What We Didn’t Cover: The Present and...

What We Didn’t Cover:The Present and Future

Reading for this lecture:

Lecture Notes and cited papers.

Last Lecture: Natural Computation & Self-Organization, Physics 256 (Spring 2014); Jim Crutchfield

1Sunday, May 18, 14

Topics & Applications...


Course narrative:

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ModellerSystem

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Forms of Chaos: Deterministic sources of novelty Mechanisms that produce unpredictability Sensitive dependence on initial condition Sensitive dependence on parameter

How random?

How structured?

Measurement Theory: Partitions Optimal Instrument:

max

{P}hµ

min{P}

Cµ

Universal model:

Pattern defined Causal Architecture Intrinsic Computation

! ! Machine

Cµ,E,T,�,⌅

�max

, H(L), hµ,

G,R

2Sunday, May 18, 14



What we didn’t cover:

ε-Machine Enumeration Infinite ε-Machines & Generalized Hidden Markov Models Hierarchical ε-Machine Reconstruction Quantum Processes & Quantum Machines Rate Distortion Theory & Optimal Causal Inference Cellular Automata Computational Mechanics Spin Systems in 1D and 2D Optimal Instrument Design Complexity-Entropy Diagrams

3Sunday, May 18, 14



What we didn’t cover ... ε-Machine Enumeration

Every process has its ε-machine presentationSpace of processes = Space of ε-Machinesε-Machines can be exactly enumerated

B. D. Johnson, J. P. Crutchfield, C. J. Ellison, and C. S. McTague, "Enumerating Finitary Processes", (2010) submitted.Santa Fe Institute Working Paper 10-11-027. arXiv:1011.0036 [cs.FL].

n\k 2 3 4 5 61 3 7 15 31 632 7 141 1,873 20,925 213,9973 78 15,598 1,658,606 136,146,5904 1,388 3,625,6385 35,1866 1,132,6137 43,997,4268 1,993,473,480

States

Alphabet

4Sunday, May 18, 14

http://www.santafe.edu/research/working-papers/year/2010


http://arxiv.org/abs/1011.0036




What we didn’t cover ... ε-Machine Enumeration

Every process has its ε-machine presentationSpace of processes = Space of ε-Machinesε-Machines can be exactly enumerated

B. D. Johnson, J. P. Crutchfield, C. J. Ellison, and C. S. McTague, "Enumerating Finitary Processes", (2010) submitted.Santa Fe Institute Working Paper 10-11-027. arXiv:1011.0036 [cs.FL].

n\k 2 3 4 5 61 3 7 15 31 632 7 141 1,873 20,925 213,9973 78 15,598 1,658,606 136,146,5904 1,388 3,625,6385 35,1866 1,132,6137 43,997,4268 1,993,473,480

States

Alphabet

Unknown unknowns!

4Sunday, May 18, 14







What we didn’t cover ... Infinite ε-Machines & Generalized Hidden Markov Models

D. Upper, Ph.D. Dissertation (Berkeley 2007).J. P. Crutchfield, "Calculi of Emergence:", Computation, Dynamics, and Induction", Physica D 75 (1994) 11-54.

5Sunday, May 18, 14



What we didn’t cover ... Hierarchical ε-Machine Reconstruction

J. P. Crutchfield, "Calculi of Emergence:", Computation, Dynamics, and Induction", Physica D 75 (1994) 11-54.

6Sunday, May 18, 14



What we didn’t cover ...

Quantum Processes & Quantum Machines

K. Wiesner and J. P. Crutchfield, "Computation in Finitary Stochastic and Quantum Processes", Physica D 237:9 (2008) 1173-1195.

7Sunday, May 18, 14



What we didn’t cover ... Rate Distortion Theory & Optimal Causal Inference

I[ !X 2;"X 5 R]

I[" X5 ;R]

0.0

3.5

0.5

3.02.52.01.51.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14|

P(00)

P(01)P(10)

P(11)

P(!X "x )|

3 states4 states

6 states

2 states

5 states

1 stateInfeasible

Feasible

2 5

minPr(R|

�X)

�I[�X;R] + �I[

�X;⇥X |R]

⇥

S. Still, J. P. Crutchfield, and C. J. Ellison "Optimal Causal Inference: Estimating Stored Information and Approximating Causal Architecture", CHAOS 20:3 (2010) 037111.

8Sunday, May 18, 14



What we didn’t cover ... Cellular Automata Computational Mechanics:

0

Time

99286Site0

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Time

99286Site0

J. E. Hanson and J. P. Crutchfield, "Computational Mechanics of Cellular Automata: An Example", Physica D 103 (1997) 169-189.

9Sunday, May 18, 14



What we didn’t cover ... Spin Systems in 1D and 2D

: GeneralizedOrder Parameter!ED. P. Feldman and J. P. Crutchfield, "Structural Information in Two-Dimensional Patterns:

Entropy Convergence and Excess Entropy", Physical Review E 67 (2003) 051104.

10Sunday, May 18, 14



What we didn’t cover ... Optimal Instrument Design

C. C. Strelioff and J. P. Crutchfield, "Optimal Instruments and Models for Noisy Chaos", CHAOS 17 (2007) 043127.




Analogous to Thermodynamic Phase Diagram (gas, liquid, solid). But uses only intrinsic computation properties.A wide diversity of Complexity-Entropy Diagrams.

RandomnessEntropy

StructuralComplexity

Predictable Unpredictable

Simple

Complex

D. P. Feldman, Carl S. McTague, and J. P. Crutchfield, "The Organization of Intrinsic Computation: Complexity-Entropy Diagrams and the Diversity of Natural Information Processing", CHAOS 18:4 (2008) 53-73.

What we didn’t cover ... Complexity-Entropy Diagrams: Analyze a class of processes




Logistic Map





Logistic Map

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0 0.2 0.4 0.6 0.8 1

Exce

ss E

ntro

py E

Entropy Rate hµ

One-Band RegionTwo-Band Region





0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

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Exce

ss E

ntro

py E

I

Entropy Density hµ

2D Ising Spin System





Complexity-Entropy Diagram: Analyze a class of processes ...

0.0 0.2 0.4 0.6 0.8 1.0 1.2hµ

0.0

0.5

1.0

1.5

2.0

2.5

E

E vs hµ

N = 5N = 4N = 3N = 2N = 1

ε-Machines




Computational Mechanics Applications by Others:

Chaotic Dynamical SystemsSymbolic DynamicsStatistical Mechanical Models: Spin systems (Ising, glasses, ...)Cellular AutomataHidden Markov ModelsChaotic CrystallographyHydrodynamics: Dripping faucet, turbulenceQuantum Dynamical SystemsSingle Molecule Dynamics


Molecular Dynamics Spectroscopy:

C.-B. Li, H. Yang, & T. Komatsuzaki, Proc. Natl. Acad. Sci USA 105:2 (2008) 536–541.






Computational Mechanics Research, Ongoing:

Novel materialsQuantum chaotic dynamics & measurement effectsContinuous processesSpatiotemporal processesInteractive learningNetwork dynamicsNeurobiological processesMultiagent systemsDistributed roboticsEvolutionary dynamics...




Computational Mechanics Research ... Complex Materials & ε-Machine Spectral Reconstruction

ε-MSR

D. P. Varn, G. S. Canright, and J. P. Crutchfield, "ε-Machine spectral reconstruction theory: A direct method for inferring planar disorder and structure from X-ray diffraction studies", Acta Cryst. Sec. A 69:2 (2013) 197-206.




Computational Mechanics Research ...

Novel materials: Use eMSR to design Run eMSR backwards

Hypothesis: Structure key to properties.But we can now analyze structure.

E.g., Design polytypes = Exotic semiconductors




Computational Mechanics Research ...Spectral Decomposition of Intrinsic Computation:

Exact, closed-form expression of all information measures using ε-machine presentation.

How?

Mixed states + Eigenspectrum of ε-machine.

J. P. Crutchfield, C. J. Ellison, P. Riechers in preparation.





hµ(L) = H⇥XL�1|XL�1

0 , S0 ⇠ µ0(�)⇤

= H⇥XL�1|

�RL�1|R0 = µ0(�)

�⇤

Recall mixed state expression:

Track analytically using spectral decomposition:


Need only track evolution of mixed states via .

TL =X

�2⇤T

�LT�

Projection operators:

TL

T� =Y

⇣2⇤T⇣ 6=�

T � ⇣I

�� ⇣

Eigenvalues: ⇤T ⌘ {� 2 C : det(�I � T ) = 0}

Spectral decomposition of intrinsic computation ...




Computational Mechanics Research ...Spectral decomposition of intrinsic computation ...

Excess entropy:


Synchronization information:

Transient information:T ⌘1X

L=1

L [hµ(L)� hµ]

=X

�2⇤T|�|<1

1

(1� �)2h�⇡|T�|H({T (s)})i

S =

(P�2⇤T|�|<1

11�� h�⇡|T�|H[⌘]i if h⇡T |H[⌘]i = 0

1 otherwise

E ⌘1X

L=1

[hµ(L)� hµ]

=X

�2⇤T|�|<1

1

1� �h�⇡|T�|H({T (s)})i





ε-Transducers: Input-output functions! ε-Machines: Behaviors and distributions over them Now, mappings on stochastic behaviors Optimal, minimal, unique presentation of mappingsNoisy communication channel:

Input process

Output process

Joint process

N. Barnett, C. J. Ellison, J. P. Crutchfield, in preparation.

. . . , (X,Y )�1, (X,Y )0, (X,Y )1, . . .

. . . , X�1, X0, X1, . . .

. . . , Y�1, Y0, Y1, . . .

Input OutputChannel




Computational Mechanics Research ...ε-Transducers ...

Transitions: x|p|y x is input symbol, y is output symbol, and p is probability

Identity channel.

All-to-fair channel.

Memoryless channels:





Delay-by-1 Channel:Input is order-1 Markov, output nonsynchronizing.

Memoryful Channel:

Memoryless channels ...

Z Channel: Fuzz out ones, transmit zeros.





Even-to-Fair: Map Even Process to Fair Coin.Random bits from Even Process, but fuzz out pairs of 1s.

Feedback-NOR: NOR of the previous output & current input.Strictly sophic on input, nonsynchronizing on output, but order-1 Markov on joint symbols.

Memoryful channels ...




Computational Mechanics Research ... Interactive Learning:

World

Agent

Model Policy

ActionsObservations

The Feedback Loop





The Evolution of Language:

Game of Telephone:

M0 ↵N0 = a0 a1 · · · aN

M1 ↵N1 = a0 a1 · · · aN

M2 · · ·

Generatealleles

Generatealleles

Infer✏-machine

Infer✏-machine

J. P. Crutchfield and S. Whalen, "Structural Drift: The Population Dynamics of Sequential Learning", PLoS Computational Biology 8:6 (2012) e1002510.

‘‘Send Three- and Four-Pence, We’re Going to a Dance’’

‘‘Send reinforcements we’re going to advance’’

Heard over radio during WWI:

Intended message:

Mathematical Theory of Sequential Learning





The Evolution of Language ...Mathematical Theory of Sequential Learning

Genetic Drift

t1(p0) = � 1

p0[4Ne(1� p0) log(1� p0)]

0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

Initial Probability p

Time(Generations)

Fixation (MC) Fixation (t1 Theory)

Deletion (t0 Theory)

0 N = 10

(Theory: Kimura)

J. P. Crutchfield and S. Whalen, "Structural Drift: The Population Dynamics of Sequential Learning", PLoS Computational Biology 8:6 (2012) e1002510.




Computational Mechanics Research:

The Evolution of Language ...

Structural Drift

GMP SubspaceAlternating

Process

M0 = GMP

Semantic Creation

0 50 100 150 200 250

0.4

0.6

0.8

1

Time (Generations)

Pr[Heads]

Golden Mean Even Process

Biased Coin

BA1 |H

1 |T

A

1 |H

Sequential learning as a diffusion in space of ε-Machines




Computational Mechanics Research:

The Evolution of Language ...

translationparty.com




Computational Mechanics Research:Emergence of Evolutionary Selection

Self-Organizing NetworksEvolutionary Theory: From the Primordial Soup to Natural Selection

TR

TB TA

TC

TD

J. P. Crutchfield and O. Goernerup, “Objects That Make Objects: The Population Dynamics of Structural Complexity”, J. Royal Society Interface 3 (2006) 345-349.




Computational Mechanics Research:Emergence of Evolutionary Selection

Self-Organizing NetworksEvolutionary Theory: From the Primordial Soup to Natural Selection

TR

TB TA

TC

TD

J. P. Crutchfield and O. Goernerup, “Objects That Make Objects: The Population Dynamics of Structural Complexity”, J. Royal Society Interface 3 (2006) 345-349.

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position

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positon

|M|

�

5




The Message:

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ModellerSystem

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How Nature is Organized is How Nature Computes


Further:

Computational Mechanics Archive: http://csc.ucdavis.edu/~cmg/

Dynamics of Learning Research Group Mail List: http://lists.csc.ucdavis.edu/mailman/listinfo/dynlearn




http://lists.csc.ucdavis.edu/mailman/listinfo/dynlearn

http://lists.csc.ucdavis.edu/mailman/listinfo/dynlearn

What We Didn’t Cover: The Present and...

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