B2kn34ekz/S2457

[Colonizor48]

This rule is interesting. It seems to be smack in the middle of a phase transition between class 1, class 2, class 3, and class 4. And minor changes to it can result in all 3 even though the rule itself is seemingly class 3. Most small changes result in a class 1 or class 2 rule. The rule itself is class 3 but has some simple structure, and has initial conditions where it does not blow up. And grows somewhat slowly and chaotically. It seems hard to determine if a pattern will stop or go forever.

The way in which the rule grows is also seemingly chaotic, rather then expanding forever, a blob will grow and shrink seemingly randomly, but seems to on average get bigger. It kind of reminds me of the collatz conjecture in a way, just in reverse.


Examples of rules 1 or 2 transitions away from this one:
B2kn34ekz/S24j57 - A boring but seemingly class 4 rule. On the boundry between class 2 and 4
B2kn34ekz/S2-e457 - A class 3 rule that is seemingly even closer to this phase transition. It grows even slower.
B2kn34ekz/S2k457 - A class 2 rule that rapidly converges to still lifes
B2kn34ekz/S2-k457 - A class 3 rule that also explodes slowly, and seems to shrink locally quite a lot
B2kn34ekz/S2-ek457 - A class 2 rule that is barely class 2. Random soups seem to shrink slowly, but seemingly always or almost always evolve into still lives.
B2kn3e4ekz/S2457 - A very boring class 1 rule

Studying this rule and it's adjacent rules could give a better understanding of the "phase transition" phenomenon that occurs with a lot of seemingly difficult to analyze sequences(for example, the boundary between P and NP, kn+b sequences for odd k and b(collatz is a specific case of this, seemingly on a phase transition between chaos and stability), and in general the difference between hard and easy problems.

[confocaloid]

A high-period oscillator in a small torus might work as a PRNG for some purposes. But few patterns I ran to completion all settled into low-period result (at most p2).
This takes over 4.5 megaticks to settle into empty 10x10 torus:

Code: Select all

x = 10, y = 10, rule = B2kn34ekz/S2457:T10,10
4ob4o$2b3o2b2o$2o2b4obo$bob2ob2o$bob2obob2o$bobobo2bo$o2bob3o$obob2ob
2o$o2b5obo$2b3o2b2o!

This did not settle into any low period after 25 megaticks:

Code: Select all

x = 12, y = 12, rule = B2kn34ekz/S2457:T12,12
7b2o$4b3obo$4bo2b2o$4bo5bo$4bo5bo$7b4o$6b2o$5bo$5bo!

Examples of rules 1 or 2 transitions away from this one:
B2kn34ekz/S24j57 - A boring but seemingly class 4 rule. On the boundry between class 2 and 4
B2kn34ekz/S2-e457 - A class 3 rule that is seemingly even closer to this phase transition. It grows even slower.
B2kn34ekz/S2k457 - A class 2 rule that rapidly converges to still lifes
B2kn34ekz/S2-k457 - A class 3 rule that also explodes slowly, and seems to shrink locally quite a lot
B2kn34ekz/S2-ek457 - A class 2 rule that is barely class 2. Random soups seem to shrink slowly, but seemingly always or almost always evolve into still lives.
B2kn3e4ekz/S2457 - A very boring class 1 rule

Studying this rule and it's adjacent rules could give a better understanding of the "phase transition" phenomenon that occurs with a lot of seemingly difficult to analyze sequences(for example, the boundary between P and NP, kn+b sequences for odd k and b(collatz is a specific case of this, seemingly on a phase transition between chaos and stability), and in general the difference between hard and easy problems.

I'm not sure how to comment on classes and the boundary between P and NP, but even a single "isotropic toggle" can have significant effect. Out of these rules, only B2kn34ekz/S2-e457 (toggled S2e), B2kn34ekz/S2-k457 (toggled S2k) are 1 "isotropic toggle" away (counting as one "toggle" addition/removal of either of the 102 conditions described by Hensel notation). B2kn34ekz/S2-ek457 is 2 isotropic toggles away (S2e, S2k). On the other hand B2kn34ekz/S24j57 is 12 toggles away (remove 12 out of 13 subconditions of S4), B2kn34ekz/S2k457 is 5 toggles away (remove 5 out of 6 subconditions of S2), B2kn3e4ekz/S2457 is 9 toggles away (remove 9 out of 10 subconditions of B3).

[Colonizor48]

A high-period oscillator in a small torus might work as a PRNG for some purposes. But few patterns I ran to completion all settled into low-period result (at most p2).
This takes over 4.5 megaticks to settle into empty 10x10 torus:

Code: Select all

x = 10, y = 10, rule = B2kn34ekz/S2457:T10,10
4ob4o$2b3o2b2o$2o2b4obo$bob2ob2o$bob2obob2o$bobobo2bo$o2bob3o$obob2ob
2o$o2b5obo$2b3o2b2o!

This did not settle into any low period after 25 megaticks:

Code: Select all

x = 12, y = 12, rule = B2kn34ekz/S2457:T12,12
7b2o$4b3obo$4bo2b2o$4bo5bo$4bo5bo$7b4o$6b2o$5bo$5bo!

Examples of rules 1 or 2 transitions away from this one:
B2kn34ekz/S24j57 - A boring but seemingly class 4 rule. On the boundry between class 2 and 4
B2kn34ekz/S2-e457 - A class 3 rule that is seemingly even closer to this phase transition. It grows even slower.
B2kn34ekz/S2k457 - A class 2 rule that rapidly converges to still lifes
B2kn34ekz/S2-k457 - A class 3 rule that also explodes slowly, and seems to shrink locally quite a lot
B2kn34ekz/S2-ek457 - A class 2 rule that is barely class 2. Random soups seem to shrink slowly, but seemingly always or almost always evolve into still lives.
B2kn3e4ekz/S2457 - A very boring class 1 rule

Studying this rule and it's adjacent rules could give a better understanding of the "phase transition" phenomenon that occurs with a lot of seemingly difficult to analyze sequences(for example, the boundary between P and NP, kn+b sequences for odd k and b(collatz is a specific case of this, seemingly on a phase transition between chaos and stability), and in general the difference between hard and easy problems.

I'm not sure how to comment on classes and the boundary between P and NP, but even a single "isotropic toggle" can have significant effect. Out of these rules, only B2kn34ekz/S2-e457 (toggled S2e), B2kn34ekz/S2-k457 (toggled S2k) are 1 "isotropic toggle" away (counting as one "toggle" addition/removal of either of the 102 conditions described by Hensel notation). B2kn34ekz/S2-ek457 is 2 isotropic toggles away (S2e, S2k). On the other hand B2kn34ekz/S24j57 is 12 toggles away (remove 12 out of 13 subconditions of S4), B2kn34ekz/S2k457 is 5 toggles away (remove 5 out of 6 subconditions of S2), B2kn3e4ekz/S2457 is 9 toggles away (remove 9 out of 10 subconditions of B3).

Wait I'm sorry I got that mixed up. But I am interested in looking at the space of all cellular automata and trying to determine if there is any sort of pattern or continuity between adjacent cellular automata. I want to eventually define the space of all Isotropic rules as a metric space with a well defined notion of distance.

[confocaloid]

But I am interested in looking at the space of all cellular automata and trying to determine if there is any sort of pattern or continuity between adjacent cellular automata. I want to eventually define the space of all Isotropic rules as a metric space with a well defined notion of distance.

Well this is an interesting question.
Some kinds of continuity exist. For example many reactions that work in Life, work the same way in rules B38/S23 (Pedestrian Life) and B3/S234c (Conway++). So one might conclude that in a sense, those rules are "very close" to Life.
However, B38/S23 has a statorless rotating p424 glider gun and (5,2)c/190 moving objects, which do not work in B3/S23. Likewise B3/S234c has a sparky p62 pulsator oscillator that does not exist in B3/S23. And B3/S23 has commonly occurring traffic lights (which are less common in B38/S23) and Corderships (which are absent in either of the two other rules). So in another sense, those rules are "very different".

You might find these discussions interesting:
viewtopic.php?f=11&t=3374 "Developing a more detailed classification of rule behaviors"
viewtopic.php?p=65747#p65747 several replies in "Re: Gun Discussion Thread"
viewtopic.php?f=11&t=3262 "Transition dynamics"
viewtopic.php?f=11&t=5167 "How to make certain patterns more common"
viewtopic.php?f=11&t=1679 "What life-like rules are interesting and what are not?"
viewtopic.php?f=11&t=2315 "Use Smoothiness to classify rules"

[Colonizor48]

But I am interested in looking at the space of all cellular automata and trying to determine if there is any sort of pattern or continuity between adjacent cellular automata. I want to eventually define the space of all Isotropic rules as a metric space with a well defined notion of distance.

Well this is an interesting question.
Some kinds of continuity exist. For example many reactions that work in Life, work the same way in rules B38/S23 (Pedestrian Life) and B3/S234c (Conway++). So one might conclude that in a sense, those rules are "very close" to Life.
However, B38/S23 has a statorless rotating p424 glider gun and (5,2)c/190 moving objects, which do not work in B3/S23. Likewise B3/S234c has a sparky p62 pulsator oscillator that does not exist in B3/S23. And B3/S23 has commonly occurring traffic lights (which are less common in B38/S23) and Corderships (which are absent in either of the two other rules). So in another sense, those rules are "very different".

You might find these discussions interesting:
viewtopic.php?f=11&t=3374 "Developing a more detailed classification of rule behaviors"
viewtopic.php?p=65747#p65747 several replies in "Re: Gun Discussion Thread"
viewtopic.php?f=11&t=3262 "Transition dynamics"
viewtopic.php?f=11&t=5167 "How to make certain patterns more common"
viewtopic.php?f=11&t=1679 "What life-like rules are interesting and what are not?"
viewtopic.php?f=11&t=2315 "Use Smoothiness to classify rules"

I'm more intersted in general behavior more then particular structures. Such as if a cellular automata is explosive, or cyclical, or chaotic. And if it explodes, how fast does it explode? do most patterns remain bounded bellow a finite number of cells?(Some class 4 automata do not have this property, for example brian's brain) Is it turing complete? Also, this distance function should be easily generalizable to non isotropic rules as well. thinking as them as sort of space "in between" isotropic rules. Perhaps it is possible to generalize further into a continuous space, but I somewhat doubt that. Unless you say interpolate rules and somehow define that as a real number valued rule. In which case cardnality would line up with that of the reals, as you could just use a different rule each step non periodically.

[Yoel]

(Some class 4 automata do not have this property, for example brian's brain) Is it turing complete?

Brian's Brain is Turing-complete, proven by me in 2020:

viewtopic.php?&p=110762#p110762

And it is, unlike Seeds, very easily manageable.