statement
A
.
Clus
,
Var
min
P
Clus
,
P
Var
A
A
1
A
2
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