Randomness in Cellular

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CS851


Biological Computing

February 6, 2003

Nathanael Paul

Randomness in Cellular
Automata

Defining Randomness


“… only with the discoveries of this book
that one is finally now in a position to
develop a real understanding of what
randomness is.”

Some concepts of randomness


Irregular, sporadic, nonuniform,… Is there
a pattern?


Something can appear random, but its origin
can be from something quiet simple (rule
30)

Wolfram’s definition of
randomness from a New Kind of
Science


Try some standard simple programs to
detect regularities or patterns.


If no regularities are detected, then it is
highly probable no other tests will show
nonrandom behavior.


Wolfram does not consider something to
be truly random if generated from simple
rules. Should rule 30 be considered
random?

Rule 30 with different initial conditions.
Should this rule be considered random?

Does traditional mathematics fail to tell us
much about rule 30?

Wolfram’s earlier definition of
randomness (1986)


“… one considers a sequence ‘random’ if no
patterns can be recognized in it, no
predictions can be made about it, and no
simple description of it can be found.”


Calculations of pi


pi/2 =
2*2*4*4*6*6*8*8*… /




1*3*3*5*5*7*7*9…


Ch. 4 shows representation may change random
look (consider
e
)

Statistical analysis


Probabilistic CAs


Usually appear more random than
corresponding CAs


Compute quantities and compare
computations with a given average


Ex: count black squares in a sequence
and compare to ½

Randomness in initial conditions


Previous cellular automata had a single
black cell for initial condition


Consider random initial conditions


Order emerges


Wolfram’s 4 CA classes

Class 1 characteristics


Simple


Uniform final state (all black or all white)


Some examples are rules 0, 32, 128, 160,
250, 254

Class 1 Example

Class 2 characteristics


Set of simple structures


Structures remain the same or repeat
every so often


Examples include rules 132, 164, 218,
222

Class 2 Example

Class 3 characteristics


Appears random


Smaller structures can be seen some at
some level


Most are expected to be computationally
irreducible


Examples include rules 22, 30, 126

Class 3 Example

Class 4 characteristics


Has order and randomness


Smaller scale structures interacting in
complex ways


Examples include codes 1815, 2007, 1659,
2043


Recall: Codes are “totalistic” CAs where
new color depends on average of neighbors


Class 4 emerges as an intermediate class
between classes 2 and 3

Class 4 Example

Exceptions


Totalistic automata that don’t seem to fit
into just one class


Codes 219, 438, 1380, 1632

Initial condition sensitivity


Each class responds differently to a change
in its initial conditions


Response types


Class 1 changes always die out


Changes continue on but are localized for
Class 2


Uniform rate of change affecting the
whole system seen in Class 3


Class 4 has nonuniform changes

Class 1

Class 2

Class 3

Class 4

Claim


Differences in responses of classes show
each class handles information in a different
way


Fundamental to our understanding of nature

Class 2


Repetitive behavior


No for support long
-
range communication


Lack of long
-
range communication makes
systems of limited size forcing
repetitiveness

Observing systems of limited
behavior


Limiting the size forces repetivness


Period of repetition increases with size of
system


With n cells, there are at most 2
n

possible
states (maximum period of 2
n
)


Modulus

Repetition as a function of
system size

Rule 90

Rule 30

Rule 110

Rule 45

Class 3 randomness


Randomness exists even without random
initial conditions


Different initial conditions can produce
random behavior or nested pattern behavior
in the same rule (rule 22)


Some rules need the random initial
condition to exhibit randomness (90) and
some rules don’t (30)


“Instrinsic Randomness”


Do systems like rule 22 or rule 30 have
intrinsic randomness?


Do these examples prove that certain
systems have intrinsic randomness and do
not depend on initial conditions?


Special initial conditions can make class 3
systems behave like a class 2 or even a class
1 system (rule 126)

Rule 22 with
different initial
conditions

Rule 22 with
another set of
initial conditions

Rule 22
appearing
random with
different initial
conditions

Class 4 structures


Certain structures will always last


Any way to predict the structures of a given
rule and initial conditions?


One can find all structures given a period,
but prediction is another matter


Attractors


Sequences of cells restricted as iterations
progress, even with random initial
conditions


Networks examples

Types of Networks


Classes 1 and 2


Never have more than t
2

nodes after t
steps


Classes 3 and 4


Allowed sequences of cells becomes
more complicated


Number of nodes increases at least
exponentially

Class 3 and 4 Exceptions


Increase in network complexity not seen in
special initial conditions for rules 204, 240,
30, and 90


Onto mappings defined


Any other initial conditions than
“special” initial conditions rapidly
increase in complexity

Final thoughts…


Tests may be done to show randomness, but
a new test could reveal a regularity…


Ch. 4 shows different representations have
varying degrees of randomness


Random CAs look random, but does a
representation exist that will show a
pattern?