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Transition dynamics

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Transition dynamics

Postby googleplex » February 6th, 2018, 1:51 pm

Certain transitions change rule dynamics, so I thought I would figure out how transitions (eg. b1a, s4k) effect dynamics of rules (eg. if it's explosive or not)

Most of these will have exceptions.

Most rules with B2a have naturally occuring infinte growth: (this example has a sawtooth!)
x = 6, y = 9, rule = B2a4/S356
2b2o$bo2bo$o4bo$o4bo$4bo$3bo$3bo$$3bo!


Rules without b01ce2ace3ai can't have spaceships, period. No exceptions.
Rules with b2ic4ic6i can't have gutter symmetry.

as an example, here's the question mark in CGoL
x = 6, y = 9, rule = B3/S23
2b2o$bo2bo$o4bbbbo$ob4bob$4bbo$3bobobbo$3bo$$3bo!
Last edited by googleplex on February 6th, 2018, 10:09 pm, edited 1 time in total.
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Re: Transition dynamics

Postby Rhombic » February 6th, 2018, 2:29 pm

One of the most interesting things about transitions is that they impact the rule dynamics depending on what rule they are in.

For instance, B7c makes B3/S23 explosive, whereas B7c has 0 impact in B2/S becaue it never appears.
While this is a very extreme example, it means that certain transitions (B3j, B4a, S2a, S1e) have extremely different impacts of the behaviour of a rule depending on the previous behaviour of that rule based on the following factors:
  • The abundance of a given neighbourhood. In B3/S23, B4c seldom appears, so B34c/S23 is almost identical to Life.
  • What new patterns arise from the common patterns using that transition. For instance, B8 is very rare in Life,
    but the dynamics of B38/S23 are mostly observed in how there are no traffic lights.
  • Self-sustainability of neighbourhoods. The probability that in a randomly generated pattern, a given allowed transition will make it MORE likely for that transition to happen in the future. Or if, on the other hand, a given transition makes it more difficult for other transitions to appear in the future. This explains why certain rules become LESS explosive after adding transitions. This is probably the hardest point to analyse mathematically.
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Re: Transition dynamics

Postby Macbi » February 6th, 2018, 2:38 pm

Does anyone know any rules of thumb to determine if a strobing rule is explosive or not?
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Re: Transition dynamics

Postby googleplex » February 7th, 2018, 10:19 am

also, some survival conditions don't directly influence still lives, for instance adding S7e to life doesn't make still lives where one cell has 7 neighbors. however, if you remove b3i and add S4r5c:
x = 3, y = 3, rule = B3-i/S234r5i7e
obo$3o$3o!

EDIT: here's one in a totalistic rule:
x = 9, y = 6, rule = B3/S23457
2obobob2o$bob3obo$2ob3ob2o$$9o$obobobobo!

Looks like some artifact.

so you would think that S7 doesn't really affect life.
But it does.
x = 3, y = 3, rule = B3/S237
obo$3oooo$3o!

adding s7 to life turns it into what in my opinion is like a methuselah that never stabilizes.

of all the S678 additions to life, the only one that makes new still lives is S6i:
x = 7, y = 5, rule = B3/S236i
2o3b2o$obobobo$2b3o$obobobo$2o3b2o!
Last edited by googleplex on February 7th, 2018, 11:43 am, edited 1 time in total.
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Re: Transition dynamics

Postby A for awesome » February 7th, 2018, 10:45 am

googleplex wrote:of all the S678 additions to life, the only one that makes new still lives is S6i:
x = 7, y = 5, rule = B3/S236i
2o3b2o$obobobo$2b3o$obobobo$2o3b2o!

x = 4, y = 4, rule = B3/S236k
2obo$b3o$3o$ob2o!

It has a somewhat unfortunate appearance, but it's the only counterexample I can find.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

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Re: Transition dynamics

Postby googleplex » February 7th, 2018, 10:59 am

A for awesome wrote:
googleplex wrote:of all the S678 additions to life, the only one that makes new still lives is S6i:
house fuse house

very bad no good horrible still life

It has a somewhat unfortunate appearance, but it's the only counterexample I can find.


Again, nearly all of these will have counter examples.
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Re: Transition dynamics

Postby googleplex » February 7th, 2018, 11:00 am

I'm writing a simple program to make random rules for me so that I don't have to randomly search them.
How are interesting rules like snowflakes and diamonds found anyways?
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Re: Transition dynamics

Postby KittyTac » February 7th, 2018, 11:07 am

googleplex wrote:I'm writing a simple program to make random rules for me so that I don't have to randomly search them.
How are interesting rules like snowflakes and diamonds found anyways?


Probably luck. A script would immensely help with research.
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Re: Transition dynamics

Postby googleplex » February 7th, 2018, 11:11 am

Some rules are a still life with patches of activity expanding the still life:
x = 6, y = 9, rule = b2cei3ai/s01c234i
2b2o$bo2bo$o4bo$o4bo$4bo$3bo$3bo$$3bo!


EDIT:
I wonder if this rule is Apgsearch-able.
Last edited by googleplex on February 7th, 2018, 11:29 am, edited 1 time in total.
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Re: Transition dynamics

Postby googleplex » February 7th, 2018, 11:20 am

KittyTac wrote:
googleplex wrote:I'm writing a simple program to make random rules for me so that I don't have to randomly search them.
How are interesting rules like snowflakes and diamonds found anyways?


Probably luck. A script would immensely help with research.


Because it would be nice if someone wrote a program that, given certain requirements for a rule, (eg. b2i, s4c) runs all polyplets up to ~8 bits and censuses the rule, and if it meets certain requirements, deems the rule interesting.
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Re: Transition dynamics

Postby 77topaz » February 7th, 2018, 4:32 pm

googleplex wrote:Some rules are a still life with patches of activity expanding the still life:
b2cei3ai/s01c234i


EDIT:
I wonder if this rule is Apgsearch-able.


No, I don't think it's apgsearchable. :P Catagolue does not like rules with very large interconnected still lifes - they can even overload the servers. And that's assuming the algorithm would even be able to separate the still lifes from the linear growths; if not, you'd just have every pattern in the census as "ov" or "zz_LINEAR".
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Re: Transition dynamics

Postby vyznev » February 8th, 2018, 7:37 am

googleplex wrote:Certain transitions change rule dynamics, so I thought I would figure out how transitions (eg. b1a, s4k) effect dynamics of rules (eg. if it's explosive or not)

In a certain literal sense, of course, all the dynamics of CA rules arise from their transitions. Sometimes they just arise more directly, and from a smaller set of transitions, than in others.

For example, (non-strobing) rules with B1c trivially explode, and cannot support still lifes, spaceships or oscillators, since any finite pattern will expand diagonally at lightspeed. Rules with B1e or B2a also tend to explode, since either a single cell or a pair of adjacent cells will expand orthogonally at lightspeed, but these rules can support non-exploding patterns. But if you put both B1e and B2a in the same rule, explosion is again guaranteed.

Personally, I've found rules on the 4-cell von Neumann neighborhood to be a nice starting point for this kind of analysis, both because there are fewer possible transitions to consider (only 10 for totalistic rules, or 12 for general isotropic rules) and because the lack of triangles in the cell adjacency graph makes things simpler (since two adjacent cells share no neighbors besides each other). Indeed, all the isotropic rules on the 4-cell neighborhood can be classified as follows:

  • For rules without B0, empty space is stable:
    • In rules with B1 (and without B0), all patterns explode at lightspeed. In rules without B1 (or B0), patterns cannot expand outside their initial bounding box (but may still have interesting dynamics within that box, e.g. as in the rule below).
      x = 80, y = 80, rule = B24/S23V
      80o$3o76bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo
      $o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o
      78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo
      $o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o
      78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo
      $o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o
      78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$o78bo$80o!
  • For rules with B0 and S4, empty space will flip once and then stabilize. These rules all have equivalent non-B0 duals obtained by swapping and bitwise complementing the birth and survival transitions, which effectively amounts to just swapping the labels on the dead and live cell states.

    (Indeed, all two-state CA rules have such duals, although a few rules are fully state-symmetric and thus self-dual. On the von Neumann neighborhood, these state-symmetric rules are themselves subject to a different kind of duality, where swapping but not complementing the birth/survival transitions yields a different rule with equivalent dynamics if every second cell on the lattice is swapped. This second kind of duality doesn't work on the 8-cell Moore neighborhood, though, since it requires the adjacency graph to be bipartite.)
  • Rules with B0 and without S4 are strobing. For the 4-cell neighborhood, these rules are the only ones that can have spaceships, although not all of them do.
    • Strobing rules with S3 and no B1 are explosive, just like non-strobing rules with B1.
    • Patterns in strobing rules with B1 and no S3 are confined to their bounding boxes, just like in non-strobing rules without B1.
    • Again, strobing rules with B1 and S3 are equivalent to their duals with neither.
    • Out of the remaining rules (with B0 and no B1/S34), those with B2e are also explosive, as the corners of any pattern with expand diagonally at c/2 (which is the maximum diagonal movement speed on the 4-cell neighborhood).
    • That leaves a total of 32 distinct totalistic (or 128 isotropic) rules, plus their duals, that are in principle capable of supporting spaceships on the 4-cell neighborhood. Several of those indeed do:
x = 2, y = 6, rule = B02/SV
bo$2o3$2o$bo!

x = 4, y = 10, rule = B02/S1V
2bo2$2b2o$o2bo$b3o$b3o$o2bo$2b2o2$2bo!

x = 9, y = 5, rule = B024/S1V
3o2$4b5o2$2bo!

x = 9, y = 10, rule = B03/S1V
6bo$5b3o$6bo2$7bo$7b2o$6bo$5bo$o5bo$4bo!

x = 14, y = 5, rule = B04/S1V
ob2o5b3o$7b2o3bo$4b4o2bo2bo$7b2o3bo$9b3o!


Macbi wrote:Does anyone know any rules of thumb to determine if a strobing rule is explosive or not?

Well, obviously, any strobing rule with S7c and without B1c must explode diagonally at lightspeed, just like non-strobing B1c rules do. Similarly, S6a7e without B1e2a makes the corners of any pattern expand orthogonally at lightspeed, just like having B1e2a does for non-strobing rules.

Perhaps more interestingly, these can be mixed: a strobing rule with S7c and no B1e2a, or S6a7e and no B1c, will also explode, with the corners spreading out at (2,1)c/2 in alternating diagonal and orthogonal steps:

x = 1, y = 1, rule = B01e2345678/S6a7e
o!

Conversely, just like non-strobing rules need at least one of the transitions in B1ce2ac3i for patterns to escape their bounding box, a strobing rule will need to lack at least one transition from both B1ce2ac3i, or to have at least one of S5i6ac7ce, for patterns not to stay confined in boxes. And similarly, just like patterns in non-strobing rules need at least one of B1ce2ae3a to escape their bounding diamond, in strobing rules they need to lack at least one of B1ce2ae3a or to have at least one of S5a6ae7ce.

In general, the most obvious difference between strobing and non-strobing rules, of course, is that adding more birth transitions tends to stabilize strobing rules, whereas it tends to destabilize non-strobing rule. Also, since the "active" cell state alternates between generations, there's no such clear distinction between (non-)birth and survival transitions: adding, say, B3e to a randomly chosen strobing rule has, on average, the same effect as removing S5e. (Or, more precisely: adding B3e to a strobing rule has exactly the same effect as removing S5e from its dual rule.)
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