computational mechanics of ecas, and machine metrics
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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