Properties Of Finite Cellular Automata

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Properties Of Finite Cellular Automata

• By Joseph Seaborn
• Advisor Dr. Mummert

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Definitions

• Cellular Automata Consists of D-dimensional grid
  of cells that take on values from a finite
  alphabet which are updated in discrete time
  steps according to the neighboring cells and an
  update rule.
• State A configuration of values from the
  alphabet in the grid at a certain time.

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The Finite Alphabet

• -The set of non-negative integers less than
  n.
• 0,1,2,,n-1
• Example Calculations in 0,1,2,3,4
• 34 7 (1)5 2 2
• 1234 10 (2)5 0 0
• 134

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1-Dimensional Update Rule examples

• xab
• (Pascals triangle)
• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• xabc

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Special Case P-rule

• P5
• P7

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Finite/Infinite Cellular Automata(1-Dimensional
Examples using )

• Finite Length8
• Each row is a state. This is just a compilation
  of a lot of states.
• Infinite
• Pascals triangle is infinite.

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More Definitions

• Cycle The sequence of states that the automaton
  eventually repeats.
• Transient States The states that occur before
  the cycle begins.

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Cycle and Transient States

• Alphabet , Length8, using adjacent 2-rule

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Main objective thus far

• To understand properties such as cycle length and
  number of transient states which Finite Automata
  have when working with different alphabets,
  lengths and update rules.

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Main Tool Pascgalois
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Pascgalois
Blaise Pascal
Éveriste Galois
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Auxiliary Tool Python Programming

• To address limitations with Pascgalois
• Limit to length of Automaton
• Lack of a probabilistic update rule

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Some interesting properties

• There are some cellular automata that, given
  any state in the automata, they produce the
  trivial cycle of length 1.
• Examples P-Rule, Length , Alphabet
  
• p2 p3
  p5

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A formula

• For p2, the maximum number of transient rows for
  any given starting state is
• when using length and
  .
• Length
• Alphabet
• Number of
  Transient
• States

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More Generally

• For P-rule, alphabet-gt , and Length
• A more general formula is given by
  

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Example with P3
Alphabet-gt
, Length
m4, n3, p3
Number of Transient rows
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Probabilistic Update Rule

• Instead of just one update rule, we have a set of
  rules from which to choose based on a
  probability.
• Goal To understand how Automaton act when their
  update rules are slightly changed.

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Python

• State Generator
• Expected Value Approximator

Alphabet-gt
Length15
3-rule or 5-rule
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Why Study Automata

• To Understand
• Its fun.
• Computer Science
• Occurrences in nature

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Possible Ways to Proceed

• Entropy
• Correlations between bi/multinomial theorems and
  update rule actions.
• Coefficients of
  

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Sources

• Bardzell, Michael J. and Kathleen Shannon.
  "Patterns in Pascal's Triangle - with a Twist."
  Journal of Online Mathematics and its
  Applications. Nov 2003. 22 Jul 2008
  lthttp//mathdl.maa.org/mathDL/4/?pacontentsavi
  ewDocumentnodeId493gt.
• "Cellular Automaton." Wikipedia. 23 July 2008.
  Wikimedia. 23 Jul 2008 lthttp//en.wikipedia.org/w
  iki/Cellular_automatagt.
• Churchill, Aaron. The Life and Death of a Finite
  1-D Cellular Automata. 5 May 2006
• D'amico, Michele. "On Computing the Entropy of
  Cellular Automata." Theoretical Computer Science
  (2003) 1629-1646.
• "Majority Problem(Cellular Automaton)."
  Wikipedia. 23 June 2008. Wikimedia. 23 Jul 2008
  lthttp//en.wikipedia.org/wiki/Majority_problem_2
  8cellular_automaton29gt.
• Miller, Nicole. "The Evolution Homomorphism and
  Permutation Actions on Group Generated Cellular
  Automata." Complex Systems 15.2 (2004) 121-136.
  22 Jul 2008
• "Rule 110." Wikipedia. 23 July 2008. Wikimedia.
  23 Jul 2008 lthttp//en.wikipedia.org/wiki/Rule_11
  0gt.
• Weisstein, Eric W. "Cellular Automaton." From
  Mathworld-A Wolfram Web Resource.
  http//mathworld.wolfram.com/CellularAutomaton.ht
  ml