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2D Four Colour cellular Automaton

(Surface explorations)

NKS-2006

Dr Robert H Barbour

Unitec

New Zealand

Wolfram Context

Open Problems and Projects.

- Six different mentions of ‘Gray Code’
- One mention of exploring more than two colours in a cellular automaton.

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

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.

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.

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.

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.

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.
- Langton’s Ant (demo Ant Farm here) 2D two colour CA.
- Integer Sequence A102358 (Visit iterations) (Barbour, 2005) and Integer Sequence A102369 (Iteration Intervals) (Barbour, 2005).

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

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

Interaction Cycle

Becoming true, being explored

unstable

01

true

false

00

11

stable

10

unstable

Becoming false, being forgotten

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

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

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

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

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)

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)

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.

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

References

- Barbieri,M., F. De Martini, G. Di Nepi, P. Mataloni (2003) Experimental Detection of Entanglement with Polarized PhotonsarXiv:quant-ph/0307003 v1 1 Jul 2003
- Barbour, R.H. (2004) A099423.Online Encyclopaedia Integer of Sequences. http://www.research.att.com/projects/OEIS?Anum=A099423, 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?Anum=A094266
- Barbour, R.H. & J Chapman (2004) A094867 Online Encyclopaedia of Integer Sequences. http://www.research.att.com/projects/OEIS?Anum=A094867
- 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, King’s 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

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