Further Investigations by
Langton
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2-D CAs
K=8
N=5
64 ¥ 64 lattice
periodic boundary conditions
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1
Transient Length (I, II)
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2
Transient Length (III)
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3
Shannon Information
(very briefly!)
• Information varies directly with surprise
• Information varies inversely with
probability
• Information is additive
• \The information content of a message is
proportional to the negative log of its
probability
I{s} = -lgPr{s}
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Entropy
• Suppose have source S of symbols from
ensemble {s1, s2, …, sN}
• Average information per symbol:
Â
N
N
Pr{sk }I{sk } =Â Pr{sk }(-lgPr{sk })
k=1
k=1
• This is the entropy of the source:
N
H{S} = -Â Pr{sk } lgPr{sk }
†
k=1
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Maximum and Minimum
Entropy
• Maximum entropy is achieved when all
signals are equally likely
No ability to guess; maximum surprise
Hmax = lg N
• Minimum entropy occurs when one symbol
is certain and the others are impossible
No uncertainty; no surprise
Hmin = 0
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Entropy Examples
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Avg. Transient Length vs. l
(K=4, N=5)
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Avg. Cell Entropy vs. l
(K=4, N=5)
H(A) =
K
Âp
k
lg pk
k=1
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†
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Avg. Cell Entropy vs. l
(K=4, N=5)
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Avg. Cell Entropy vs. l
(K=4, N=5)
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Avg. Cell Entropy vs. Dl
(K=4, N=5)
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Avg. Cell Entropy vs. l
(K=4, N=5)
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Avg. Cell Entropy vs. Dl
(K=4, N=5)
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Entropy of Independent Systems
• Suppose sources A and B are independent
• Let pj = Pr{aj} and qk = Pr{bk}
• Then Pr{aj, bk} = Pr{aj} Pr{bk} = pjqk
H(A,B) = Â Pr( a j ,bk ) lgPr (a j ,bk )
j,k
= Â p j qk lg( p j qk ) = Â p j qk (lg p j + lgqk )
j,k
j,k
= Â p j lg p j + Â qk lgqk = H(A) + H(B)
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j
k
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Mutual Information
• Mutual information measures the degree to
which two sources are not independent
• A measure of their correlation
I( A,B) = H ( A) + H (B) - H ( A,B)
• I(A,B) = 0 for completely independent
sources
† • I(A,B) = H(A) = H(B) for completely
correlated sources
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Avg. Mutual Info vs. l
(K=4, N=5)
I(A,B) =
H(A) + H(B)
– H(A,B)
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Avg. Mutual Info vs. Dl
(K=4, N=5)
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Complexity vs. l
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Schematic of
CA Rule Space vs. l
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Fig. from Langton, “Life at Edge of Chaos”
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Additional Bibliography
1. Langton, Christopher G. “Life at the Edge
of Chaos,” in Artificial Life II, ed. Langton
et al. Addison-Wesley, 1992.
2. Emmeche, Claus. The Garden in the
Machine: The Emerging Science of
Artificial Life. Princeton, 1994.
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Slime Mold
(Dictyostelium discoideum)
“Dicty”
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Amoeba Stage
• Single cell
• Lives in soil
• Free moving
• Engulfs food
(bacteria)
• Divides asexually
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Amoebas
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Aggregation Stage
• Triggered by
exhaustion of food
• Aggregate by
chemotaxis
• Form expanding
concentric rings and
spirals
• Up to 125 000
individuals
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Stream Formation Stage
• As density increases,
begin to adhere
• Begin to form mound
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Mound Stage
• Cells differentiate
• Some form an elongated finger
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Slug Stage
•
•
•
•
Prestalk elongates, topples, to form slug
Behaves as single organism with 105 cells
Migrates; seeks light; seeks or avoids heat
No brain or nervous system
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Culmination Stage
• Cells differentiate into base, stalk, and spores
• Prestalk cells form rigid bundles of cellulose &
die
• Prespore cells (at end) cover selves with cellulose
& become dormant
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Fruiting Body Stage
• Spores are dispersed
• Wind or animals carry
spores to new territory
• If sufficient moisture,
spores germinate,
release amoebas
• Cycle begins again
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Complete Life Cycle
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Emergent Patterns
During Aggregation
• a-c. As aggregate,
wave lengths shorten
• d. Population divides
into disjoint domains
• e-f. Domains contract
into “fingers”
(streaming stage)
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fig. from Solé & Goodwin
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Belousov-Zhabotinski Reaction
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Hodgepodge Machine
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Demonstration of
Hodgepodge Machine
Go to hodgepodge machine
applets at CBN website
or unix program at course website
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Universal Properties
• What leads to these expanding rings and
spirals in very different systems?
• Under what conditions do these structures
form?
• What causes the rotation?
• These are all examples of excitable media
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