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5.4.2. Boolean rules
Without short-cuts, the units of our model have exactly three inputs. In this case,
we can write the truth table of Boolean functions (or rules) of three-inputs in the
following form [Wolfram (1994)]:
i
1
(t)
1
0
1
0
1
0
1
0
i
(t)
1
1
0
0
1
1
0
0
(5.3)
i
2
(t)
1
1
1
1
0
0
0
0
i
(t + 1)
b
7
b
6
b
5
b
4
b
3
b
2
b
1
b
0
where b
j
is the output for each of the eight possible combinations of inputs, and can
take values 0 or 1. The rules are designated by the decimal number that corresponds
to the binary number b
7
b
6
b
5
b
4
b
3
b
2
b
1
b
0
; for example, the Boolean function 11101000
is rule 232 [Wolfram (1994); Kaplan and Glass (1997)]. This rule returns as an
output the value in the majority among the inputs, and hence it is also named the
\majority" rule.
In the following, we focus our attention on rules that can be generalized to an
arbitrary number of inputs. In general, a three-input rule cannot be generalized to
any number of inputs. An important class of Boolean rules that is mostly excluded
by the generalizable Boolean rules considered above is the class of canalizing func-
tions [Wolfram (1994)]. In a canalizing rule, the output value of the rule is solely
determined by one input, the canalizing variable. Three of the rules we study here
are canalizing: 1, 19, and 50. As we have reported [Amaral et al. (2004)], two of
these rules lead to a broad range of dynamical behaviors. However, the majority
rule, which is not canalizing, also displays a broad range of dynamical behaviors,
suggesting that canalizing variables are not necessary to obtain such diverse dy-
namics.
In order to dene the generalizable Boolean functions, we replace the inputs of
the neighbors in a 3-input rule by an average input
n
(t):
n
(t) =
1 + sign(
i
1
+
i
2
1)
2
;
(5.4)
where we dene sign(0)0; a value of 1=2 for the state
n
is what we call a
Tie. The average input
n
(t) can be generalized to include an arbitrary number of
inputs:
P
j=k
i
j=1
1 + sign
2
i
j
k
i
n
(t) =
:
(5.5)
2
The Boolean rules of three inputs that are generalizable are symmetric rules that
have b
1
= b
4
and b
3
= b
6
:
1
1
Only in this case it does not matter where the Tie comes from, a 1 on the right and a 0 on the
left or just the other way around.
