Randomness in Cellular

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Biological Computing

February 6, 2003

Nathanael Paul

Randomness in Cellular

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

Wolfram’s definition of
randomness from a New Kind of

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

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*… /


Ch. 4 shows representation may change random
look (consider

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


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,

Class 2 Example

Class 3 characteristics

Appears random

Smaller structures can be seen some at
some level

Most are expected to be computationally

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,

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


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


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

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

Observing systems of limited

Limiting the size forces repetivness

Period of repetition increases with size of

With n cells, there are at most 2

states (maximum period of 2


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

Rule 22 with
another set of
initial conditions

Rule 22
random with
different initial

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


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

Networks examples

Types of Networks

Classes 1 and 2

Never have more than t

nodes after t

Classes 3 and 4

Allowed sequences of cells becomes
more complicated

Number of nodes increases at least

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