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Computer Science Graduate Student Conference 2011

”On the Edge of Chaos and Possible Correlations Between Behavior and Cellular Regulative Properties”

Stefano Nichele2011, May 30th

Stefano Nichele, 2011

Cellular automata modeling two species of gastropod Chris King, University of Auckland - MATHS 745 2009

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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

• 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

• Countable array of discrete cells i

• Discrete-time update rule Φ

(operating in parallel on local neighborhoods

of a given radius r)

• Alphabet: σit {0, 1,..., k- 1 } ≡ ∈ A

• Update function: σit + 1 = Φ(σi - r

t , …., σi + rt)

• State of CA at time t: st ∈ AN (N=number of cells)

• Global update Φ: AN → AN

• st = Φ st - 1

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Wikipedia, Conway’s Game of Life, 26/05/2011

Example 1 - Conway’s Game of Life

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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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Class 1

Evolution ”dies”

Irreversible

Outcome is determined with probability 1

Wolfram, 1984

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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 4Complex localized structures

Non-reversible The current site values could have arisen from more than one previous configuration

Only class with non-trivial automataChaotic 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:tot

q1

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

• developmental complexity

• structural complexity

• CA attractor length

• CA trajectory length

• CA transient length.

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Preliminary Results

3x3 CAexperiments are not finalized

early results show a correlation between genomic composition and developmental properties

State space:3x3 = 3^9 = 196836x6 = 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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