2D Four Colour cellular Automaton

1
2D Four Colour cellular Automaton
(Surface explorations)
NKS-2006 Dr Robert H Barbour Unitec New Zealand
2
Wolfram Context

• Open Problems and Projects.
• Six different mentions of Gray Code
• One mention of exploring more than two colours in
  a cellular automaton.

3
Problem

• A space is to be searched
• Progress reports are required from time to time
• The search process must be replicable and
  reversible
• The algorithm describing the search behaviour
  must be as simple as possible

4
A context model solution as a Cellular automaton

• Cellular automata require two entities, and
• a display system,
• agreed rules for managing entity behaviour
• a recording system is useful.

5
Motivation

• To identify a spatial search algorithm that could
  model event time.
• To identify a means of externalising the
  algorithm to study its behaviour.
• To implement the algorithm in a suitable
  computing environment linking Gray Code and CA.

6
The Entities

• We have two entities.
• A two dimensional matrix of cells much like an
  unshaded chess board.
• A space searching virtual Ant that moves in
  single steps from one cell to the next.

7
The Algorithm

• Move a Vant (virtual ant) from one cell to the
  next.
• On leaving a cell change the cell colour in the
  following sequence white-green-black-yellow.
• On leaving a cell turn left 90o if the cell
  colour was black or yellow and right 90o if the
  cell colour was white or green.
• Record the cell colour and the ant coordinates.

8
Binary Logic

• Represents true and false conditions
• A basis of digital computing representation
• Does not represent sequences of change well
• 0,1,0,1,0,1,0,1,0,1 provides no parsimonious way
  of distinguishing between sets of interactions.
• Langtons Ant (demo Ant Farm here) 2D two colour
  CA.
• Integer Sequence A102358 (Visit iterations)
  (Barbour, 2005) and Integer Sequence A102369
  (Iteration Intervals) (Barbour, 2005).

9
Quaternary Logic

• Quaternary Logic adds representations that
  disambiguate the direction through sequences .
• Two bits are required for the four
  representations (a Gray Code.).
• 00, 01, 11, 10, 00, note the single bit change

10
Quaternary Logic applied

• Quaternary logic/Gray Code allows the reporting
  of specific changes in cell visits.
• Cells in the grid unexplored (or forgotten) 00
• Some cells explored (data added) 01
• All cells explored (data complete) 11
• Some cells forgotten (data lost) 10

11
Interaction Cycle

Becoming true, being explored
unstable
01
true
false
00
11
stable
10
unstable
Becoming false, being forgotten
12
Interaction Sequence

Unstable
01
stable
00
00
11
Unstable
10
Interaction Sequence
13
Change or not?
11
Change
01
01
Change occurs, or not, in sequence Either towards
or away from 11 (or learning about cells space)
00
00
01
No Change
00
14
The Binary tree of change

00
Completed sequence
10
11
11
Change
11
01
01
01
00
00
01
11
01
No Change
01
00
No change
00
00
15
Interaction World lines

• World line traces the actual status sequence
  through the possible worlds in an interaction.

11
11
11
01
01
01
World Line 00010101
00
00
01
11
World Line 00000111
01
01
00
00
00
16
Demo here

• This cellular automaton uses colours to
  distinguish where in the cycle of visits the Ant
  has reached on any iteration
• White 00 unexplored
• Green 01 exploring
• Black 11 explored
• Yellow 10 forgetting

17
Probability of a particular outcome during a
particular iteration

• From any particular start an assessment may be
  made of the probability of a particular outcome
  by enumerating the possibilities during each
  iteration.
• The first five iterations generate an integer
  sequence
• 00, 01, 11, 00
• 1, 0, 0, 0
• 1, 1, 0, 0
• 1, 2, 1, 0
• 1, 3, 3, 1
• 2, 4, 6, 4
• Leading to the conclusion that the most likely
  single outcome after the fifth iteration is 11
  or true.
• (see Integer Sequence A094266)

18
Probability of a particular outcome over a number
of iterations.

• The cumulative totals from the columns of the
  four alternatives gives the changing
  probabilities going forward from a particular
  status.
• 00, 01, 11, 01 representations.
• 1, 0, 0, 0. false
• 2, 1, 0, 0. false
• 3, 3, 1, 0. false
• 4, 6, 6, 1. moving to true
• 6, 10, 10, 5. moving to true
• 12, 16, 20, 15. true or agreement
• 28, 28, 36, 35. true but moving away
• (see Integer Sequence A099423)

19
Interaction Model in use

• Space is searched by repeated cell visits.
• The pattern of visits can be exteriorised using
  the recent path function and the cell visits
  function.
• The cellular automaton regularly returns to
  base
• The status of visited cells in the searched grid
  is known simultaneously on iterations 4, 8, 32,
  64, 416, 832 that is Integer Sequence A094867,
  the six completed single colour squares.

20
Summary

• Search Status refers to the aggregate of cell
  visits having the same attributes (colour or some
  other marker) in the searched space.
• Two bits provides the representation, while
  quaternary logic provides the underlying
  reasoning.
• Unpredicted emergent regularities in some Integer
  sequences and unpredicted completed squares in
  the Ant Farm.
• Relationship between Gray Code and CA shown

21
References

• Barbieri,M., F. De Martini, G. Di Nepi, P.
  Mataloni (2003) Experimental Detection of
  Entanglement with Polarized Photons
  arXivquant-ph/0307003 v1 1 Jul 2003
• Barbour, R.H. (2004) A099423.Online Encyclopaedia
  Integer of Sequences. http//www.research.att.com/
  projects/OEIS?AnumA099423, A102358, A102369,
• Barbour, R.H. (2005) LQTL. in Beziau
  Costa-Leite, UNILOG-2005 Handbook.
• Barbour, R.H. L.D. Painter (2004) A094266
  Online Encyclopaedia of Integer Sequences.
  http//www.research.att.com/projects/OEIS?AnumA09
  4266
• Barbour, R.H. J Chapman (2004) A094867 Online
  Encyclopaedia of Integer Sequences.
  http//www.research.att.com/projects/OEIS?AnumA09
  4867
• Beziau, J-V A. Costa-Leite (eds) (2005)
  Handbook of the First World Congress and School
  on Universal Logic UNILOG'0 2005, Montreux
  Switzerland http//www.uni-log.org5 March 26th -
  April 3rd
• Chapman, J. (2004) Personal Communication.
• Endriss, U. (2003) Modal Logic of Ordered Trees.
  Unpublished PhD Thesis, Kings College, London.
• Ganguly, N. et al., (2003) A survey of Cellular
  Automata. Technical Report Centre for High
  Performance Computing, Dresden University of
  Technology, December 2003.
• Gray, F. (1953) Pulse code communication, March
  17, U.S. patent no. 2,632,058.
• Hazelhurst, S. (1996) Compositional Model
  Checking of Partially ordered state spaces. DPhil
  Thesis University of British Coilumbia.
• Prior, A. (1967). Past Present and Future Oxford
  University Press.
• Sarkar P. (2000), A Brief History of Cellular
  Automata. ACM Computing Surveys. Vol. 32 No. 1
  March.
• Wolfram, S.(2002) A New Kind of Science, Wolfram
  Media, Champaign, Illinois