Toward a Better Understanding of...
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Toward a Better Understanding ofComplexity
——————————————Definitions of Complexity, Cellular Automata as
Models of Complexity, Random Boolean Networks
Christian [email protected]
Department of Computer Science
University of Calgary
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Complexity
Definitions
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Definitions of Complexity
F Information– The capability of a system to “surprise” an observer, i.e.
to provide information.
F Time-Computational Complexity– The time an algorithm needs to solve a problem.
F Space-Computational Complexity– The amount of memory an algorithm needs to solve a
problem.
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Definitions of Complexity (2)
F Effective Complexity– The degree of order (instead of randomness) of a
system.
F Entropy– The complexity of a system is equal to the thermo-
dynamical measure of its disorder.
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Definitions of Complexity (3)
F Fractal Dimension– The “fuzziness” of a system, measuring the degrees of
details a system reveals on arbitrary scales.
F Hierarchical Complexity– The diversity of different layers, which a hierarchical
system is composed of.
F Mutual Information– The degree to which a part of a system has information
about other involved system constituents.
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Definitions of Complexity (4)
F Information Distance– The differences among parts of a system.
F Grammatical Complexity– The degree of universality a language must have to
describe a system (regular, context-free, context-sensitive, universal Turing machine)
F …
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Cellular Automata
Random BooleanNetworks
Classifier Systems
SwarmSystems
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Cellular Automata
Global Effects from LocalRules
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Cellular Automata
F The CA space is a lattice of cells with aparticular geometry.
F Each cell contains a variable from a limitedrange (e.g., 0 and 1).
F All cells update synchronously.
F All cells use the same updating rule,depending only on local relations.
F Time advances in discrete steps.9
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One-dimensional finite CA architecture
time
F K = 5 localconnectionsper cell
F Synchronousupdate in discretetime steps
A. Wuensche: The Ghost in the Machine, Artificial Life III, 1994. 10
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Cellular Automata:Local Rules — Global Effects
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Time Evolution of the ith Cell
Ci(t+1) = f (Ci -[ K / 2]
( t ) ,..., Ci -1( t) ,Ci
( t ),Ci +1( t ) ,..., Ci +[ K / 2]
( t ) )
With periodic boundary conditions:
x < 1: Cx = CN+ x x > N : Cx = Cx - N
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Value Range and Update Rules
F For V different states (= values) per cellthere are VK permuations of values in aneighbourhood of size K.
F The update function f can be implementedas a lookup table with VK entries, givingVVK
possible rules.
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Example Update Rule
F V = 2, K = 3
F The rule table for rule 30:
111 110 101 100 011 010 001 000 0 0 0 1 1 1 1 0
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CA Demos
F Evolvica CA Notebooks
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Cellular Automata
Models of Complexity(Stephen Wolfram Approach)
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Analysis of Cellular Automata
F … as Discrete Dynamical Systems
– Discrete idealization of partial differential equations
– Set of possible (infinite) CA configurations forms aCantor set
– CA evolution may be viewed as a continuous mappingon this Cantor set
– Entropies, fractal dimensions, Lyapunov exponents
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Analysis of Cellular Automata
F … as Information-Processing Systems
– Parallel-processing computer with a simple gridarchitecture.
– Initial configuration is processed by theevolution of the cellular automaton.
– What types of formal languages are generated?
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Four Classes of Patterns
F Wolfram classifies CAs according to the patternsthey evolve:
– 1. Pattern disappears with time.– 2. Pattern evolves to a fixed finite size.– 3. Pattern grows indefinitely at a fixed speed.– 4. Pattern grows and contracts irregularly.
– 3/text.html: Fig. 1
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Four Qualitative Classes
F 1. Spatially homogeneous stateF 2. Sequence of simple stable or periodic structuresF 3. Chaotic aperiodic behaviourF 4. Complicated localized structures, some
propagating
• 85-cellular/7/text.html: Fig. 3 (first row)
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Classes from an InformationPropagation Perspective
F 1. No change in final state
F 2. Changes only in a finite region
F 3. Changes over an ever-increasing region
F 4. Irregular changes
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Degrees of Predictability for theOutcome of CA Evolution
F 1. Entirely predictable, independent ofinitial state
F 2. Local behavior predictable from localinitial state
F 3. Behavior depends on an ever-increasinginitial region
F 4. Behavior effectively unpredictable
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Suggested Explorations:
F Natural CA-like Phenomena (CBN, 15.5)F Stephen Wolfram’s vs. Chris Langton’s CA
Classification (CBN, 15.2, 15.3)F Conway’s Game of Life (CBN, 15.4)F Self-Similarity and Fractal Geometry (CBN,
Ch. 5)F L-Systems and Fractal Growth (CBN, Ch.
6)
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Suggested Explorations: (2)
F Fractals (CBN, Ch. 9):– Simplicity and Complexity
F Nonlinear Dynamics in Simple Maps– Logistic map, stability vs. instability,
bifurcations, chaos (CBN, Ch. 10)
F Strange Attractors– Hénon-Lorenz attractor (CBN, Ch. 11)
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Modeling Excitable Media
• Slime mold growth,
• star formation in spiral disk galaxies,
• cardiac tissue contraction,
• diffusion-reaction chemical systems and
• infectious disease epidemics
– would seem to be quite dissimilar systems.
– Yet, in each of these cases, various spatiallydistributed patterns, such as concentric andspiral wave patterns, are spontaneously formed.
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Modeling Excitable Media
• The underlying cause of the formation of these self-organized, self-propagating structures is that theseare excitable media, consisting of spatiallydistributed elements which can
– become excited as a result of interacting with neighboringelements,
– subsequently returning incrementally to the quiescent state
– in which they are again receptive to being excited.
• The excited-refractory-receptive cycle thatcharacterizes these systems can be modelled usingmulti-state cellular automata.
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2-D CA:Emergent Pattern Formation in Excitable Media
Neuron excitation
Neuron excitation (relaxed)
Hodgepodge
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Cellular Automata
Random BooleanNetworks
Classifier Systems
SwarmSystems
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Complex Systems
EmergentBehaviours and Patterns
fromLocal Interactions
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Stevens et al., 1988
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Nuridsany & Pérennou, 1996
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Ernst, 1998
Nuridsany & Pérennou, 1996
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What to Learn from Ant Colonies asComplex Systems
F Fairly simple units generate complicatedglobal behaviour.
F “If we knew how an ant colony works, wemight understand more about how all suchsystems work, from brains to ecosystems.”(Gordon, 1999)
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Emergence in Complex Systems
F How do neurons respond to each other in away that produces thoughts?
F How do cells respond to each other in a waythat produces the distinct tissues of agrowing embryo?
F How do species interact to producepredictable changes, over time, inecological communities?
F ...
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Complexity through EmergenceExamples of Swarm Systems ...
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References
F Bonabeau, E., Dorigo, M., and Theraulaz, G. (1999). SwarmIntelligence: From Natural to Artificial Systems. New York, OxfordUniversity Press.
F Ernst, A. M., ed. (1998). Digest: Kooperation und Konkurrenz,Heidelberg, Spektrum Akademischer Verlag.
F Gordon, D. (1999). Ants at Work. New York, The Free Press.F Hölldobler, B., and Wilson, E. O. (1990). The Ants. Cambridge,
MA, Harvard University Press.F Nuridsany, C., and Pérennou, M. (1996). Microcosmos: The
Invisible World of Insects. New York, Stewart, Tabori & Chang.F Resnik, M. (1997). Turtles, Termites, and Traffic Jams.
Cambridge, MA, MIT Press.F Stevens, C. F., et al. (1988). Gehirn und Nervensystem.
Heidelberg, Spektrum Akademischer Verlag.