cs851 – biological computing february 6, 2003 nathanael paul randomness in cellular automata
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CS851 – Biological Computing
February 6, 2003
Nathanael Paul
Randomness in Cellular Automata
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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.”
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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)
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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?
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Rule 30 with different initial conditions. Should this rule be considered random?Does traditional mathematics fail to tell us much about rule 30?
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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)
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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
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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
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Class 1 characteristics
• Simple
• Uniform final state (all black or all white)
• Some examples are rules 0, 32, 128, 160, 250, 254
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Class 1 Example
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Class 2 characteristics
• Set of simple structures
• Structures remain the same or repeat every so often
• Examples include rules 132, 164, 218, 222
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Class 2 Example
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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
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Class 3 Example
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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
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Class 4 Example
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Exceptions
• Totalistic automata that don’t seem to fit into just one class
• Codes 219, 438, 1380, 1632
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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
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Class 1
Class 2
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Class 3
Class 4
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Claim
• Differences in responses of classes show each class handles information in a different way
• Fundamental to our understanding of nature
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Class 2
• Repetitive behavior
• No for support long-range communication
• Lack of long-range communication makes systems of limited size forcing repetitiveness
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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
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Repetition as a function of system size
Rule 90
Rule 30
Rule 110
Rule 45
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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)
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“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)
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Rule 22 with different initial conditions
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Rule 22 with another set of initial conditions
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Rule 22 appearing random with different initial conditions
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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
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Attractors
• Sequences of cells restricted as iterations progress, even with random initial conditions
• Networks examples
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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
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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
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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?