Use Smoothiness to classify rules

For discussion of other cellular automata.
shouldsee
Posts: 406
Joined: April 8th, 2016, 8:29 am

Re: Use Smoothiness to classify rules

Post by shouldsee » August 26th, 2016, 6:09 pm

Hi guys,

I made another attempt to classify rulespace by calculating spatial mutual information. The idea is that if entangling is present, neighboring cells should assciates with each other by sharing mutual information. During the calculation, I assign a new state to each cell according to its 3-gen history (2^3=8 possible states). Then I calclulate the mutual info between 2 neighboring cells (1 on the left, 1 on the right) at each step. The entropy at each step is also included. An additional input entropy is recorded to further resolve the rulespace.

From the result, it's pretty clear that complex rules exhibit an excess of spatial mutual information.

## matlab 3d figure
Attachments
mi_rulespace_2d.jpg
mi_rulespace_2d.jpg (35.53 KiB) Viewed 7592 times
mi_rulespace.jpg
mi_rulespace.jpg (33.33 KiB) Viewed 7592 times

Bullet51
Posts: 536
Joined: July 21st, 2014, 4:35 am

Re: Use Smoothiness to classify rules

Post by Bullet51 » August 28th, 2016, 6:14 am

shouldsee wrote:The idea is that if entangling is present, neighboring cells should assciates with each other by sharing mutual information.
There seems to be some issues on biphasic rules like this:

Code: Select all

x = 60, y = 60, rule = B4678/S35678:T60,60
13bo3b22o$9bob28o$7bob30o$5bob33o$2b38o$42o5b7o2b2obo$43o2b15o$60o$60o
$60o$60o$60o$60o$60o$60o$60o$60o$47obo2bobobob3o$45o13b2o$42obo15bo$
40o19bo$ob38o$4b33obo$6b28obo$6b24ob2o$7b2ob19o$11b17o$13bob13o$16b11o
$20b3o4$53b4o2bo$o51b8o$3o48b9o$3o46bob9o$3o44bob11o$4o43b13o$4o40b16o
$6o37b17o$6o37b17o$8o33b19o$8o4bo27b20o$15o25b20o$15ob3o18bob21o$23o
13b24o$24o11b25o$24o11b25o$23o11b26o$15ob2o2b2o11b27o$10o22b28o$obobob
ob2o17b2o2b28o$25b26ob3o$23b23ob3o$22b23o$22b23o$20b24o$20b21obo$18b
22o!
The entropy for each cell is 1 bit. Since there is strong "entangling", the mutual information for two adjacent cells is also ≈1 bit. Such data may lead to results that the rule is somewhat complex, but in fact, it's just a voter rule.

The issue also apply to B0 rules:

Code: Select all

x = 60, y = 60, rule = B014/S2:T60,60
20b3o$20b4obo$19b7o$13b2o3b9o$17b12o$15b15o4bo$15b15ob3o2bo$15b17ob4o$
14b25o14b2o$14b24o$13b26obo11b2o$12b29obo8b2o$12b31o7b4o$10b34o5b3obo$
9bob41obo$8b44obo$9b43obo$2bo5b45o3bo$3o4b27ob10obo3b4o2b4o$32obo2bob
2ob3o8b8o$31o9b3o10b7o$30o11bo11b7o$29o21bob8o$28o21b11o$29o21b10o$29o
21b10o$28o22b10o$28o11b2o9b10o$29o9b3o9b10o$26obo11b4o7b10o$25o12bob3o
7b11o$25obo9bob5o8b9o$26o8bo2b5o9b9o$27o5bob7obo6b11o$28o5b9o3bo4b10o$
45obo2b11o$45o3b12o$47ob12o$45o2b13o$60o$60o$19o2b39o$7ob10o4b38o$5ob
2ob8o6b2ob34o$6o4b8o10b32o$3o6b8o12b9o2b20o$obo8b7o10b9o4b19o$11b8o11b
6o5b18o$12b7o11b5o5b18o$12b9o9bob2o6b19o$12b8o13bo6b17o$11b10o19b17o$
11b10o18b17o$12b11o18b15o$12b10o19bob13o$12b11o19b15o$12b11o21b7o4bo$
13b10o23b4o$13b3ob6o24bo$19b4o!
Still drifting.

shouldsee
Posts: 406
Joined: April 8th, 2016, 8:29 am

Re: Use Smoothiness to classify rules

Post by shouldsee » August 28th, 2016, 7:54 am

Bullet51 wrote: The rule B378S126 may be a good sample to test. How many states are there near Bunc=0?
Good news. With my newest criteria, I have successfully detected B3578/S126 as complex rule.
Bullet51 wrote: The entropy for each cell is 1 bit. Since there is strong "entangling", the mutual information for two adjacent cells is also ≈1 bit. Such data may lead to results that the rule is somewhat complex, but in fact, it's just a voter rule.
I agree that the spatial domain of complex behaviour is an important factor. We are naturally interested in glider-behavior on a blank background, and it's true that complex behavior on a ring is not as interesting.

Hi guys,

I want to announce my most recent attempt. Instead of calculating mutual information between spatial neighbors, this time I measured mutual information between neighbors in time, i.e. between history states of a cell. The input NH of each cell is recorded over time to represent a random variable. H(NH_0) is the input entropy of current universe. H(NH_-4) is the entropy of 4 steps back. H(NH_0,NH_-4) is the joint entropy between distribution of current universe and 4 steps earlier. I(NH_0,NH_-4) is deduced to be H(NH_0)+H(NH_-4)-H(NH_0,NH_-4). An lag of 4 is taken here to reduce the periodicity in the statistics.

This mutual information comes out to be a better measure than the last one and achieve a better resolution. I plotted (H(NH_0),I(NH_0,NH_-4)) for each soup evolution. To extract stats, mean, total variance and covariance are calculated. During the final visualisation of rulespace, mean and covariance is plotted.

It can be seen clearly from the result that the complex rules occupy a region with reduced temporal mutual information (tMI_input), and a small covariance. This is consistent with the fact that successive steps of complex rules are usually distinguishable, while those of chaotic rules are indistinguishable (contains the same information over and over). A small covariance implies that H_input and tMI_input are decoupled/ some of them is too stable to show any coupling.

Apart from the complex rules, other regions are also indicative of the behavior of corresponding rules. The line of H=I implies H is completely redundant and no new information is generated over time, i.e. the information content decays over time. The chaotic core near H=4,I=0 corresponds to that no information is passed on over time, all information are completely new. Under such definition, complex rules pass on some existing information to the next generation, while generating some novel information as well.

Other features include a curve on the H=I plane, and a dense region connecting chaotic core to H=I plane. I can't interpret any of them yet.

EDIT: The second feature is seemingly related to the fact that emergent behavior is more abundant on an irregular background than a regular background.
Attachments
tMI_rulespace.jpg
tMI_rulespace.jpg (63.8 KiB) Viewed 7544 times
all points here has a Covariance<0.25
all points here has a Covariance<0.25
tMI_rulespace_2d_thresholded.jpg (64.32 KiB) Viewed 7544 times
tMI_rulespace_2d.jpg
tMI_rulespace_2d.jpg (75.77 KiB) Viewed 7544 times

Bullet51
Posts: 536
Joined: July 21st, 2014, 4:35 am

Re: Use Smoothiness to classify rules

Post by Bullet51 » August 30th, 2016, 2:15 am

There seems to be some issue on the cov statistic: Different H-I behavior may yield similar cov values.
H-I graph of B3/S125
H-I graph of B3/S125
6.png (4.06 KiB) Viewed 7520 times
H-I graph of B3/S2368
H-I graph of B3/S2368
B3S2368.png (4.94 KiB) Viewed 7520 times
Is there any statistics that can capture the non-linear behavior of the H-I graphs?
Still drifting.

shouldsee
Posts: 406
Joined: April 8th, 2016, 8:29 am

Re: Use Smoothiness to classify rules

Post by shouldsee » August 30th, 2016, 4:52 am

Bullet51 wrote:There seems to be some issue on the cov statistic: Different H-I behavior may yield similar cov values.
Is there any statistics that can capture the non-linear behavior of the H-I graphs?
It's indeed a pain-in-the-ass to extract stats from a phase diagram.

First, you can try calculate total variance (trace of covariance matrix) along-side covariance. Secondly, you can apply a random walk model and yield some statistics from it. One way it to plot a return map according to H-I phase diagram, which would reflect the volume experienced by the trajectory.

On the other hand, I did try to apply MI as a distance between 2 different universe. This method is capable of resolving some linear/near-linear oscillation, but is very prone to noise.

I want to mention the first figure specifically. It's clear how the ring behavior masked the along-side complex behaviour in the return map/distance matrix.
Attachments
B012348S0236_soup57.jpg
B012348S0236_soup57.jpg (115.58 KiB) Viewed 7502 times
B3S23 _soup1.jpg
B3S23 _soup1.jpg (130.54 KiB) Viewed 7510 times
B012S1256_soup5.jpg
B012S1256_soup5.jpg (111.35 KiB) Viewed 7510 times

shouldsee
Posts: 406
Joined: April 8th, 2016, 8:29 am

Re: Use Smoothiness to classify rules

Post by shouldsee » September 1st, 2016, 8:00 am

Here are some snapshots of spatial mutual information. Each point is associated with a 2-d shift vector, and its color determined by the value of the mutual information associated. By summing the value near vector(0,0), one can estimate the local entangling at any instantaneous moment.

The problem of making such plot in 3D is that time is not bounded, and one can not easily shift a time-space pattern to itself.

Point (0,0) is manually centred in the plots.
Attachments
B3S23 _soup3.jpg
B3S23 _soup3.jpg (120.01 KiB) Viewed 7476 times
B0123456S0246_soup30.jpg
B0123456S0246_soup30.jpg (161.18 KiB) Viewed 7476 times
B012S1256_soup1.jpg
B012S1256_soup1.jpg (173.16 KiB) Viewed 7476 times

Bullet51
Posts: 536
Joined: July 21st, 2014, 4:35 am

Re: Use Smoothiness to classify rules

Post by Bullet51 » September 4th, 2016, 7:26 am

The H and I time series differ much from convectional time series, mainly from the auto-correlation profile:
Sample autocorrelation for B37S23
Sample autocorrelation for B37S23
2.png (5.26 KiB) Viewed 7448 times
Sample autocorrelation for B3S023
Sample autocorrelation for B3S023
3.png (3.64 KiB) Viewed 7448 times
Still drifting.

shouldsee
Posts: 406
Joined: April 8th, 2016, 8:29 am

Re: Use Smoothiness to classify rules

Post by shouldsee » September 4th, 2016, 7:55 am

Bullet51 wrote:The H and I time series differ much from convectional time series, mainly from the auto-correlation profile:
The attachment 2.png is no longer available
The attachment 3.png is no longer available
Indeed. The perservation of information is a central topic in CA.

I have recently develop the idea of "vicinity counting". That is, counting every pair of cells separated by a spatial-temporal distance of 1. Calculate this entropy to be H_vic. The mutual info follows as 2*H_x-H_vic. By taking MI of a spatial-temporal block,the dynamics is better analysed.

Edit: Vincinty counting can be done with or without directionality. For every base direction, there are two postivity. We can either only count along one direction of each vector (with direction), or along both (without direction).
Attachments
vic_rulespace_2d.jpg
vic_rulespace_2d.jpg (72.15 KiB) Viewed 7445 times
vic_rulespace.jpg
vic_rulespace.jpg (74.09 KiB) Viewed 7445 times

shouldsee
Posts: 406
Joined: April 8th, 2016, 8:29 am

Divergence of trajectories is dependent on the initial densi

Post by shouldsee » September 28th, 2016, 4:53 am

The divergence here is visualised by computing std(H_input) over a set/collection of configurations. The higher the std, the more diverged is this set.(I have just started measure theory so I am not sure whether this would count as a measure over this set). Here, I recursively take the divergence of a 500-point sample after each step of mapping, which started as a random collection with a fixed density p0.

One can choose to normalise the divergence by the mean(H_input) of the set. This needs more discussion.

From preliminary results, this measurish variable is capable of characterising the dynamics.

PS: The colormap of the pictures are not normalised yet.
PPS: The previous plots appear to be faulty. They are NOT fully fixed yet.
Attachments
B024S0123_soup19.jpg
B024S0123_soup19.jpg (130.91 KiB) Viewed 7341 times
B3S23_soup91.jpg
B3S23_soup91.jpg (135.9 KiB) Viewed 7341 times
B15S23478 _soup19.jpg
B15S23478 _soup19.jpg (128.2 KiB) Viewed 7341 times

wildmyron
Posts: 1274
Joined: August 9th, 2013, 12:45 am

Re: Use Smoothiness to classify rules

Post by wildmyron » December 7th, 2016, 2:30 am

I am curious how the rule below fits into the classification scheme you have been working on here.

There are no known gliders but it seems possible that there are gliders of a sort on some periodic tiling.

Code: Select all

x = 90, y = 90, rule = B345678/S012678:T90,90
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo$
90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
o$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obo$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobo$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobo$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobo$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobo$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobo$44o2b44o$o
bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobobobobobobobo
bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo$90o
$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobobobobobobo
bobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobobobobobobo
bobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobo$
90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
o$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obo$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobo$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobo$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobo$90o$obobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobo$90o$obobobobob
obobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobo$90o$obobobobobobobobobobobobobobobobobobobobobobobobobobobob
obobobobobobobobobobobobobobobobo$90o!
The latest version of the 5S Project contains over 221,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.

shouldsee
Posts: 406
Joined: April 8th, 2016, 8:29 am

Re: Use Smoothiness to classify rules

Post by shouldsee » February 13th, 2017, 8:24 pm

wildmyron wrote:I am curious how the rule below fits into the classification scheme you have been working on here.

There are no known gliders but it seems possible that there are gliders of a sort on some periodic tiling.
HI wildmyron,

Unfortunately I can't remember exactly how I undertook the classification earlier on. I remember heavily relied on the notion of mutual information which I had just touched upon. The algorithm was largely based on statistical inference and I am not quite sure whether they were really valid.

I might not able to re-classify them under the earlier notation (since I was hugely disappointed by them after realised that I can be quite biased when interpreting my result). The pleasure I had was limited to the discovery of new exciting rules containing gliders, but it was more like a manual annotation rather than a machine one. Thus, I regret to tell you I might not be able to redo the classification under the earlier framework (because it was too, too messy from my current perspective).

Apologies for not being helpful.
Feng
Last edited by shouldsee on February 13th, 2017, 9:01 pm, edited 1 time in total.

shouldsee
Posts: 406
Joined: April 8th, 2016, 8:29 am

Re: Use Smoothiness to classify rules

Post by shouldsee » February 13th, 2017, 8:56 pm

Meanwhile, I want to discuss some recent ideas I was testing on a ECA exploratory toolkit: expt branch

The story began with I spent some good time learning density matrix renormalisation group(DMRG) as posted by Nishino in 1995. I was particularly interested in its application to 2D-ising model. DMRG is fascinating in that it can deal with an infinite system with good approximation. But to me, the biggest gift is to become familiar with density matrix(DM) as a linear algebraic object. Importantly, DMRG uses an earlier technique called transfer matrix as introduced by Onsager, which allows one express partition function as a product of matrixes and thus a big matrix itself. Yvan pointed out that the eigenvector of a transfer matrix correspond to a stationary distribution (which hugely clarify my confusion).

I then tried to construct some statistical model as a layered state vectors, where adjacent layer interacts with an energy defined by how likely the lower layer is the image of higher layer under the projection of a cellular automata. The larger the difference, the higher the energy, and less likely the configuration would appear. A stringency constant was introduced (reciprocal of temperature, denoted K), As K->+Inf, only the trajectory produced by the specified CA will have a non-zero probability. The model respond to K in some obscured fashion. Strangely, the transfer matrix give a uniform distribution for any K. This means at any layer, no preference is given to any particular state. This is reasonable given that the partition for any layer is a sum over all possible choices of the lower layer, and this result of such operation does not depend on the identity of the higher layer. Well, that means we can't simply apply TM technique in this fashion.

During the process, I tried to marginalise over the higher layer and compute the entropy of the resultant distribution. Unfortunately back then I did not have reliable rule-generator. (Mostly of times I believed I was investigating rule 172 only to realise it actually is rule 110), so I have yet to test the usefulness of this entropy. The observation is this entropy does depend on K, unlike the stationary TM eigenvector.

More ideas came to me during the process:
1. Ising model effectively uses hamming distance as energy function. What would happen if we replace it with mutual information instead?

Accidentally, while I was searching for new ideas, I bumped into Derrida plot again. (I met it earlier in DDLab, but didn't understand it since the manual wasn't clear). This time with some concentration I was able to use it to classify 1D ECA to some extent. I hope this can be readily generalised to 2D. It's important to note these plots aren't strict Derrida plots. Derrida plots take average along the second dimension (marginalised), while here the whole joint distribution is displayed. In such way, more information is kept.

Here are some demos from the tool box. The observation is that, if there is strong correlation in the lower plot, then the underlying CA can't afford complex dynamics. The problem remains: a chaotic rule can't be distinguished from a complex rule. I plan to resolve the issue by abstracting a heatmap into a single number, then use this distance/correlation function to visualise the auto-correlation profile of the distance set.

Image
Image
Image
Last edited by shouldsee on February 14th, 2017, 7:11 pm, edited 1 time in total.

shouldsee
Posts: 406
Joined: April 8th, 2016, 8:29 am

Re: Use Smoothiness to classify rules

Post by shouldsee » February 14th, 2017, 7:02 pm

I've manged to adapt the aforementioned methodology, namely replacing the joint distribution with a correlation value. In this way, a metric is defined between two timepoints/horizons |t_1,t_2|=1-corr(hamm(t_1),hamm(t_2)), where hamm(t) is random variable denoting the hamming distance between 2 trajectories at time t.

I've managed to calculate an atlas of such plot for 256 ECA rules. Please see CorrProfile_ECA

I've also hand-picked chaotic/complex rules according to the profile (s for short dynamics, ss for supershort dynamics):

Code: Select all

25
30
32s
40s
41
45
54
57s
60
75
86
89
90
96ss
97
101
102
105
106
107
110
120
121
122s
124
126s
129s
135
136s
137
146s
147
149
150
151s
153
160s
161s
165
169
182s
183s
192ss
193
195
225

demos:
Image
Image
Image
Image
Image

wildmyron
Posts: 1274
Joined: August 9th, 2013, 12:45 am

Re: Use Smoothiness to classify rules

Post by wildmyron » February 14th, 2017, 10:07 pm

shouldsee wrote:
wildmyron wrote:I am curious how the rule below fits into the classification scheme you have been working on here.

There are no known gliders but it seems possible that there are gliders of a sort on some periodic tiling.
HI wildmyron,

Unfortunately I can't remember exactly how I undertook the classification earlier on. I remember heavily relied on the notion of mutual information which I had just touched upon. The algorithm was largely based on statistical inference and I am not quite sure whether they were really valid.

I might not able to re-classify them under the earlier notation (since I was hugely disappointed by them after realised that I can be quite biased when interpreting my result). The pleasure I had was limited to the discovery of new exciting rules containing gliders, but it was more like a manual annotation rather than a machine one. Thus, I regret to tell you I might not be able to redo the classification under the earlier framework (because it was too, too messy from my current perspective).

Apologies for not being helpful.
Feng
No problem,

I mentioned it as a curiousity as the rule seemed to exhibit some similar dynamics to other rules you are exploring, but I have no need for an answer to the question I posed. Thank you for responding.

Good luck with your ECA exploration and analysis
The latest version of the 5S Project contains over 221,000 spaceships. Tabulated pages up to period 160 are available on the LifeWiki.

shouldsee
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Re: Use Smoothiness to classify rules

Post by shouldsee » February 24th, 2017, 4:13 pm

shouldsee wrote:
Bullet51 wrote:
And how on earth did you find such amazing rules?
Thank you for you appreciation. The short answer is: I don't know. The discovery of checkerlife is merely an accident: I was trying to stabilise [B12e3eijkr4ejqrtwz5cn6-ce7e/S01c2in3aejkr4ejkqtyz5ckq6-a], and attempted various modifications, and some mod give up to puffers and guns.
The discovery of [B12e3eijkr4ejqrtwz5cn6-ce7e/S01c2in3aejkr4ejkqtyz5ckq6-a], however, is due to my ongoing screening of non-totalistic rules. The screen is semi-automated: I used a matlab script to generate random rules and plot their dynamics into a pairwise distance map (aka profile), along with its space-time trajectory beside it. These rules are then ranked according to their profiles (specifically, the average of the profile plot). Then I manually inspected the top ranked rules, with some criteria. Usually the most interesting rules are of top 50-300 of each 10000 rules, but it's largely empirical (As long as the profile is not extreme, there is always space for interesting dynamics). I would say there is certain logic behind this method but it's not formally established yet.

The script is included in the expt branch of CA_tfmat repository, it current supports both non-totalistic rules, totalistic rules, ECA and coupled logistic maps. The major script is derrida_general.m. Generate an example through samplesys.m to get familair with the idea. I also included manual annotation under CA_tfmat/Annotations.

Another important factor to consider is the vast rulespace of the non-totalistic rules. Whereas totalistic rulespace has 2^18 possibilities, nt-rules (nt for non-totalistic) has 2^102 possibilities. (1d semi-totalistic has 2^4 possibilities). Thus it's not too surprising that nt-rules exhibit a richer diversity than T-rules. (From another perspective, NT-rules are subject to less symmetry than T-rules).

Coming back to the methodology: The general derrida plot (GDP) is intimately related to the concept of the attractor basin. The GDP contained information about deviation at 2 time points: whether two closely related configs diverge, or two diverged configs coalesce. These relations can be treated statistically, and the observation is that the covaraince between deviation_t1 and deviation_t2 is an important indicator of the dynamics. The GDP merely listed the covariance for every (t1,t2) combination. Consider a rule with a strong early-onset attractor, then we will see an early intensive covariant region on GDP. If the rule project the configs into a small number of basins, the GDP will show an early plateau. There is much to reason about GDP, and I think it might even be possible to extract information about the dominant attractor basin from the randomly generated ensemble. Interestingly, if you start with a random soup in checkerlife, the interesting glider dynamics is absent. On one hand, one can argue that GDP captures more information about a rule than glider-based criteria do. On the other hand, GDP seem to diverge from my perception of dynamics -- two dyanmically different rule might show a similar GDP (Although this similarity is often limited to coarse level. i.e. if one expand the GDP into distribution heatmap, then there is still perceviable difference, but that precludes efficient ranking of rules).

Somehow I feel it's not appropriate to rank rules in a linear fashion. Instead, a hierarchial organisation maybe more appropriate. Take checkerlife as an example, there are 102 possible mutations one can make. Some mutations abolish the dynamics completely. while others only incur minor changes. Personally I am mostly intrigued by the self-organised gliders from randomness, but once the glider became somehow replicative/regenerative, it would fill up the space too rapidly to result in a gaseous phase. How can we describe these relations with the language of attractor basins? I don't know. But we can at least figure out some incremental changes along a mutation path. More specifically, the hamming distance between 2 given rulestring is distorted. Starting from B3/S23, B23/S23 changes the dynamics greatly with 1 bit fliped, whereas B368/S238 endures 3 fliped bits but bear greatly similarity to B3/S23. Thus it'd be great if we can find a universal way to quantify the dynamical difference between 2 given rules given their rulestring, or at least spot the important bits of a given rulestring. (obviously not all 102bits in a nt-rule is important)

The kind of image generated is like this (the color might be inverted depending on how the distance is defined, d=1-corr(x,y) or d=cov(x,y))Image

Additionally, I wrote a Chinese blog with some nice graphics embedded to illustrate the idea of generalised derrida plot.



I want to discuss yet another piece of experience I had: the easiest way to find a novel interesting CA is not with any program, but human brain. It's incredibly difficult to write a automated evaluation function, but it's incredibly easy to look at the dynamics and say: "Hmm, I think this is amazing". Take some boring rule, mutate the bits randomly, and there is a good chance you find a rule with spontaneous gliders in 10 iterations. The problem with NT-rule/T-rule is that, the rulespace is simply too large to be completely searched with human brain, and some complex rules posess a larger territory while some complex rules are very stringent about mutations.

Essential Elements for a glider:
1. A self-sustainable stable background/agar. (All 0's, zebrastrip, checkerboard).
2. A transmitting pattern.

General descriptor for a rule:
1. Behavior from a random soup:
Does the evolution leaves debris/still lives behind it?
Does the evolution leaves dislocation/defects behind it?
Is the evolution sparky?
Does the evolution create correlation between space?

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Re: Use Smoothiness to classify rules

Post by shouldsee » February 28th, 2017, 8:58 pm

An incomplete ATLAS of outer semi-totalistic rulespacem, with TCA_id indexed. Rules are grouped according to similarity in dynamics.
Generated with dstgraph.m in CA_tfmat/expt

PS: What's the best file format to exchange graph structure?
Image
tca_atlas.jpg
tca_atlas.jpg (218.71 KiB) Viewed 6693 times
Attachments
tca_atlas.fig.tar.gz
matlab fig in archieve
(113.44 KiB) Downloaded 117 times

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Re: Use Smoothiness to classify rules

Post by Apple Bottom » March 1st, 2017, 8:50 am

shouldsee wrote:PS: What's the best file format to exchange graph structure?
Graphviz is fairly universal (the file format, not the program), but there's a variety of other formats as well. Here's an overview from the Gephi website.
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Re: Use Smoothiness to classify rules

Post by shouldsee » March 2nd, 2017, 5:41 pm

visulisation with vis.js

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Re: Use Smoothiness to classify rules

Post by shouldsee » March 5th, 2017, 10:10 am

Snapshot of rules similar to B3/S23 (6152)
Edge with similarity higher than exp(-4) is retained.
Image
The applet can be found at here hi-def image at here

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Re: Use Smoothiness to classify rules

Post by shouldsee » March 5th, 2017, 7:13 pm

Guys, I'm detecting something special about 6152/B3_S23.
Firstly, adding B8 or S8 largely perserves its dynamic.
ie B3_S23 ~ B3_S238 ~ B38_S23 ~ B38S238

Secondly, these four rules are susceptible to B7 or S7 similarly. In other words, B3_S237 is slightly different from B3_S23, so is other 2 rules. And B3_S237's dynamics is also largely perserved when subject to B8 | S8.

Notably, B7|S7 changes the dynamics more than B8|S8.
Thus, I here coin B8|S8 as group A mutation, B7|S7 as group B mutation. And B3_S23 family has a hierarchial structure of "a2b2". This can be visualised in the following group plot. Thicker edges correspond to more similar dynamics.
6152_a2b2_phd.png
6152_a2b2_phd.png (154.56 KiB) Viewed 6615 times
When physics engine of vis.js is enabled, the graph relax to the following shape
6152_a2b2_ph.png
6152_a2b2_ph.png (192.74 KiB) Viewed 6615 times
Intriguingly, B3_S23 family is not the only rule family exhibit such phenomena. Other instances are available, like 131764, 185845, 193384. Unfortunately, their dynamics does not coincide with 6152/B3_S23. Nevertheless, such structure should deserve a deeper explanation.
193384_a2b2_phd.png
193384_a2b2_phd.png (197.62 KiB) Viewed 6615 times
Hierarchial structures other than a2b2 are also available, such as a1b3(195827) a1*(235503).

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Re: Use Smoothiness to classify rules

Post by shouldsee » March 8th, 2017, 7:26 pm

I have updated my git repository at shouldsee/CA_tfmat (expt branch), adding densi_fluct.m using density fluctuation to probe complex dynamics
it should be fairly straight-forward to run profile_script.m and profile_script_nt.m (You also need to define $repos in your environment variable to direct the output of pictures)
Accompanied is some examples from densi fluct. Note the algorithm is not particularly good at identifying glider, but it does give you sparky rules. The fluctuating rules are particularly intriguing.

Much more can be done about this algorithm, and it is superior to any of my previous implementation.

For example this one has 3 meta-stable dynamics.

Code: Select all

x = 172, y = 119, rule = B12a3en4cintwyz5aejky6-in7c8/S1e2cin3-akr4-eijkr5eijry6-a78
b2o4b2o4b2o2b4obobob2obob2ob2obobob2ob2obob2ob2ob2ob2obobob2ob4obob6ob
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obobo5b2o$bo2b3o3bobo2bobo14bobo7bobo4bobobo2bobo7b3o7bo3bo4b3o6b2o20b
2o9bo3b2o2b2o5bo5bo3bo3bo3bo2bo7bo$obob3obobo2b2obobo5b3obo2bobob2obob
2obob2o2bobobobobob2o3b4o4b3obobo4bob3obo3bo2bo3b4o8b2ob4ob2o4b3o2bob
2ob6o5b3ob3ob3ob3obob3obo$2ob6ob2o5b3o3b2obob2obobo2b2obo2b3o3bobo5b3o
3bobobobob2o5bo7b3obo4b2ob2ob4ob2o4b2o6bobo2b5ob3o2bo5b4ob6o5b3o2bobob
o2b3o$2bobobob2o2bo5bo2b2ob2obobobo3bo2b4obob3obo2b2o3bob2o2bo2b2ob2o
2bo2b7o2bo4bob5o4b2o5b3o4bo2bo2b2ob3o2bobob2ob2o4bo4b6obo6bobo3bobo$2b
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$bo2bo2bobo2b3o3b3obobob2ob2obo3b7o2bobobobobobo2bob2ob2ob4ob2o2bob5ob
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$o2b2ob6obo2b11obobobobob6obobob3obob3obo3b2ob4o2bobob2o2b3o2bobo2bob
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ob2ob3ob2obobobo3bob7ob14ob2obob5obobob4ob3obob4o2b2o2bob4o2b6o4bo$b2o
bob3o2bobob3ob9ob3obobobobo2bobob4obobo2b4ob7obobo2b2ob2o2bob3o3b4o2b
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obo2b2o2bo2b3ob3o$ob5o2bobobobob3obo3bob2o3b2ob3ob2ob5o3bob2ob3obob6ob
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BlinkerSpawn
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Re: Use Smoothiness to classify rules

Post by BlinkerSpawn » March 8th, 2017, 11:23 pm

shouldsee wrote:...we can at least figure out some incremental changes along a mutation path. More specifically, the Hamming distance between 2 given rulestrings is distorted. Starting from B3/S23, B23/S23 changes the dynamics greatly with 1 bit flipped, whereas B368/S238 endures 3 flipped bits but bears great similarity to B3/S23. Thus it'd be great if we can find a universal way to quantify the dynamical difference between 2 given rules given their rulestring, or at least spot the important bits of a given rulestring. (obviously not all 102bits in a nt-rule is important)
There's no real way to do this purely theoretically, because - as I'm sure you realized - the Hamming distance treats all bits equally when we know they should not be. (e.g. B0 definitely changes dynamics much more than S8)
However, in my mild experimentation with non-totalistic rules there is one thing I have noticed, that being cognizant of the relative frequency of certain neighborhoods gives one greater control over dynamics, which leads to my little rookie's suggestion:
Instead of expressing dynamic difference purely on the difference between rulestrings, weight the changed neighborhoods based on the frequency in which they appear naturally in the two rules. (I say check both rules to preserve transitiveness of the dynamic difference and give a more refined estimate)
This may be difficult for rules with large differences in dynamics that cause very disparate "transition profiles", but it looks like it could still be a better approach that a naive Hamming comparison.
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Re: Use Smoothiness to classify rules

Post by shouldsee » March 9th, 2017, 5:27 am

BlinkerSpawn wrote: There's no real way to do this purely theoretically, because - as I'm sure you realized - the Hamming distance treats all bits equally when we know they should not be. (e.g. B0 definitely changes dynamics much more than S8)
However, in my mild experimentation with non-totalistic rules there is one thing I have noticed, that being cognizant of the relative frequency of certain neighborhoods gives one greater control over dynamics, which leads to my little rookie's suggestion:
Hi BlinkerSpawn,

In a way, it is a good thing that bits contribute differently to the dynamics. This is both an inevitable, and exploitable phenomena. Think about it, say you have a coupon-picker rule, B/S012345678, how many mutations could it take? Most possibly it can have 6-7 bits mutated but still giving unchanged dynamics. However, if we are looking at B3/S23 instead, it is far more sensitive to rulestring mutation, and we are only allowed to touch B8,S8. It is this difference that I want to exploit here.
BlinkerSpawn wrote: Instead of expressing dynamic difference purely on the difference between rulestrings, weight the changed neighborhoods based on the frequency in which they appear naturally in the two rules. (I say check both rules to preserve transitiveness of the dynamic difference and give a more refined estimate)
This may be difficult for rules with large differences in dynamics that cause very disparate "transition profiles", but it looks like it could still be a better approach that a naive Hamming comparison.
Just to clarify, I am not "expressing dynamic difference" with difference between rulestrings, I used correlation to do that. I simulate say 500 soups on a 20x20 torus for 11 steps, and collapse all trajectories into 500x20x20x11 bits, and check the correlation between the 2 sets of trajectories (one from each rulestring). The hamming distance is merely there to restrict searching depth to nearby rules (so that we only calculate similarity between superficially/naively related rules! )

I hope this clarifies a bit.

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Re: Use Smoothiness to classify rules

Post by BlinkerSpawn » March 9th, 2017, 9:37 am

shouldsee wrote:
BlinkerSpawn wrote: Instead of expressing dynamic difference purely on the difference between rulestrings, weight the changed neighborhoods based on the frequency in which they appear naturally in the two rules...
Just to clarify, I am not "expressing dynamic difference" with difference between rulestrings, I used correlation to do that. I simulate say 500 soups on a 20x20 torus for 11 steps, and collapse all trajectories into 500x20x20x11 bits, and check the correlation between the 2 sets of trajectories (one from each rulestring). The hamming distance is merely there to restrict searching depth to nearby rules (so that we only calculate similarity between superficially/naively related rules! )

I hope this clarifies a bit.
Well, my response was geared more toward where you said
Thus it'd be great if we can find a universal way to quantify the dynamical difference between 2 given rules given their rulestring, or at least spot the important bits of a given rulestring.
but yes, that clarifies the search process a little bit.
I still think it would be a better determinant of what constitutes a "superficially related rule", considering that even though specific profiles (what I think you call trajectories) can vary between rules, certain transitions are basically always more or less important than others (e.g. 2a or 2e as opposed to 2n). So you could say that a bitflip in 3a or somesuch may or may not constitute a "superficial variation" and better fine-tune your search.
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Re: Use Smoothiness to classify rules

Post by shouldsee » March 9th, 2017, 10:45 am

BlinkerSpawn wrote: I still think it would be a better determinant of what constitutes a "superficially related rule", considering that even though specific profiles (what I think you call trajectories) can vary between rules, certain transitions are basically always more or less important than others (e.g. 2a or 2e as opposed to 2n). So you could say that a bitflip in 3a or somesuch may or may not constitute a "superficial variation" and better fine-tune your search.
Let me explain this idea in a little bit more detail. The idea originates really in bioinformatics, where you have important sequences conserved across evolution, because they are of functional importance. The whole point of enforcing hamming distance to search superficially realted rules is to simulate the process of random mutations. i.e. we assume only a single-base flip is allowed. Say we take B3/S23, and allow it to randomly mutate and kill off the progeny with non-similar dynamics (die off due to poor biological functionality) . Then we should get a bunch of sequences/rulestrings with closely related dynamics. It is by looking the conservation of individual bits (across this bunch of sequences) that we infer their relative importance towards the wanted dynamics. As for hamming distance/superficially metric, it's merely saying "the rulestring is only allowed to change 1 bit at a time“, and has absolutely no connection towards predicting the resultant dynamics.

I have nearly abandon the idea of "looking at a rulestring and infer the type of resultant dynamics". We are simply not at that stage yet. All I am trying to do now is to construct a reliable metric to characterise the dynamics. If you look at the examples.tar.gz, you should see the profiles correspond well with the amount of density fluctuation, and that's how my script can be useful. i.e. there are 2^102 rules out there in nt-rulespace, it's simply not possible to manually review them one by one.

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