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)* 1 0,1 A B A B 1 D C 1

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? (3) Use T’ to construct M’ = [T]out 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.

Distinguishing Between Sources I #54 (1110)* #54 (0001)* #160 (0)*

Distinguishing Between Sources II # 80 (1,0,W)* RR-XOR

Machine Metrics I 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 … 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)| 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 metric between 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 b b a

Examples A B A B (A,B) (B,A) (A,B) dH(A,B) = (A,B) = (B,A)

Machine Metrics II Hausdorff ‘metric’ on length n words 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) } w2 w1 ** Can take weighted averages or lim n   dn,H(M1,M2) **

Example - ECA 18 and 3 periodic Domains #18 (0W)* #54 (1110)* #54 (0001)* #160 (0)*

Distance to ECA domain (0W)* (vs. periodic domains) 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-H d-AMD Rule 80 0.909 0.3 0.115 (10W)* 1.0 0.229 RR-XOR 0.926 0.4 0.187

ECA 25 vs. ECA 9 d1 = 0.6 dH = 0.2 d-AMD = 0.08 25 9

RR0 vs. RR-XOR d1 = 0.883 dH = 0.2 d-AMD = 0.092 RR0 RR-XOR

Metric Correlations