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GECCO 2011 Graduate Student Workshop

”Discrete Dynamics of Cellular Machines: Specification and Interpretation”

Stefano Nichele2011, July 12th

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 are simple!)

• CA Classes

• Edge of Chaos (CA are complex!)

• Research Questions

• Experimental Setup

• Preliminary Results

• Conclusion and Future Work

• Bibliography

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Introduction

• Motivation: using biological inspiration from evolution and development towards hardware capable of unconventional computation

• Artificial Developmental Systems: analyzed and evaluated by viewing the system as a discrete dynamic system. The development process is treated as series of discrete events, each representing a point in time on the developmental path from zygote to multi-cellular organism

• Which theoretical and experimental approaches can be used to find methods for the specification of input data and interpretation of the discrete dynamics of cellular machines?

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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 – no central controller•local interactions with neighbours•global behavior: emergent

Szedő Gábor, MBE_MIT, 2000

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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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Developmental System

• A CA can be considered as a developmental system, in which an organism can develop (e.g. grow) from a zygote to a multi-cellular organism (phenotype) according to specific local rules, represented by a genome (genotype).

• The genome specifications and the gene regulatory information control the cells’ growth and differentiation.

• The behavior of the CA is represented by the emergent phenotype, which is subject to shape and size modification, along the developmental process.

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Research Questions

1. Is there any relationship between computation performed by CA and their regulatory input and cellular actions?

2. Is it possible to develop an organism of a given complexity?

3. Can we predict the developmental behavior of the phenotype from the genotype composition?

Investigation of the correlations between Cellular Automata (CA) behavior (development process) and cellular regulative properties (genome information)

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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, 1990)

tot

q1

k

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Developmental lambda:

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Experimental Setup - 1

• Minimalistic developmental system• 3 cell types (type 0: quiescent, type 1

and type 2 for multicellularity)• All possible 35 = 243 regulatory input

combinations are represented in a development table

• 2D CA, von Neumann neighborhood

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Experimental Setup - 2

• 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 = 19 6836x6 = 3^36 = 1,5 x 10 ^ 17

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Conclusion and Future Work

• Now running 4 by 4, 5 by 5 and 6 by 6 (Tufte and Nichele, 2011 – IN PRESS).

• Promising results: λ can be an indicator of how the organism will develop.

• Other complexity measures? Growth rate, change rate, structural complexity.

• λ can be used to drive evolution in desired parts of the search space.

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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.”

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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

[16] G. Tufte and S. Nichele . On the Correlation Between Developmental Diversity and Genomic Composition, to appear in GECCO ’11: Proceedings of the 20th annual conference on Genetic and evolutionary computation, New York, NY, USA (2011). ACM. (IN PRESS)

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