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 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?