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Cellular automata modeling two species of gastropod
Chris King, University of Auckland - MATHS 745 2009
Computer Science Graduate Student Conference 2011
”On the Edge of Chaos and Possible Correlations
Between Behavior and Cellular Regulative Properties”
Stefano Nichele
2011, May 30th
Stefano Nichele, 2011
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Agenda
• Introduction: Cellular Automata
• Formal Definition
• CA Classes
• Edge of Chaos
• Experimental Setup
• Preliminary Results
• Summary
• Bibliography
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von Neumann architecture
• 1 complex processor
• tasks executed sequentially
cellular computing
• myriad of small and unreliable parts: cells
• simple elements governed by local rules
• cells have no global view
Szedő Gábor, MBE_MIT, 2000
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ALife – key principles
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Population: collection of simple elements
No central controller
Each element only knows the status of few neighbours
No rules for the global behaviour, only local rules
Global behaviour: emergent from the local behaviours
INTERACTION: the behaviour seems alive
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Cellular Automaton
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Countable array of discrete cells i
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Discrete-time update rule Φ
(operating in parallel on local neighborhoods
of a given radius r)
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Alphabet: σit ∈ {0, 1,..., k- 1 } ≡ A
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Update function: σit + 1 = Φ(σi - rt , …., σi + rt)
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State of CA at time t: st ∈ AN (N=number of cells)
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Global update Φ: AN → AN
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st = Φ s t - 1
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Example 1 - Conway’s Game of Life
Wikipedia, Conway’s Game of Life, 26/05/2011
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Example 2 – 2nd law of thermodynamics
• From ordered and simple initial conditions, according to the
second law of thermodynamics, the entropy of a system
(disorder and randomness) increases and irreversibility is quite
probable (but not impossible, as stated by Poincaré’s theorem of
reversibility)
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Wolfram, 1984
Class 1
Evolution ”dies”
Irreversible
Outcome is
determined with
probability 1
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Class 2
Fixed point or
periodic cycle
Some parts of initial
state are filtered-out
and others are
propagated forever
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Class 3
Chaotic behavior
Completely reversible
Not random, not noise
Reversible if and only if, for every current
configuration of the CA, there is exactly
one past configuration
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Class 4
Complex localized
structures
Non-reversible
The current site values could have arisen
from more than one previous configuration
Only class with nontrivial automata
Chaotic behavior is considered to be trivial
because it is not random and thereby it is
completely reversible
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Computation at the Edge of Chaos
A region in the CA rule space where there is a phase transition
between ordered and chaotic behavioral regimes
Langton:
q
  1
tot
  1
1
k
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Questions
• Is there a relation between the genomic composition
(cellular regulative properties) and the emergent
behaviour?
• If yes, is it possible to construct the development
rules (genotype) in such a way to obtain a desired
behaviour (phenotype)?
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Experimental Setup - 1
CA can be considered as a developing organism
Developmental system:
• an organism can develop (e.g. grow)
• zygote
multi-cellular organism (phenotype)
• development rules: genome (or genotype)
• gene regulation information control the cells’ growth and differentiation
• behavior of CA: emergent phenotype
(subject to size and shape modifications)
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Experimental Setup - 2
• Minimalistic developmental system
• 3 cell types: multicellularity (2 types of
cells + dead cell)
• All regulatory input combinations are
represented in a development table of
243 (35) configurations
• Neighborhood
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Experimental Setup - 3
• Investigation in all the λ space
• possible correlation between
– properties of the developmental mapping
– behavior of the automata
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developmental complexity
structural complexity
CA attractor length
CA trajectory length
CA transient length.
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Preliminary Results
experiments are not finalized
3x3 CA
early results show a
correlation between genomic
composition and
developmental properties
State space:
3x3 = 3^9 = 19683
6x6 = 3^36 = 1,5 x 10 ^ 17
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Conclusions
• In many studies regarding CA and in particular their
development process and the produced computation,
artificial organisms have shown remarkable abilities
of self-repair, self-regulation and phenotypic plasticity
• Our guess is that many of the CA rules which lead to
these behaviors are in the frozen or ordered region
(Wolfram’s classes 1 or 2) and not in the “edge of
chaos”
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Wolfram, 2000
“This mollusk is essentially running a
biological software program.
That program appears to be very
complex.
But once you understand it, it's actually
very simple.”
"Within 50 years, more pieces of
technology will be created on the basis
of my science than on the basis of
traditional science. People will learn
about cellular automata before they
learn about algebra."
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Bibliography
[1] C. Langton. Computation at the Edge of Chaos: Phase Transitions and Emergent Computation. Physica D Volume 42 (1990) pp. 12-37
[2] S. Wolfram. Universality and Complexity in Cellular Automata. Physica D Volume 10 Issue 1-2 (1984) pp. 1-35
[3] N. Packard. Adaptation Toward the Edge of Chaos. Dynamic Patterns in Complex Systems. Kelso, Mandell, Shlesinger, World Scientific,
Singapore Press (1988), ISBN 9971-50-485-5 pp. 293-301
[4] M. Mitchell, J. Crutchfield, P. Hraber. Dynamics, Computation and the “Edge of Chaos”: A Re-Examination. Santa Fe Institute Working
Paper 93-06-040, Complexity: Metaphors, Models and Reality, Addison-Wesley (1994) pp.497-513
[5] J. Crutchfield and K. Young. Computation at the Onset of Chaos. Complexity, Entropy and Physics of Information, Addison-Wesley (1989)
[6] J. Miller. Evolving a Self-Repairing, Self-Regulating, French Flag Organism. Gecco 2004. Springer-Verlag Lecture Notes in Computer
Science 3102, (2004) pp. 129-139
[7] G. Tufte and P. Haddow. Extending Artificial Development: Exploiting Environmental Information for the Achievement of Phenotypic
Plasticity. Springer Verlag Berlin Heidelberg, ICES 2007 – LNCS 4684, (2007) pp. 297-308
[8] S. Ulam. Los Alamos National Laboratory 1909-1984. Los Alamos: Los Alamos Science. Vol. 15 special issue, (1987) pp. 1-318
[9] J. Von Neumann. Theory and Organization of complicated automata. A. W. Burks, (1949) pp. 29-87 [2, part one]. Based on transcript of
lectures delivered at the University of Illinois in December 1949.
[10] J. H. Holland. Genetic Algorithms and Adaptation. Technical Report #34 Univ. Michigan, Cognitive Science Department (1981)
[11] E. Berlekamp, J. H. Conway, R. Guy. Winning ways for your mathematical plays. Academic Press, New York, NY (1982)
[12] M. Sipper, The Emergence of Cellular Computing, Computer, vol. 32, no. 7, (1999) pp. 18-26, doi:10.1109/2.774914
[13] M. Mitchell, P. T. Hraber and J. T. Crutchfield. Revisiting the Edge of Chaos: Evolving Cellular Automata to Perform Computation. Complex
Systems, vol. 7, (1993) pp. 89-130
[14] A. Turing. On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society,
Series 2, 42, (1936) pp 230–265
[15] S. Wolfram. A New Kind of Science. Wolfram Media Inc, (2002), 1197pages – ISBN 1-57955-008-8
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