Emergence - Mathematics and Computer Science

rucksackbulgeAI and Robotics

Dec 1, 2013 (3 years and 10 months ago)

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Self
-
Organization

and

Emergence


Emergence in Graphs



Graphs

Isomorphic graphs

Vertices map to vertices and edges map to
edges such that the relationships between
vertices and edges are preserved.

Consider a fully
-
connected graph
with 25 randomly
placed vertices
(i.e., K
25
)…

Consider a fully
-
connected graph
with 25 randomly
placed vertices
(i.e., K
25
)…



But, what if we
care about the
placement of
vertices?

For example,
consider K
25

with
vertices arranged
on a circle…


What do you see?

Arranged
vertices

and

emergent
patterns

Key idea: Systematic circular
arrangement of vertices in a fully
connected graph results in the
visual perception of a collection
of concentric circles formed by
the intersection of the edges.
These circles represent an
emergent property of the graph
(as they are not explicitly
specified in the construction
parameters).

http://www.sacred
-
destinations.com/italy/venice
-
san
-
marco
-
photos/slides/interior
-
cc
-
petunia2323

Spandrels?

Some Questions


1.

Under what circumstances will visual patterns emerge?


2.

What mechanisms exist for detecting emergent patterns?


3.

Are there rules for predicting pattern emergence?


4.

What about reverse
-
engineering?


5.

Are there any practical ramifications?




Under what circumstances will
visual patterns emerge?

Key factors:

Arrangement of vertices

# of vertices

# of edges

Arrangement of edges

Vertex proximity (i.e., size of graph)




Reminder: this is a matter of visual perception!

K
25


K
24


Odd or even number of vertices…

24 vertices, 138 random
connections (50% of all
possible edges)

24 vertices, 207 random
connections (75% of all
possible edges)

Number of edges…

24 vertices with 4 random
edges per vertex

Arrangement of edges…

24 vertices with 4 regularly
spaced edges per vertex

Vertex proximity

K
25


Vertex proximity

K
25


Vertex proximity

K
25


Vertex proximity

K
25


Vertex proximity

K
101



zoom x 1

Vertex proximity

K
101



zoom x 3

Vertex proximity

K
101



zoom x 5

Vertex proximity

K
101



zoom x 7

Vertex proximity

K
101



zoom x 9

Vertex proximity

K
101



zoom x 19

Vertex proximity

K
101



zoom x 29

Vertex proximity

K
101



zoom x 51

A Simple Cellular Automaton

Imagine a one
-
dimensional grid of cells with possible states 0 or 1




The state of each cell changes based on its current state and the
state of its two immediate neighbors


There are thus 8 parent combinations: 000, 001, 010,…, 111.


There are 2
8

= 256 possible rule sets

(e.g. 000→0/1, 001→0/1, …, 111→0/1 with child at the center).


All cells change state simultaneously


Successive generations are displayed one under

the other and the resulting pattern is observed.

0

0

1

0

1

1

1

0

1

1

Emergence!

Emergent properties in CA: Cellular automata
illustrate the idea that complex (emergent)
behavior can arise from a simple rule set

http://ourworld.compuserve.com/homepages/DP5/pattern1.htm

http://www.ucmp.berkeley.edu/people/klf/MicroGalleryLarge_files/Radiolarian2.jpg

http://static.open.salon.com/files/zebra
-
picture1220629421.jpg

“Principle of Computational Equivalence: all processes, whether they are produced
by human effort or occur spontaneously in nature, can be viewed as computations.”

(Wolfram p.715,
A New Kind of Science
)

Crystallization

Self
-
Assembly

(Pelesko)

http://www.canyonsworldwide.com/crystals/pictures/CrystalCavesRick.jpg

http://upload.wikimedia.org/wikipedia/commons/thumb/7/78/Devils_postpile_NM.jpg/800px
-
Devils_postpile_NM.jpg

http://upload.wikimedia.org/wikipedia/commons/4/46/Devils_Tower_CROP.jpg

Self
-
Assembly

(Pelesko)

Diffusion Limited Aggregation

Algorithm

Seed the center of a large, rectangular lattice

Loop


Add particle at boundary of system


Take particle on a random walk until it is adjacent to an occupied position


Add the particle to the aggregate at its current position

Until constraint(s)


http://www.its.caltech.edu/~atomic/snowcrystals/frost/frost.htm

http://www.its.caltech.edu/~atomic/snowcrystals/frost2/frost2.htm

Swarm
Intelligence

Hofstadter,
Gödel, Escher, Bach
, p.358

Swarm: “loosely structured collection of
interacting agents”

(Santa Fe Institute)

An autonomous agent is a system situated
within and a part of an environment that
senses that environment and acts on it,
over time, in pursuit of its own agenda and
so as to effect what it senses in the future.”

(Franklin and Graesser, 1996)

Boids

(Craig Reynolds)



http://www.red3d.com/cwr/boids/

Gravity Waves Over Iowa

Cosmological Emergence

Exploring Motion Demos

Conway's Game of Life (1968)
-

(cf. Levy)


2D; 2 states/cell (alive/dead); 8 neighbors/cell


Living cell survives if 2 or 3 neighbors are alive


Living cell dies of overcrowding if > 3 living neighbors


Living cell dies of exposure (loneliness) if < 2 neighbors


Cell is "born" if exactly 3 neighbors are alive






Game of Life

Demo

or
http://www.math.com/students/wonders/life/life.html

http://www.indianastandardsresources.org/admin/library/codon_acid.pdf


-
helix:





60







60


(3.6 amino acids per 360


turn


-
獴牡湤猺r





135





135


Secondary Structure

Fundamental Concepts of Bioinformatics
: Krane and Raymer

Protein Folding

http://cnx.org/content/m11461/latest/protein_folding.jpg

http://www.google.com/imgres?imgurl=http://upload.wikimedia.org/wikipedia/commons/a/a9/Protein_folding.png&imgrefurl=http
://commons.wikimedia.org/wiki/Image:Protein_folding.png&h=440&w=994&sz=69&tbnid=ZFo3e4RHz0sJ::&tbnh=66&tbnw=149
&prev=/images%3Fq%3Dprotein%2Bfolding%2Bimages&hl=en&usg=__z6Rs6HBySUrszdZoVlm6j7y
-
G
-
E=&sa=X&oi=image_result&resnum=2&ct=image&cd=1

http://www.cs.ucl.ac.uk/staff/d.jones/t42morph.html

Protein folding videos:

http://www.youtube.com/watch?v=swEc_sUVz5I


http://www.youtube.com/watch?v=meNEUTn9Atg&feature=related


http://www.youtube.com/watch?v=AlfvWESPyZY&feature=related

Self
-
Assembly

(Pelesko)

http://www.answers.com/topic/micelle

http://simscience.org/membranes/advanced/essay/micelle.html

Micelles

Self
-
Assembly

(Pelesko)

Packing parameter = v/(a
0
l
c
) where v is the volume of the hydrocarbon
chain; l
c

is the max effective length that chains can assume; a
0

is the
surface area occupied by the head group in a given structure


Draw a colored equilateral triangle


Repeat indefinitely for every colored triangle
currently in existence


Connect the midpoints of each side to form
four separate triangles


Color the center triangle black


Problem: Construct a Sierpinski Triangle

Algorithm

Create an equilateral triangle with vertices
A
,
B
,
C

Select a random starting point
p

within the triangle

LOOP UNTIL satisfied or bored


generate a random integer
r

from 1 to 3


IF
r

= 1: draw a line from
p

to vertex
A


IF
r

= 2: draw a line from
p

to vertex
B


IF
r

= 3: draw a line from
p

to vertex
C


set
p

= the center of the new line

END LOOP


Sierpinski Triangle Via Randomness

(random processes can generate predictable results)


Sensitivity to Initial Conditions


Consider the 1D CA examined earlier: Note that changing the starting cell states
can result in changes to the final outcome…


For example, consider rule 22 with a random start state versus a start state
consisting of a single white cell (1) surrounded by all black cells (0)…













A New Kind of Science

(Wolfram), pp. 231,235

An Algorithm for Everything?

From Brain Stuff to…

mage source:

http://domino.watson.ibm.com/comm/pr.nsf/pages/rscd.n
eurons_picb.html/$FILE/NeuronsInAColumn1_s.bmp

http://www.brainbasedbusiness.com/uploads/neuron.jpg

http://www.sciencemuseum.org.uk/on
-
line/brain/images/1
-
1
-
8
-
0
-
0
-
0
-
0
-
0
-
0
-
0
-
0.jpg

http://www.cirrusimage.com/hymenoptera_Great_Golden_Digger_Wasp.htm

mage source:

http://domino.watson.ibm.com/comm/pr.nsf/pages/rscd.neuro
ns_picb.html/$FILE/NeuronsInAColumn1_s.bmp

mage source: http://www.mabot.com/brain/

10
11

neurons 10
14
-
10
15

connections

http://www.brainbasedbusiness.com/uploads/neuron.jpg

http://www.chm.bris.ac.uk/webprojects2006/Cowlishaw/300px
-
Action
-
potential.png

How does meaning arise from mindless mechanisms?

http://www.russianlegacy.com/catalog/images/matryoshka_traditional/ND50
-
001.jpg

What’s inside that makes you
conscious/intelligent?

Huse, S. (1993).
The Collapse of Evolution
. BakerBooks: Grand Rapids, MI, p. 53

Presumed Lack of Sufficient Time

Huse, S. (1993).
The Collapse of Evolution
. BakerBooks: Grand Rapids, MI, p. 53

Presumed Lack of Sufficient Time

mechanism?

mechanism?

mechanism?

mechanism?

mechanism?

1.
The Primordium

2.
Making a Non
-
uniform Universe

3.
The Emergence of Stars

4.
The Periodic Table

5.
Planetary Accretion

6.
Planetary Structure

7.
The Geospheres

8.
The Emergence of Metabolism

9.
Cells

10.
Cells with Organelles

11.
Multi
-
cellularity

12.
The Neuron

13.
Animalness

14.
Chordateness

15.
Vertebrates

16.
From Fish to Amphibians

17.
Reptiles

18.
Mammals

19.
The Niche

20.
Arboreal Mammals

21.
Primates

22.
The Great Apes

23.
Hominization

24.
Tool making

25.
Language

26.
Agriculture

27.
Technology and Urbanization

28.
Philosophy

29.
The Spirit

Note that even our theories have an emergent character. For example, consider
Morowitz comments (p.8): “Science starts with the mind, both as the perceiver
of sensations and the postulator of constructs… This, of course, presents an
epistemic circle. One starts with the mind as the primitive and goes around the
circle of constructs in an effort to explain mind.”

The Emergence of Everything

(Morowitz, p. 8)



Key Pruning Principles


1.
A given set of (low
-
level) building blocks can only
go together so many ways based on physical,
chemical, environmental, etc. constraints.


2.
Self
-
organization is a basic principle of nature
(creating
pockets of order

that buck the trend
toward increasing entropy).




rules



disorder


order


3.

Learning means ordering (arranging).





One example cited in the literature
1

is that even
something as simple as randomly generating the
word “evolution” from the 26 letters of the English
alphabet is prohibitively unlikely.


Probability is (1/26)
9

= 1/26
9
≈ 1 / 5,429,503,700,000


But evolution doesn’t just put random things together!


Presumed Inadequacy of Random Processes

1
Huse, S. (1993).
The Collapse of Evolution
. BakerBooks: Grand Rapids, MI, p. 84

Consider random generation of the string “evol” (probability is
(1/26)
4

=

1 / 456976:


rgvh

lyvn

qxlg

bill

qjit

wjzy



yymn

otfo

xkjr

lwyk

epug

ggoq

evol

Target string generated on trial # 149932.


Several other tests resulted in 250561, 520139, and 261924 trials…

Presumed Inadequacy of Random Processes

Human Origins: Alternative Perspective

http://embryo.soad.umich.edu/

Consider the 2
nd

Law of Thermodynamics…

Self
-
organization: “the study of the way in which systems
made up of simpler elements, governed by simple dynamical
principles, spontaneously develop organization or patterns of
order at a higher level.”
1


Emergence: A scenario in which complex patterns and
behaviors arise from relatively simple collections of rules.
(“The whole is more than the sum of its parts…”) Morowitz
calls emergence “the opposite of reduction.”
2





1
Werbos, P.

Origins: Brain & Self Organization
, p.18

2
Morowitz, H. (2002).
The Emergence of Everything
. Oxford University Press: New York, p.14

If I can write the equations for something is it an
emergent property?


Is emergence real or just perceived?


What is the relationship between determinism
and randomness in self
-
organizing systems?


How can meaning and purpose emerge in self
-
organizing systems?


How can God operate through emergence?