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Coolout Conjecture Counterexamples

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Coolout Conjecture Counterexamples

Postby wwei23 » June 24th, 2017, 10:20 am

I know of four so far:
x = 16, y = 12, rule = B3/S23
2o2b2o4b2ob2o$ob2obo4bobobo6$6bo$7o3b2o2b2o$o10bo2bo$o3bo6bo2bo$5o7b2o
!
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby toroidalet » June 24th, 2017, 10:46 am

another one:
x = 5, y = 3, rule = B3/S23
o3bo$bobo$obobo!
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Re: Coolout Conjecture Counterexamples

Postby BlinkerSpawn » June 24th, 2017, 10:48 am

I'm not so certain that the bottom two count in the first post count as "internally consistent" but that's just me looking at the problem.
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » June 24th, 2017, 1:34 pm

The conjecture refers to patterns where all the cells survive on to Generation 1. So it would look like this:
x = 7, y = 3, rule = B3/S23
bo3bo$obobobo$bobobo!
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby A for awesome » June 24th, 2017, 2:23 pm

A counterexample on both sides:
x = 6, y = 2, rule = B3/S23
2ob3o$2o2bo!
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: Coolout Conjecture Counterexamples

Postby wwei23 » June 24th, 2017, 2:27 pm

It's nice, but the right side can be stabilized with a pre-block and the top can be stabilized with a much larger induction coil(Though the left can't be stabilized. You tied my low-cell record, congratulations!).
x = 34, y = 16, rule = B3/S23
4b2o2b2o14b2o2b2o$2o3bo2bo3b2o6b2o3bo2bo3b2o$o2bo6bo2bo6bo2bo6bo2bo$2b
3o4b3o10b3o4b3o$6b2o18b2o$4b2o2b2o14b2o2b2o$3bo2b2o2bo12bo2b2o2bo$4b2o
2b2o14b2o2b2o$6b2o18b2o$4bo4bo14bo4bo$4b6o14b6o2$4b2ob3o14b2ob3o$4b2o
2bo2bo12b2o2bo$10b2o18bo$29b2o!
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » June 24th, 2017, 2:42 pm

Found a much smaller induction coil:(Left side can't be stabilized.):
x = 21, y = 8, rule = B3/S23
7b2o10b2o$o2bobo2bo3bo2bobo2bo$4ob3o4b4ob3o2$2ob3o6b2ob3o$2o2bo7b2o2bo
2bo$6bo11b2o$5b2o!
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » June 26th, 2017, 10:38 am

Try apgsearching B3-i/S23 and close variants, that's how I found one counterexample.
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » June 27th, 2017, 9:47 am

Three more:
x = 21, y = 7, rule = B3/S23
3bo$2b3o5b3o3b2ob2o$bo3bo3bobobo3bobo$2o3b2ob2o3b2o3bo$bo3bo3bo3bo4bo$
2b3o5b3o4bobo$3bo7bo4b2ob2o!

Edit: the third one has a smaller counterexample in it:
x = 6, y = 3, rule = B3/S23
2b2o$2o2b2o$o4bo!

EDIT OF EDIT:
The first and second are really the same with a smaller counterexample in it:
x = 7, y = 5, rule = B3/S23
bo3bo$2o3b2o$bo3bo$2b3o$3bo!
Last edited by wwei23 on July 8th, 2017, 7:57 pm, edited 1 time in total.
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » July 3rd, 2017, 7:27 pm

Two more:
x = 16, y = 6, rule = LifeHistory
2.A$.3A$A3.A6.A3.A$A3.A6.2A.2A$.3A9.A$2.A!
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby BlinkerSpawn » July 3rd, 2017, 8:29 pm

I finally got around to updating the Coolout Conjecture wiki entry but I'm not sure about how it looks.
Does it look ok to everyone else? :?
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » July 3rd, 2017, 8:43 pm

I think that you are completely right. It can easily be shown that 2 by 5 is the smallest counterexample-there are only 512+256+64+4=836 patterns to search, easy-even by hand!
Edit:
I meant in a smaller bounding box.
Last edited by wwei23 on July 4th, 2017, 8:11 pm, edited 1 time in total.
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby gmc_nxtman » July 3rd, 2017, 9:55 pm

wwei23 wrote:I think that you are completely right. It can easily be shown that 2 by 5 is the smallest counterexample-there are only 512+256+64+4=836 patterns to search, easy-even by hand!


It is true that 2*5 is the smallest but there are actually 1024 possible patterns in a 2*5 bounding box.
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Re: Coolout Conjecture Counterexamples

Postby BlinkerSpawn » July 3rd, 2017, 11:19 pm

gmc_nxtman wrote:
wwei23 wrote:I think that you are completely right. It can easily be shown that 2 by 5 is the smallest counterexample-there are only 512+256+64+4=836 patterns to search, easy-even by hand!


It is true that 2*5 is the smallest but there are actually 1024 possible patterns in a 2*5 bounding box.

True, but some can be easily shown to be unfeasible even before evaluation. Not all the patterns need to be looked at.
Also, I showed this not by looking at 2*5 patterns but by showing that there were no 2*4 or 3*3 patterns to beat the 2*5 provided; see the first reference.
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » July 22nd, 2017, 11:39 am

Three more:
x = 25, y = 6, rule = LifeHistory
5A5.5A5.5A$A3.A5.A3.A5.A3.A$A3.A5.A3.A5.A3.A$5A5.A3.A5.A3.A$10.5A5.A
3.A$20.5A!

At first it seems like it is nonextendable for 3n+1, but it is:
x = 5, y = 7, rule = LifeHistory
5A$ABDBA$AB.BA$A3.A$AB.BA$ABDBA$5A!
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » August 7th, 2017, 10:45 am

Skew!
x = 8, y = 8, rule = B3/S23
4bo$3bobo$b3obo$o4b2o$b2o4bo$2bob3o$2bobo$3bo!
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » August 14th, 2017, 7:42 pm

Another:
x = 6, y = 9, rule = B3/S23
5bo$3b3o$2bo$o2bo$4o2$3o$o2bo$2b2o!

Infinitely extendible:
x = 37, y = 6, rule = B3/S23
o2bo6bo3bo5bo4bo4bo5bo$4o6b5o5b6o4b7o3$4o6b5o5b6o4b7o$o2bo6bo3bo5bo4bo
4bo5bo!
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby fluffykitty » August 16th, 2017, 5:06 pm

I think a clarification on "internally consistent" is required. I interpret it as meaning "all non-border cells do not change state".
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Re: Coolout Conjecture Counterexamples

Postby wwei23 » August 17th, 2017, 9:18 am

My interpretation is "no living cells die at the next generation."
Replicator!
x = 3, y = 3, rule = B3/S234y
2bo$3o$bo!
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Re: Coolout Conjecture Counterexamples

Postby dvgrn » August 17th, 2017, 10:12 am

wwei23 wrote:My interpretation is "no living cells die at the next generation."

I think the original definition is more subtle than that. It seems as if it would be okay for living cells in a candidate counterexample to die at T=1. There just has to be a way to save each such individual cell from dying, by placing neighbor cells outside the boundary of the candidate pattern.

Internally consistent with being part of a still life definitely implies that non-boundary ON cells can't turn OFF at T=1 -- but also that non-boundary OFF cells can't turn ON at T=1.

Coolout Conjecture counterexamples show that very often, even though you can stabilize each individual cell considered separately, you can't always stabilize an arbitrary combination. Given that we have a minimum 2x5 counterexample, it's easy to generate an arbitrary number of additional larger counterexamples, but that doesn't add anything to the proof.
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Re: Coolout Conjecture Counterexamples

Postby fluffykitty » August 17th, 2017, 12:27 pm

I wonder if there's a finite set of "minimal unstabilizable patterns" where every unstabilizable pattern contains at least one. Kind of like how K5 and K3,3 are the forbidden minors of planar graphs.
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