Randomness in Cellular
Automata
CS851 – Biological Computing
February 6, 2003
Nathanael Paul
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 2n possible
states (maximum period of 2n)
• 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 t2 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?