Patterns across rules

[Tropylium]

I'd like to introduce a notation I've been using in tracking the compatibility of patterns across rules. This is a simple extension of the usual B/S notation: I simply add two more letters, A ("abstain") and D ("death") for their negations, signifying what rules must be absent for the rule to function. Though it would be IMO more natural to call the death conditions the "rules", since these are what actually changes the pattern in some way, not the survival conditions. Hence, I list these in the order BA/DS.

The number of rule conditions a pattern has I call its rank; a pattern with rank X functions in 2^(18-X) rules. (Or perhaps I should rather use the figure 18-X?)

Some simple examples.
—The simplest still life is the dot, which is A01/S0, and the only rank-3 pattern AFAICT.
—The simplest oscillator is AFAICT the p2 duplet, which is B2A01/D1 (rank 4)
—The simplest spaceships are AFAICT the two symmetric lightships, one of them B2A013/D01, the other ("moon" as I saw the Life Without Death lexicon call it) B2A013/D12.
And some examples from Life. For brevity, these all come with implicit A012:
—The block is the most robust still life in Life, requiring only S3.
—Some still lifes get by with A4/S2: the tub, the mango, the aircraft carrier, and all lakes. This is a complete listing, I think?
—Most simple still lifes seem to be A4/S23 or A45/S23. The hat is the simplest requiring all of A456. (A78 never comes up with still lifes.) I think the only ones with A5/S23 are the boat and the longship, and the only one with A6/S23 the ship. For A45/S2 there are the loaf, beehive, and various small lakes (also a complete listing?); for A46/S23, a few things like table on table, or shiptie.
—Of the small p2 oscillators, the blinker is B3/D1S2, the toad is B3/D14S2, the beacon is B3A4/D4S23, and the clock is B3A4/D1S3. The only other oscillator in Life with a rank of 8 or less I've noted so far is the bipole, which is B3A4/D1S23.

Can anyone tell if I am retracing territory here? How much work along these lines has been done before? David Eppstein has a site indexing spaceships and replicators across rules, but I've not seen anything for oscillators, nor anything on this topic on the wiki aside from the occasional enumeration of rules supporting a pattern.

And few questions that may be relevant for the Life enthusiast:
—What is the non-p2 oscillator with the smallest rank? Monogram, pulsar, octagon II and jam are rank 11 (in different ways). Are any rank 10?
—What about spaceships? The glider ranks at 10 which I don't think can be beaten, the LWSS at 13, and there are rank 12 examples as well, like the minimal c/3. Is rank 11 possible?
—Conversely, what are the smallest patterns with a high rank? Figure 8 at rank 16 and the Schick engine at rank 18 seem like good contenders.
—The *WSSes are the only complex patterns I'm aware of that do not require A4. Are there others?
—I'm unable to come up with any A56 still lifes. Any sketches for a proof of their non-existence?

[ebcube]

This would be a Life A56 still life, wouldn't it?

Code: Select all

x = 6, y = 6, rule = B3/S23
4bo$3bobo$3b2o$b2o$obo$2o!

[knightlife]

This would be a Life A56 still life, wouldn't it?

Code: Select all

x = 6, y = 6, rule = B3/S23
4bo$3bobo$3b2o$b2o$obo$2o!

That one requires A4 to be a still life

[knightlife]

This would be a Life A56 still life, wouldn't it?

Code: Select all

x = 6, y = 6, rule = B3/S23
4bo$3bobo$3b2o$b2o$obo$2o!

That one requires A4 to be a still life

So it is A012456 ?

[Tropylium]

Yeah, the tie-point has 4-cell neighborhoods.

A general issue is that ties/bridges or tails, which generally come handy in stabilizing still lifes, impart A4. Any still lifes divisible in more than one island require A4 as well (not hard to proov).

It is possible to create an A56 banner (mixed chickenwire/zebra stripes agars are A56 as well)

Code: Select all

----------OOO-…
---------O---O
--------O-OOO-
--------O-O---
------OOO-O---
-----O---O----
----O-OOO-----
----O-O-------
--OOO-O-------
-O---O--------
O-OOO---------
…

But coming up with an end part gets problematic. Yet I don't see any general reason why the boundary of a still life could not be accomplished by A0125/S23 alone; the elevener, for example, only has two A4 points.

Coming up with still lifes not requiring S2 other than the block is also surprizingly difficult; this might be proovable since the S2 cells of nearly-S3 still lifes tend to accumulate near the exterior, especially the corners.

-Also, on the list of Life patterns with small ranks: phoenixes are B3A012/D01 for a rank of 6.

[dakra]

The rules are usually expressed as actions, and depend on whether a cell starts out alive or dead. An alternative expression does not. It reports the next generation state, without any indication of whether a cell is born or dies.

Consider these Conway's-Life-equivalent rules. They use the count of all living cells in a 3x3 block, meaning a cell and its immediate neighbors:

• 3 -> ALIVE
• 4 -> STAY AS IS
• Otherwise -> DEAD

Note that these are future state rules, not action rules, such as BeBorn and Die.

How do they fit into this scheme?

[confocaloid]

Consider these Conway's-Life-equivalent rules. They use the count of all living cells in a 3x3 block, meaning a cell and its immediate neighbors:

• 3 -> ALIVE
• 4 -> STAY AS IS
• Otherwise -> DEAD

Intuitively, one can say that the part "3" in the first rule is a living condition (indeed, it is a condition for the cell to be alive in the next generation). The condition is "there are currently exactly three alive cells in the 3-by-3 neighbourhood".

The part "4" in second rule is a condition for the cell to remain in the same state in the next generation. The condition is "there are currently exactly four alive cells in the 3-by-3 neighbourhood".

The part "Otherwise" in the last rule is a condition for the cell to be dead in the next generation. The condition is "the number of currently alive cells in the 3-by-3 neighbourhood is neither 3 nor 4".

(See also my post in another thread viewtopic.php?p=192539#p192539 for a reply in the context of that thread.)

Several common rulestring notations only list explicitly the living conditions. For example, "B3/S23" can be intuitively read as the set {B3, S2, S3}, with three different nonoverlapping conditions for the cell to live in the next generation:

• B3: the cell is currently dead and has exactly three alive neighbours
• S2: the cell is currently alive and has exactly two alive neighbours
• S3: the cell is currently alive and has exactly three alive neighbours

So the "empty rulestring" B/S tells "the cell never lives in the next generation".

On the other hand, RuleLoader ruletables assume "stay as is" by default. So the empty ruletable tells "the cell never changes in the next generation".

Of course different notations / specifications can be useful in different situations.