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Cellular Automata Cellular Automata and beyond and beyond The World of Simple Programs The World of Simple Programs Christian Jacob AI AI Department of Computer Science University of Calgary CPSC 601.73 Winter 2003 Emergent

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 1

Cellular Automata Cellular Automata

and beyond and beyond … … The World of Simple Programs The World of Simple Programs

AI AI

Christian Jacob

Department of Computer Science University of Calgary

CPSC 601.73 — Winter 2003

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 2

Cellular Automata Random Boolean Networks Classifier Systems Lindenmayer Systems

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 3

Cellular Automata Cellular Automata

Global Effects from Local Rules Global Effects from Local Rules

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 4

Cellular Automata Cellular Automata

  • The CA space is a lattice of cells (usually 1D, 2D, 3D)

with a particular geometry.

  • Each cell contains a variable from a limited range of values

(e.g., 0 and 1).

  • All cells update synchronously.
  • All cells use the same updating rule (in uniform CA),

depending only on local relations.

  • Time advances in discrete steps.
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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 5

One-dimensional Finite CA Architecture One-dimensional Finite CA Architecture

time

  • Neighbourhood size:

K = 5 local connections per cell

  • Synchronous

update in discrete time steps

  • A. Wuensche: The Ghost in the Machine, Artificial Life III, 1994.
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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 6

Time Evolution of Cell Time Evolution of Cell i i with with K K-Neighbourhood

  • Neighbourhood

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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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 7

Value Range and Update Rules Value Range and Update Rules

  • For V different states (= values) per cell there are VK

permuations of values in a neighbourhood of size K.

  • The update function f can be implemented as a lookup

table with VK entries, giving VVK possible rules. 00000: 1 … V 00001: _ 00010: _ … 11110: _ 11111: _ VK

1.3 ¥ 10154 512 9 2 3.4 ¥ 1038 128 7 2 4.3 ¥ 109 32 5 2 256 8 3 2

Vv^K vK K v

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 8

Cellular Automata: Local Rules Cellular Automata: Local Rules — — Global Effects Global Effects

Demos

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 9

History of Cellular Automata History of Cellular Automata

  • Alternative names:

– Tesselation automata – Cellular spaces – Iterative automata – Homogeneous structures – Universal spaces

  • John von Neumann (1947)

– Tries to develop abstract model of self-reproduction in biology (from investigations in cybernetics; Norbert Wiener)

  • J. von Neumann & Stanislaw Ulam (1951)

– 2D self-reproducing cellular automaton – 29 states per cell – Complicated rules – 200,000 cell configuration – (Details filled in by Arthur Burks in 1960s.)

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 10

History of Cellular Automata (2) History of Cellular Automata (2)

  • Threads emerging from J. von Neumann’s work:

– Self-reproducing automata (spacecraft!) – Mathematical studies of the essence of

  • Self-reproduction and
  • Universal computation.
  • CAs as Parallel Computers (end of 1950s / 1960s)

– Theorems about CAs (analogies to Turing machines) and their formal computational capabilities – Connecting CAs to mathematical discussions of dynamical systems (e.g., fluid dynamics, gases, multi-particle systems)

  • 1D and 2D CAs used in electronic devices (1950s)

– Digital image processing (with so-called cellular logic systems) – Optical character recognition – Microscopic particle counting – Noise removal

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 11

History of Cellular Automata (3) History of Cellular Automata (3)

  • Stansilaw Ulam at Los Alamos Laboratories

– 2D cellular automata to produce recursively defined geometrical

  • bjects (evolution from a single black cell)

– Explorations of simple growth rules

  • Specific types of Cas (1950s/60s)

– 1D: optimization of circuits for arithmetic and other operations – 2D:

  • Neural networks with neuron cells arranged on a grid
  • Active media: reaction-diffusion processes
  • John Horton Conway (1970s)

– Game of Life (on a 2D grid) – Popularized by Martin Gardner: Scientific American

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 12

Stephen Wolfram Stephen Wolfram’ ’s World of CAs s World of CAs

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 13

Stephen Wolfram Stephen Wolfram’ ’s World of CAs s World of CAs

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 14

Stephen Wolfram Stephen Wolfram’ ’s World of CAs s World of CAs

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 15

Stephen Wolfram Stephen Wolfram’ ’s World of CAs s World of CAs

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 16

Example Update Rule Example Update Rule

  • V = 2, K = 3
  • The rule table for rule 30:

111 110 101 100 011 010 001 000 0 0 0 1 1 1 1 0 See examples ...

128 64 32 16 8 4 2 1 16 8 4 2 + + + = 30

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 17

CA Demos CA Demos

  • Evolvica CA Notebooks
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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 18

Four Wolfram Classes of CA Four Wolfram Classes of CA

  • Class 1:

A fixed, homogeneous, state is eventually reached (e.g., rules 0, 8, 128, 136, 160, 168).

168 136 160

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 19

Four Wolfram Classes of CA Four Wolfram Classes of CA

  • Class 2:

A pattern consisting of separated periodic regions is produced (e.g., rules 4, 37, 56, 73).

73 37 56 4

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 20

Four Wolfram Classes of CA Four Wolfram Classes of CA

  • Class 3:

A chaotic, aperiodic, pattern is produced (e.g., rules 18, 45, 105, 126).

126 45 105 18

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 21

Four Wolfram Classes of CA Four Wolfram Classes of CA

  • Class 4:

Complex, localized structures are generated (e.g., rules 30, 110).

110 30

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 22

Class 4: Rule 30 Class 4: Rule 30

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 23

Class 4: Rule 110 Class 4: Rule 110

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 24

Further Classifications of Further Classifications of CA Evolution CA Evolution

  • Wolfram classifies CAs according to the patterns they 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.

  • Qualitative Classes

– 1. Spatially homogeneous state – 2. Sequence of simple stable or periodic structures – 3. Chaotic aperiodic behaviour – 4. Complicated localized structures, some propagating

–3/text.html: Fig. 1

–85-cellular/7/text.html: Fig. 3 (first row)

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 25

Further Classifications of CA Evolution (2) Further Classifications of CA Evolution (2)

  • Classes from an Information Propagation Perspective

– 1. No change in final state – 2. Changes only in a finite region – 3. Changes over an ever-increasing region – 4. Irregular changes

  • Degrees of Predictability for the Outcome of the CA Evolution

– 1. Entirely predictable, independent of initial state – 2. Local behavior predictable from local initial state – 3. Behavior depends on an ever-increasing initial region – 4. Behavior effectively unpredictable

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 26

2-D CA: Emergent Pattern Formation in Excitable Media 2-D CA: Emergent Pattern Formation in Excitable Media

Neuron excitation Neuron excitation Neuron excitation (relaxed) Neuron excitation (relaxed) Hodgepodge Hodgepodge

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 27

Random Boolean Random Boolean Networks Networks

Generalized Cellular Automata Generalized Cellular Automata

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 28

[S. Kauffman: At Home in the Universe]

Crystallization of Connected Webs Crystallization of Connected Webs

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 29

Random Nets Demo

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 30

Random Network Architecture Random Network Architecture

Network at time t Network at time t+1 wiring scheme pseudo neighbourhood

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 31

Time Evolution of the Time Evolution of the i- i-th th Cell Cell

  • Cell i is connected to K cells wi1, wi2, …, wiK; with wij from {1,…, N}.
  • NK possible alternative wiring options.
  • Update rule for cell i:

Ci

(t+1) = fi(Cwi1 (t) ,Cwi2 (t ) ,..., CwiK (t ) )

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 32

Wiring/Rule Schemes Wiring/Rule Schemes

  • A random network of size N with neighbourhood size K

can be assigned alternative wiring and rule schemes.

  • Example:

V = 2, N = 16, K = 15; S = 2832 .

S = (N

K ) N ¥ (V VK ) N

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 33

States and Cycles States and Cycles

State Cycle 1 State Cycle 2 State Cycle 3

System State Following State

[S. Kauffman: Leben am Rande des Chaos]

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 34

Kauffman Kauffman’ ’s Random Boolean Networks s Random Boolean Networks

http://members.rogers.com/fmobrien/experiments/boolean_net/BooleanNetworkApplet_both.html

Boolean functions represented by shades of green. Stuart Kauffman used this network to investigate the interaction of proteins within living systems. Binary values that have changed are white. Unchanged values are blue. These networks settle very quickly into an oscillatory state.

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 35

[A. Wuensche, Discrete Dynamics Lab]

Attractor Attractor Cycles Cycles

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 36

[A. Wuensche, Discrete Dynamics Lab]

Basin of Attraction Field Basin of Attraction Field

Nodes: n =13; Connectivity: k = 3; States: 213 = 8192

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 37

[A. Wuensche, Discrete Dynamics Lab]

Basin of Attraction Field Basin of Attraction Field

Nodes: n =13; Connectivity: k = 3; States: 213 = 8192 68 984 784 1300 264 76 316 120 64 120 256 2724 604 84 428

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 38

Calculating Pre-Images Calculating Pre-Images

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 39

Calculating Pre-Images (2) Calculating Pre-Images (2)

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 40

Mutations on Random Boolean Ntworks Mutations on Random Boolean Ntworks

[A. Wuensche 98]

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 41

Attractor Attractor = Cell Type ? = Cell Type ?

  • From the set of all possible gene activation patterns, the

regulatory network selects a specific sequence of activations over time.

  • A differenciated cell doesn’t change its type any more.

– Hence, only a constrained set of genes is active – = state cycle – = attractor?

[S. Kauffman: Leben am Rande des Chaos]

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 42

Cell Types Cell Types vs vs. . Attractors Attractors

Number of Attractors / Cell Types Amount of DNA of a single chromosome set of a cell (in g)

Number of cell types Number of attractors

Bacteria Yeasts Sponges Jellyfish Nematodes Human Algae

[S. Kauffman: Leben am Rande des Chaos]

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 43

Intermediate Architectures Intermediate Architectures

Cellular automata Random networks Homogeneous rule Varying degrees of random wiring Homogeneous wiring template Varying degree of rule mix

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 44

Cellular Automata Random Boolean Networks Classifier Systems Lindenmayer Systems

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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 45

References References

  • Holland, J. H. (1992). Adaptation in Natural and Artificial Systems.

Cambridge, MA, MIT Press.

  • Kauffman, S. A. (1992). Leben am Rande des Chaos. Entwicklung und
  • Gene. Heidelberg, Spektrum Akademischer Verlag:

: 162-170.

  • Kauffman, Stuart A., (1993), The Origins of Order: Self-Organization and

Selection in Evolution. (pp. 407-522), New York, NY; Oxford University Press.

  • Kauffman, S. (1995). At Home in the Universe: The Search for Laws of

Self-Organization and Complexity. Oxford, Oxford University Press.

  • Wolfram, S. (2002). A New Kind of Science. Champaign, IL, Wolfram

Media.

  • Wuensche, A. (1994). The Ghost in the Machine: Basins of Attraction of

Random Boolean Networks. Artificial Life III. C. G. Langton. Reading, MA, Addison-Wesley. Proc

  • Proc. Vol

Vol. . XVII: XVII: 465-501.

  • Wuensche, A. (1998). Discrete Dynamical Networks and their Attractor
  • Basins. Proceedings of Complex Systems’98, University of New South

Wales, Sydney, Australia.

  • Wuensche, A. Discrete Dynamics Lab:

http://www.santafe.edu/~wuensch/ddlab.html