Information Technology Reference
In-Depth Information
n
(t)
1
T
1
T
T
0
T
0
i
(t)
1
1
0
0
1
1
0
0
(5.6)
i
(t + 1)
b
7
b
6
b
5
b
4
b
3
b
2
b
1
b
0
For the sake of clarity we will keep using the 8-bit decimal representation used for
Boolean rules with 3-inputs to label the corresponding generalizable rule of arbitrary
number of inputs [Wolfram (1994)]. However, the truth table of a generalizable rule
can be simplied as follows:
n
(t)
1
T
0
1
T
0
i
(t)
1
1
1
0
0
0
(5.7)
i
(t + 1)
b
0
5
b
0
4
b
0
3
b
0
2
b
0
1
b
0
0
where b
0
i
is the output of the function in each of the six dierent input conditions.
This representation, in terms of six possible input combinations enables a better
visual identication of the rules. In the top of Fig. 5.5 we show the truth tables in
this representation of a small subset of the generalizable Boolean functions.
Because of symmetries, there are only 2
6
= 64 independent generalizable rules.
However, some of these rules are conjugate rules and one does not need to investigate
the whole set of rules. Two conjugate rules will have identical dynamics if the zeros
and ones are switched for one of them [Wolfram (1994)]. Additionally, some rules
are self-conjugates. These are rules 23, 51, 77, 105, 150, 178, 204, and 232. Self-
conjugate rules have an inverse rule. Two rules are said to be inverse if, when
starting both with the same initial condition, they will be in the exact same state
every other time step and in the inverse state the other time steps. An example of
inverse rules is given by rules 204 (the \identity" rule) and 51 (the \negation" rule).
The identity rule will keep the initial state, whereas the negation rule will switch
between the initial state and its inverse. Other pairs of inverse rules aref23, 232g,
f77, 178g, andf105, 150g.
5.4.3. Noisy dynamics
Even in the presence of noise, not all of the 64 generalizable rules that we may
consider display uctuating time series. For example, rules 0, 4, 72, 76, 128, 132,
and 200 converge to a xed state, in which all units are in state zero (see Amaral
et al. (2004)). Additionally, the output of rules 51 and 204 depends only on the state
of
i
, so it will not be aected by noise and, as such, will not display uctuations.
Because of these facts and because conjugate rules have identical dynamics, we
nally investigated only 24 of the 64 generalizable rules. These rules are 1, 5, 18,
19, 22, 32, 33, 36, 37, 50, 54, 73, 77, 90, 94, 104, 105, 108, 122, 126, 146, 160, 164,
and 232. This process of elimination of irrelevant rules is schematized in Amaral
et al. (2004).
As an example of the noisy dynamics, the analysis and the dierent types of
behaviors we present some results in Figs. 5.1-5.4 for units operating with rule 232.
