In general, for fixed radius rules increase in order of Markov approximation
results in
smaller and more numerous
classes of rules. For fixed order of approximation, the larger the
radius the larger and more numerous the Markov classes.
The number of rules in a 0th-order class defined
by coefficient value 
is 
.
the number of such classes is 
.
The number of rules in a mean field theory class
of d-diameter rules defined by coefficient values 
is
.
The number of mean field classes is
.
There is no known general formula for the size of nth-order classes, 
,
or the number of such classes.
However, if 
is the
maximum value of a coefficient 
for d-diameter rules then there are at least
2nd-order classes.
A generous upper bound for the number of rules
in a class defined by coefficient values 
and 
is
.
Markov classes may vary widely in size.
At each order of approximation there are classes,
such as the class which contains the
rule which maps all neighborhoods to 0, which have but a solitary member.
The largest
class at 0th-order is the class
which contains rules which map exactly half of the
neighborhoods to 1.
For r=2 rules, this class contains approximately 6 x 
members. The largest 1st-order classes for r=2 rules
have approximately 6 x 
members. There are 8 of these. It is
not known how large the largest 2nd-order classes of r=2 rules are.
The largest 2nd-order classes found in the sample studied here have 120
members.
The distribution of 
of the sizes of 0th- through 3rd-order
classes sampled as above
is shown in figure 1. All 4th-order classes sampled contained only 1 rule.
This distribution is not shown.
Clearly, as the order of approximation is increased the typical size of
the classes defined decreases sharply. The average class size is approximately
1.3 x , 2.5 x 
, 13.0, 2.3, and 1.0
for 0th- through 4th-order respectively.
Figure:
Distribution of 
for 0th-order through
3rd-order classes for
r=2 rules. The total number of such rules is 
.
Total bar height in each panel is 1.
Each order of approximation
provides a strict refinement of
classes defined at previous orders, until the order of approximation
exceeds the diameter of the rules in question. This is a consequence of
the fact that blocks of the same nth-order type are also of the same
mth-order type, when 
. When the order of approximation is equal to
the diameter of the rule, each d-block is of a distinct type. Hence
there can be only one rule in a dth-order class of d-diameter rules.
This implies that the
5th-order approximation must completely classify r=2 rules. As can be
seen in figure 1,
by 3rd-order classes of r=2 rules
are too small to permit meaningful statistical analysis of the distribution of
properties of rules in a class. Hence, only 0th- through 2nd-order classes
are included in the studies below.