## Use Smoothiness to classify rules

For discussion of other cellular automata.

### Re: Use Smoothiness to classify rules

I tried to calculate eigval on a sample of similar sum-cov. Not sure whether eigval provides enough resolution. (I don't really know what I am doing now... )

The rulelist is as follows
`ind   rulename   sumcov---------with gliders1   B01345/S1235   0.09121475022   B0126/S025   0.12994891893   B0136/S013   0.15930214914   B012346/S12347   0.12040681965   B01246/S2345   0.18547128666   B3578/S2467   0.05226332367   B0246/S1234678   0.06239306558   B0123478/S13456   0.10885423989   B023/S3   0.1159036992---------without gliders10   B3568/S457   0.357853111811   B478/S0124568   0.091805374112   B47/S12356   0.06502312813   B4678/S2346   0.058118858814   B46/S34578   0.214424111115   B458/S12358   0.081449009716   B458/S03467   0.183311543417   B4578/S02357   0.135610447318   B457/S13458   0.11594388819   B457/S125678   0.043689915720   B4568/S14678   0.50890888121   B4568/S01456   0.210066344922   B4568/S012468   0.06189322323   B456/S2457   0.378556375824   B456/S13468   0.127147088425   B45/S34678   0.438327124326   B45/S13467   0.168828767627   B45/S023578   0.087322663728   B38/S2458   0.198801978629   B38/S126   0.174773281630   B38/S1258   0.148597694731   B38/S1235   0.05052525432   B38/S0137   0.112310909133   B37/S1258   0.177464218734   B37/S03467   0.182038973935   B368/S1268   0.17731840836   B368/S024678   0.154319740637   B368/S01456   0.099416722538   B3678/S234678   0.176231768439   B3678/S014678   0.21285952440   B367/S34678   0.2414952442`

Another run of statistics with eigval-wise entropy added.
`ind   rule  sumcov     ent-eigval effective gennum1   B01345/S1235   0.013576356   0.682650437   1222   B0126/S025   0.0157049499   0.043839301   1133   B0136/S013   0.0131982053   0.1878726607   624   B012346/S12347   0.0140107965   0.341077261   47.55   B01246/S2345   0.0138134507   0.0563178535   876   B3578/S2467   0.0170394302   0.984168541   1227   B0246/S1234678   0.0151378362   0.8287909057   1228   B0123478/S13456         9   B023/S3   0.0151171728   0.2377043381   12210   B3568/S457   0.0149634938   0.0449841591   6011   B478/S0124568   0.0142574139   0.1374848235   4412   B47/S12356   0.0161531784   0.1395905735   7813   B4678/S2346   0.0143078978   0.1940198251   9114   B46/S34578   0.014481117   0.163092486   82.515   B458/S12358   0.0141966238   0.0601296873   10416   B458/S03467   0.0140630358   0.0461868486   5817   B4578/S02357   0.0141081254   0.0445851141   6318   B457/S13458   0.0147691258   0.0657322013   12219   B457/S125678         20   B4568/S14678         21   B4568/S01456         22   B4568/S012468   0.0185119068   0.2313700629   12223   B456/S2457   0.0164543766   0.2653815401   112.524   B456/S13468   0.0174584745   0.5196705014   12225   B45/S34678   0.0137175951   0.0496711468   6926   B45/S13467   0.0161772229   0.0750878961   12227   B45/S023578         28   B38/S2458         29   B38/S126   0.0196320049   0.2153741964   5930   B38/S1258   0.0134766179   0.2583403383   11931   B38/S1235   0.0175393482   0.4831116569   83.666666666732   B38/S0137   0.0160080466   0.3347883123   9433   B37/S1258   0.0163332189   0.1827366728   10034   B37/S03467   0.0135733608   0.0483822069   5835   B368/S1268   0.0174716418   0.1723090065   5936   B368/S024678   0.0150004046   0.0381971307   7837   B368/S01456   0.015339473   0.1365470068   12238   B3678/S234678   0.0133595364   0.0195362846   4639   B3678/S014678         40   B367/S34678   0.0144942872   0.1663931507   83.5`
Attachments
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Last edited by shouldsee on August 12th, 2016, 6:07 am, edited 2 times in total.
shouldsee

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### Re: Use Smoothiness to classify rules

shouldsee wrote:Not sure whether eigval provides enough resolution.

It does not even discriminate:
3.png (17.1 KiB) Viewed 4989 times

By the way, how do you determine whether a rule has a glider or not?
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Bullet51

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:
shouldsee wrote:Not sure whether eigval provides enough resolution.

It does not even discriminate:
3.png

By the way, how do you determine whether a rule has a glider or not?

By manually inspecting it or sending it to apgsearch. I never really tried to use other scripts to search.

Would be glad if anyone has recommendation.

BTW, I think calculating eigenvalue is largely related to principal component analysis. We can check the largest PC's to confirm whether they are due to wave conversion or not.

A possible source for the observed large PCA/eigenvalue is the phase transition/wave conversion itself, which would cause a vectorial shift in the distribution of neighbor counts, which in turn is captured by the co-variation analysis. In this case, the largest PCA is simply a measure of time.
shouldsee

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### Re: Use Smoothiness to classify rules

Proof of concept:
PCA of neighborhood counts distribution can reflect the vectorial evolution of a universe. But more generally, it serves to visualise the phase space of a universe, for PCA's ultimate advantage is to capture variation.

Since PCA is sensitive to variation, I have smoothed the time series by a window of 6 to prevent strobing effect plagued the PC's.

A small collection of similar plots can be found here. (too big to attach)
Attachments
B012346S12347_soup12.jpg (229.66 KiB) Viewed 4862 times
B012346S12347_soup11.jpg (213.04 KiB) Viewed 4862 times
Last edited by shouldsee on August 12th, 2016, 5:02 pm, edited 1 time in total.
shouldsee

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### Re: Use Smoothiness to classify rules

shouldsee wrote:By manually inspecting it or sending it to apgsearch. I never really tried to use other scripts to search.

Would be glad if anyone has recommendation.

Gsearch may be useful:
`x = 3, y = 3, rule = B38/S24583o\$b2o\$2bo!x = 4, y = 5, rule = B38/S126bobo\$o2bo\$o\$o2bo\$bobo!x = 4, y = 5, rule = B38/S1258b3o\$ob2o\$bo\$b2o\$b2o!x = 4, y = 5, rule = B38/S0137o\$2o\$bobo\$b3o\$3bo!x = 4, y = 5, rule = B37/S12582bo\$bo\$2o\$b3o\$2bo!x = 5, y = 9, rule = B37/S034672b3o\$3b2o\$3bo\$bo\$o2bo\$bo\$3bo\$3b2o\$2b3o!x = 4, y = 5, rule = B368/S1268obo\$o2bo\$3bo\$o2bo\$obo!x = 5, y = 4, rule = B368/S024678o3bo\$2b3o\$2b3o\$o3bo!x = 5, y = 5, rule = B367/S346783bo\$3b2o\$3b2o\$3obo\$b3o!x = 4, y = 4, rule = B3678/S234678bo\$ob2o\$2b2o\$2b2o!x = 7, y = 5, rule = B3678/S014678ob2o\$3obo\$obobobo\$3obo\$ob2o!x = 9, y = 8, rule = B368/S014562bo3bo\$ob2ob2obo\$bob3obo\$4bo\$bo5bo\$3bobo\$3bobo\$3b3o!`

David Eppstein shows that most of the rules which is "essentially-not-glider-impossible" does have a glider.
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Bullet51

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### Re: Use Smoothiness to classify rules

shouldsee wrote:A possible source for the observed large PCA/eigenvalue is the phase transition/wave conversion itself, which would cause a vectorial shift in the distribution of neighbor counts, which in turn is captured by the co-variation analysis. In this case, the largest PCA is simply a measure of time.

I'm not sure what "wave conversion" means. Maybe global fluctuations of the dynamics? If the variation of neighbor counts is mostly caused by global fluctuations, all neighbor counts will correlate, thus leading a full-1 co-variation matrix.(This is exactly the case of class 1 rules.)
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Bullet51

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:
David Eppstein shows that most of the rules which is "essentially-not-glider-impossible" does have a glider.

Thanks for recommending gsearch. Yes constructing gliders manually would be possible in many rules, with low spontaneous occurrences. This in turn implies that glider constitutes either a small attractor basin or simply an unstable basin. My older interest is mainly in rules where you get gliders emitted from random soups at an acceptable frequency. This aim is gradually being replaced by characterising the cooperative/larger-scale behaviour in the CA universe. While spontaneous glider does hallmark such behaviour, other non-glider behaviour should also be included.

Imagine we have a constructed non-spontaneous glider, if what it uses are all common transitions, then it should be spontaneously emerging to some extent. One reason why it's non-spontaneous is that this local configuration is somehow prohibitive, for example it might uses a rare transition. Other possibilities exist. This local configuration might be impossible around the edge of the chaos so that the glider cannot escape chaos. Or the chaos-order edge itself has a strange topology (where the CO edge is an exploding ring between two ordered area) that disallows any observation.

Anyway, merely existence of glider seems not to be too good a criteria for describing the emergence of a rule. We should seek some other definition.
shouldsee

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:
shouldsee wrote:A possible source for the observed large PCA/eigenvalue is the phase transition/wave conversion itself, which would cause a vectorial shift in the distribution of neighbor counts, which in turn is captured by the co-variation analysis. In this case, the largest PCA is simply a measure of time.

I'm not sure what "wave conversion" means. Maybe global fluctuations of the dynamics? If the variation of neighbor counts is mostly caused by global fluctuations, all neighbor counts will correlate, thus leading a full-1 co-variation matrix.(This is exactly the case of class 1 rules.)

I had triphasic dynamics in mind when speaking of "converstion wave".
These are perfect examples where one group of neighborhoods taking territory from another group.
triphasic
`x = 22, y = 24, rule = B045678/S01568:T100,1002o4b4o2b2obob2o2bo\$b3ob2o2b2ob3ob2obo\$ob3obobo2b2o3bob3o\$b2obo2b2o5b3o3b2o\$5b2obo4b3obobo\$bo2bo3bo9bobo\$2obo2bob3obobob4obo\$b2o5b4o4b2ob2o\$b2ob2o5bo3b3o2bo\$o2b8obo3b3ob2o\$o3b2o5bobo3bob2o\$5bo5b4obob3o\$2ob2obo3bobobo4b2o\$3o2bobo2b5obobo2bo\$2obob2obobob2o4bo2bo\$3ob2ob3o3bobob2o\$7obo2bo5b5o\$ob2o3b6o2bobo\$b3o5b3ob6ob2o\$b3o2b3o2b3obo3b2o\$3b2o3bobo3b2obo3bo\$bo2b2o3b5o3b2ob2o\$2o2bobo4bob2ob6o\$o2bobob2o2b2obo3bo2bo!`

`x = 32, y = 32, rule = B013/S1236:T100,1002bo4bobo2bobo2bo3bob3o3b2o\$o3b5o5b2o2b2o2b2obob2obo\$4bobo2b2o2bo2b2o2b4o3bo3bo\$b2ob3o2bob8obobob4o2bo\$4ob3obob2o4bob4o2bobob3o\$obo2b6o4bob2o5b2o2b2o\$10b2o3bo4bo2bo3b2o\$5b2obo2bob3obob2ob2ob4o\$2b3o4b2o2b4obo2b5ob3o\$b2obob3o2bo3b2ob6o3b3o\$2o2bo4bobobobo6b4o2b2o\$2o4b2obob3o3b4o3b5o2bo\$o3bo2b2ob3o2b3o4b3o5bo\$bo2bobob3o2b3o3bo4b2ob2o2bo\$o2bo3bobo2b2obo2b5o2bo2b2obo\$ob3ob2o3bobobo4bo2bob2o2b2o\$ob2o4bo4b2o4b2o6bo2bo\$obo3b4o2b2ob7o2b2o2b2o\$ob2obobo4bo2b2o3b5ob2ob3o\$o2b4o7bobo3bo2b3o2b2obo\$4bobobo5bob5o3bo2bo3bo\$b3o5b2o3b3o2b3o6bo\$bob6o4bobo2bo5bob2o2b2o\$o5b3o5bobob2obob2o4bobo\$bo2bo2bobo3bobob3o3b4o2b2o\$2obobob2o4bobob3o2b2ob3ob3o\$bo2bo5b3o5b2obo4b3o2bo\$obo2bo2b2o3bo2bo2b3ob2o5bo\$5b3obob2o3bo2b4ob3obo\$2bo2b3obo3bob2obob2obob2o3b2o\$2o3b2ob2o4b3o2bobob3o3bo\$ob2obo2bo2bo3b5o2bo2bo2bo!`
shouldsee

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### Re: Use Smoothiness to classify rules

It seems that we should use statistics not sensible to variations, maybe some kind of local statistics.
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Bullet51

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:It seems that we should use statistics not sensible to variations, maybe some kind of local statistics.

I plan to try renormalisation at some point. CA is a perfect substrate for analysis based on different scales.

To start, we need to construct statistics at different scales. Then record their change as scale changes. Then we can get a feeling about how emergent is the rule. sum-cov is my first attempt here, and entropy of eigenvalues maybe an alternative. Once we have a statistics reliable enough, we can throw them onto different scales to characterise the emergence.
shouldsee

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### Re: Use Smoothiness to classify rules

HI guys,

I want to introduce "neighbourhood(NH) transition" based on the neighbourhood counts mentioned earlier. Say we have a cell (i,j). S(i,j) denotes the current state of (i,j), N(i,j) denotes the neighborhood of S(i,j). The updating rule can be viewed as S(t+1)=u[N(t)] where u is the updating function/mapping.

Now since we are not merely interested in updating the universe, we want to measure something out of it, especially about its temporal evolution. One way is to construct Poincare maps in the form of (A(t),B(t+1)), where A and B can be anything. Since (N(t),S(t+1)) is fixed under u, it will not be informative. But since (S(t),S(t+1)) and (N(t),N(t+1)) are not fixed, they can somewhat characterise its dynamics.

Under B/S notation, S(t) can only take two states,0/1, but N(t) can take 9*2=18 states. The resolution in N(t) return map thus allows further analysis.

Preliminary result is quite encouraging. The time-averaged return map is calculated for 20*20 torus over a maximum generation of 112. There are many hotspots/strips in the N(t) return map. I also symmetricised to make them fancy. More plots can be found here.

Kind regards
Feng

PS: As BlinkerSpawn has implied earlier,low resolution of co-variation between N(t) counts can be improved by splitting N(t) into smaller categories. The approach presented here, however, expand N(t) into temporal domain to be (N(t),N(t+1)) at a cost of O(N^t). We should manage to live through this. : )

PPS:I also remembered some other paper applying Markov model on (Sn(t),Sn(t+1)) where S is local configuration of size(n,n), which is similar to this approach here.

PPPS: The directed and undirected NHT are both mean(log2(counts)) transformed .

PPPPS: The very weird part about NHT plot is that, for a specific output NH, there exists a range of possible input NH. But not vice versa. This is very counter-intuitive, cause we would expect input-NH to put some constraint on the output-NH, not the opposite. I am still trying to reconcile this observation at the moment.
EDIT: This problem has been resolved by doing a left-right merge.
(!!)EDIT: That perception above was wrong. The NHT matrix was x-y flipped to cause the illusion mentioned above. The correct NHT matrix is NOT counter-intuitive!!
Attachments
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B4678S2346_soup38.jpg (177.89 KiB) Viewed 4862 times
Last edited by shouldsee on August 12th, 2016, 5:04 pm, edited 4 times in total.
shouldsee

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### Re: Use Smoothiness to classify rules

shouldsee wrote:I want to introduce "neighbourhood(NH) transition" based on the neighbourhood counts mentioned earlier.

That's a pretty good measure. Some more statistics can be extracted from it:
The diagonal count on S transitions shows how stable a rule is.
The diagonal count on B transitions may be an evidence of stable attractors.
What about the chaoticity of a rule?
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Bullet51

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:That's a pretty good measure. Some more statistics can be extracted from it:
The diagonal count on S transitions shows how stable a rule is.
The diagonal count on B transitions may be an evidence of stable attractors.
What about the chaoticity of a rule?

Do you plan to average your stats over time?

Diagonal count could be due to a return period of 1, or a first order stationary point. But be cautious here, there might be 2nd or 3rd or higher order stationary points that are scattered on the NHT plot, that form a closed (or nearly closed) cycle over time.

Moreover, under the framework of NHT, it's possible to throw techniques from continuous non-linear dynamics to characterise.
shouldsee

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### Re: Use Smoothiness to classify rules

shouldsee wrote:
Bullet51 wrote:That's a pretty good measure. Some more statistics can be extracted from it:
The diagonal count on S transitions shows how stable a rule is.
The diagonal count on B transitions may be an evidence of stable attractors.
What about the chaoticity of a rule?

Diagonal count could be due to a return period of 1, or a first order stationary point. But be cautious here, there might be 2nd or 3rd or higher order stationary points that are scattered on the NHT plot, that form a closed (or nearly closed) cycle over time.

Do "n-th order stationary points" correspond (roughly) to cells inside common period-n patterns/subpatterns?
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### Re: Use Smoothiness to classify rules

Do "n-th order stationary points" correspond (roughly) to cells inside common period-n patterns/subpatterns?

I have absolute no idea at this stage.

The most striaghtforward test is take a NHT plot of an oscillator.

But more generally, we are interested in near-stationary. We want things to return, but not exactly to the same point, allowing for more diversity.
shouldsee

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### Re: Use Smoothiness to classify rules

BlinkerSpawn wrote:Do "n-th order stationary points" correspond (roughly) to cells inside common period-n patterns/subpatterns?

Yes.
A result of the blinker-rich rule B3S26:
1.png (5.86 KiB) Viewed 4913 times

EDIT：The B3->S3 transition should be S1->B3
Last edited by Bullet51 on August 12th, 2016, 9:56 am, edited 1 time in total.
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Bullet51

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:
BlinkerSpawn wrote:Do "n-th order stationary points" correspond (roughly) to cells inside common period-n patterns/subpatterns?

Yes.
A result of the blinker-rich rule B3S26:
1.png

Let's do a comparison between this and chaos in B3/S26

Actually, would you mind post 1D-ised pattern along with the plot?
shouldsee

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### Re: Use Smoothiness to classify rules

shouldsee wrote:Let's do a comparison between this and chaos in B3/S26

What do you mean by chaos in B3/S26? I cannot get the idea.
shouldsee wrote:Actually, would you mind post 1D-ised pattern along with the plot?

I'm sorry about that, since I'm using a 1000x1000 grid.
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Bullet51

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:
shouldsee wrote:Let's do a comparison between this and chaos in B3/S26

What do you mean by chaos in B3/S26? I cannot get the idea.
shouldsee wrote:Actually, would you mind post 1D-ised pattern along with the plot?

I'm sorry about that, since I'm using a 1000x1000 grid.

What a gigantic universe lolllll!! I am only doing 30*30.. You computer-rich guys :/..

I meant 'chaos' as the transient pattern in the dying soup.

BTW, I tried a left-right merge of the NHT graph, this get rid of the suspicious blanks. It's possible to do a further up-down merge. The NHT collection has been updated accordingly.
Last edited by shouldsee on August 12th, 2016, 5:05 pm, edited 1 time in total.
shouldsee

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### Re: Use Smoothiness to classify rules

shouldsee wrote:I meant 'chaos' as the transient pattern in the dying soup.

Sure.
1.png (6.57 KiB) Viewed 4894 times
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Bullet51

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:
shouldsee wrote:I meant 'chaos' as the transient pattern in the dying soup.

Sure.
1.png

Did you not take the log of the counts? The colors are too faint here.
shouldsee

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### Re: Use Smoothiness to classify rules

shouldsee wrote:
Did you not take the log of the counts? The colors are too faint here.

It's not so convenient to compare the colors in a log scale.At least for me...
1.png (7.13 KiB) Viewed 4890 times
Still drifting.
Bullet51

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:
shouldsee wrote:
Did you not take the log of the counts? The colors are too faint here.

It's not so convenient to compare the colors in a log scale.At least for me...
1.png

You can try to change the colormap if you don't like the current one. Maybe try 'colorcube' or 'prism'. Log-scaled is more informative.

Also, I am not sure LR merge is really a helpful idea. It appears easier to do cobweb plots on undirected NHT plot.
shouldsee

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### Re: Use Smoothiness to classify rules

A somewhat difficult question: How much information can we extract from the transition matrix?
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Bullet51

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### Re: Use Smoothiness to classify rules

Bullet51 wrote:A somewhat difficult question: How much information can we extract from the transition matrix?

Need to try measure the degree of symmetry of NHT. We need to find dynamic with symmetry over time
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