Computational Mechanics of ECAs, and Machine Metrics Elementary - - PowerPoint PPT Presentation

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Computational Mechanics of ECAs, and Machine Metrics Elementary - - PowerPoint PPT Presentation

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Computational Mechanics of ECAs, and Machine Metrics Elementary Cellular Automata 1d lattice with N cells (periodic BC) Cells are binary valued {1,0} -- B or W Deterministic update rule, , applied to all cells simultaneously to

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Computational Mechanics of ECAs, and Machine Metrics

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Elementary Cellular Automata

  • 1d lattice with N cells (periodic BC)
  • Cells are binary valued {1,0} -- B or W
  • Deterministic update rule, Φ, applied to

all cells simultaneously to determine cell values at next time step.

  • nearest neighbor interactions only
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Example - Rule 54

000 001 010 011 100 101 110 111 0 1 1 0 1 1 0 0

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Typical Behavior of ECAs

  • Emergence of “Domains” -- spatially

homogeneous regions that spread through lattice as time progresses.

  • Largely independent of lattice size N, for N big.
  • Depends (sensitively) on update rule Φ.
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Characterizing ECA Behavior

Domains can be characterized by ε-Machines.

Rule 18 (0W)* Rule 54 (1110)*

A B

0,1

A B D C

1 1 1

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Formally Defining Domains

  • Since each ECA Domain can be characterized by a

DFA (ε-machine), domains are regular languages.

  • Def: a (spatial) domain or (spatial) domain language

Λ is a regular langauge s.t. (1) Φ(Λ) = Λ or Φp(Λ) = Λ , for some p. (temporal invariance). (2) Process graph of Λ is strongly connected (spatial homogeneity).

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Temporal Invariance?

  • Question: Given a potential domain, Λ, with

corresponding DFA, M, how do we determine temporal invariance? Can this even be done in general?

  • Answer: Yes, but somewhat involved. Steps are:

(1) Encode CA update rule as a Transducer, T. (2) Take composition T(M) = T’ (3) Use T’ to construct M’ = [T]out (4) Check if M’ = M

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How to Determine Domains

  • Visual Inspection in simple cases (#54)
  • Epsilon Machine Reconstruction
  • Fixed Point Equation
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ε-Machine Reconstruction

Several Difficulties:

  • ‘Experimental’ spatial data does not

consist entirely of domain regions. Must sort out true transitions from anomalies.

  • May be multiple domains
  • Pattern may be spatio-temporal not

simply spatial.

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Rules that worked

Rule 18 (0W)*

A B

0,1 Rule 54 (1110)*

A B D C

1 1 1 Rule 80 (00,0*,1/11…)

B D C

1 1

A A

Rule 160 (0)*

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Rules that did NOT work

Rule 144 (1000,0*)

B D C

1

A A

Rule 4, 107 No machines for 150, 180, 204 (and many others)

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Results

  • Good for entirely periodic spatial patterns, which

are temporally fixed.

  • Can reconstruct some spatial domains with

indeterminancy e.g. Rule 18 = (0W)* , Rule 80.

  • Can reconstruct some period 2 domains e.g.

Rule 54.

  • In general, difficulties for domains with lots of

‘noise’, non-block processes, low transition probabilities, and spatio-temporal processes.

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Questions from Demos

  • How to analyze patterns in space-time?
  • Minimal invariant sets - domains within

domains e.g. 000… in rule 18.

  • What does it mean for a domain to be

stable or attracting?

  • Particles and transient dynamics?
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Unit Perturbation DFAs

  • The unit perturbation language L’ of L is

L’ = { w’ s.t. ∃ w in L s.t. d(w’,w) ≤ 1}

  • Note: L regular ⇒ L’ regular

L process ⇒ L’ process

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Attractors

  • A regular language L is a fixed point attractor

for a CA, Φ, if (1) Φ(L) = L (2) Φn(L’) ⊂ L’, for all n (3) For ‘almost every’ w in L’ , Φn(w) is in L, for some n