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 determine cell values at next time step.

• nearest neighbor interactions only

Example - Rule 54

000 001 010 011 100 101 110 111

0 1 1 0 1 1 0 0

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 .

Characterizing ECA Behavior

Domains can be characterized by their state transition machines (DFAs).

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

A

B

0,1

0

A

B

D

C

1

1

1

0

Formally Defining Domains

• Since each ECA Domain can be characterized by a DFA, domains are regular languages.

• Def: a (spatial) domain or (spatial) domain language is a regular langauge s.t.

(1) () = or p() = , for some p. (temporal invariance).

(2) Process graph of is strongly connected

(spatial homogeneity).

Temporal Invariance?

• Question: Given a potential domain, , 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

How to Determine Domains

• Visual Inspection in simple cases (#54)

• Epsilon Machine Reconstruction

• Fixed Point Equation

-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.

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.

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?

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

Attractors• A regular language L is a fixed point attractor for a CA, , if (1) (L) = L (2) n(L’) L’, for all n (3) For ‘almost every’ w in L’ , n(w) is in L, for some n

• Note: If p(L) = L, but (L) L then (Lp) = Lp where Lp = {L, (L), 2(L) … p-1(L) }. And also Lp is regular. Hence, we may assume L is a fixed point and not p-periodic. The attractor is then not necessarily spatially homogeneous at each time step. It is NOT a single spatial domain, but rather a union of spatial domains each of which is periodic in time.

Comments on Attractor Definition

• This definition ensures that domain grows instead of shrinking in time at the domain/other stuff interface.

• Finite time collapse onto (NOT close to) the attractor is different for CAs then in spatially continuous systems such as DEQs or 1d-maps because you can’t ‘get within ’ without being equal, due to discreteness.

Comparison of ECAs

• Can (sort of) characterize behavior of an individual ECA.

• Can we compare the behavior of two different ECAs and measure how similar their dynamics are? And How?

• For example, in what sense is ECA 9 similar to ECA 25 (and how similar)?

Basic Strategy• Consider only asymptotic spatial patterns. • Ignore particles, transient dynamics, and even

temporal patterns. • Compare ECAs based only upon the domain

machines M1,M2. Create ‘machine metrics’. ** Note: This is now a somewhat more general

question because such metrics could be used to compare -machines for other types of processes as well.

#18 (0W)*

#54 (1110)*

#54 (0001)*

#160 (0)*

Distinguishing Between Sources I

Distinguishing Between Sources II

#18 (0W)*

# 80

(1,0,W)*

RR-XOR

Machine Metrics I

dn,1(M1,M2) = w |p1,n(w) - p2,n(w)|

dn,2(M1,M2) = (w |p1,n(w) - p2,n(w)|2)1/2

dn,(M1,M2) = Max |p1,n(w) - p2,n(w)|

Let M1, M2 be two machines with corresponding languages L1, L2 and Let p1,n(w), p2,n(w) be the probability mass function of words of length n for the languages L1, L2. We define …

Consider weighted Averages:

D(M1,M2) = n dn(M1,M2)*n , 0 < < 1

Problems with Lp Metrics

• dn,(M1,M2) 0, as n and dn,2(M1,M2) 0, as n

for M1, M2 with h(M1) > 0, h(M2) > 0.

• dn,1(M1,M2) 2 as n for any M1, M2 with h(M1) h(M2)

** Note: 2 is the maximum value for any of these metrics.**

The Hausdorff Metric

Let (X,d) be a metric space, define the Hausdorff metricbetween compact subsets of X by

(A,B) = Max Min d(a,b)

(B,A) = Max Min d(a,b)

• dH(A,B) = Max {(A,B) , (B,A) }

a

a b

b

Examples

(A,B)

A B

(B,A)

dH(A,B) = (A,B) = (B,A)

A

B

(A,B)

dH(A,B) = (A,B), (B,A) = 0

Machine Metrics II Let M1, M2 be two machines with corresponding languages L1, L2.

Hausdorff ‘metric’ on length n words

dn,H(M1,M2) = dH(L1,n, L2,n)

Averaged Min Distance “metric” on length n words

(M1,M2) = (w1 min d(w1,w2))/|L1,n|

(M2,M1) = (w2 min d(w1,w2))/ |L2,n|

dn,A(M1,M2) = Max {(M1,M2) , (M2,M1) }

** Can take weighted averages or lim n dn,H(M1,M2) **

w1

w2

Example - ECA 18 and 3 periodic Domains

#18 (0W)*

#54 (1110)*

#54 (0001)*

#160 (0)*

Distance to ECA domain (0W)*(vs. periodic domains)

d-1 d-H d-AMD

0* 0.966 0.5 0.254

(0001)* 0.952 0.2 0.156

(1110)* 1.0 0.7 0.446

Example - ECA 18 and 3 non-periodic Domains

#18 (0W)*

# 80

(1,0,W)*

RR-XOR

Distance to ECA domain (0W)*(vs. non-periodic domains)

d-1 d-H d-AMD

Rule 80 0.909 0.3 0.115

(10W)* 1.0 0.3 0.229

RR-XOR 0.926 0.4 0.187

ECA 25 vs. ECA 9

25

9

d1 = 0.6

dH = 0.2

d-AMD = 0.08

RR0 vs. RR-XOR

RR0

RR-XOR

d1 = 0.883

dH = 0.2

d-AMD = 0.092

Metric Correlations