Which Life-like rules do and don't have oscillators?

[Ntsimp]

Most Life-life rules support some oscillators, but there are a few exceptions. Obviously there must be both birth and death, so B/* is out and so is */S012345678. B0 just confuses me, and anything with B1 just explodes.
Now two living cells diagonally adjacent make a period-2 oscillator in everything from B2/ to B2345678/S02345678, or everything with B2 and without S1. I don't think B2/S1 can have oscillators.
A few minutes work can find a p2 oscillator in almost any B3 rule except B3/S123456. Are there any oscillators in this rule?
What is known about this question? Is there any impossibility proof in a non-obvious case like B3/S123456?

[137ben]

I am not sure what B0 means either...

No oscillators can exist with B1, or with s012345678, as you said. I would conjecture that almost ever other lie-like rule has oscillators, though there may be a few exceptions.

A more complicated question is which rules have spaceships. Leaving aside B0, a rule must have either b2 or b3, and cannot have B1. But there are some rules that COULD have spaceships for which none have been found, for example B345/S4567.

[calcyman]

B0 means 'birth on no neighbours', so the entire empty universe becomes live in the next generation. In rules with B0..../S....8, it is easier to analyse the dual rule, obtained by exchanging the two states. However, B0-not-S8 rules are pathological, in that the entire universe alternates between live and dead each generation. Fortunately, only one-quarter of the Life-like cellular automata fall into this category.

Golly doesn't show the strobing, because the qlife algorithm would be totally overwhelmed by an infinite number of cells. Instead, Golly alternates between two rules, if I remember correctly.

David Eppstein wrote:

Sure. On even generations, complement the output of the CA rule. On odd
generations, complement the input.

That means that in Golly, you should only copy-and-paste to generations of the same parity, or otherwise unexpected things will occur.

Eppstein found an interesting B0 rule, which supports this spaceship:

Code: Select all

x = 11, y = 15, rule = B01346/S012345
11o$8ob2o$9obo$11o$11o$11o$ob9o$2b8o$ob9o$11o$11o$11o$9obo$8ob2o$11o!

The front end and back end are identical, apart from a relative phase difference. It is known as 'amphisbaena', named after the mythological beastie.

The rule also supports a p40 oscillator:

Code: Select all

x = 22, y = 22, rule = B01346/S012345
19b3o$19bobo$19b3o$19b3o$19b3o$19b3o$19b3o$19b3o$19b3o$19b3o$19b3o$19b
3o$19b3o$19b3o$19b3o$19b3o$19b3o$17b5o$17b3obo$20o$ob16o$19o!

[Andrew]

... Golly alternates between two rules, if I remember correctly.

Yep, here are the details if anybody is interested:

If the original rule has B0 but not S8:

For even generations, whenever the original rule has a Bx or Sx, omit that
bit from the modified rule, and whenever the original rule is missing a
Bx or Sx, add that bit to the modified rule.
eg. B03/S23 => B1245678/S0145678.

For odd generations, use Bx if and only if the original rule has S(8-x)
and use Sx if and only if the original rule has B(8-x).
eg. B03/S23 => B56/S58.

If the original rule has B0 and S8:

Such rules don't strobe, so we just want to invert all the cells.
The trick is to do both changes: invert the bits, and swap Bx for S(8-x).
eg. B03/S238 => B123478/S0123467 (for ALL gens).


I'm currently adding support for bounded grids (ie. bounded plane, torus, Klein bottle, etc.) so in such grids there won't be any need for the above emulation (and HashLife will be able to do B0-not-S8 rules).

[gulabgang]

Unfortunately, much of Life's technology does not work in this rule, for example the Gosper glider gun and the twin bees shuttle.

[Dets65]

Unfortunately, much of Life's technology does not work in this rule, for example the Gosper glider gun and the twin bees shuttle.

Things as (comparatively) complex as the gosper glider gun don't work in rules other than life, usually, unless the differences are entirely trivial, like adding an S7.