## Unproven conjectures

For general discussion about Conway's Game of Life.

### Re: Unproven conjectures

77topaz wrote:
danny wrote:UPDATE: This one's been proven. Its bounding box is smaller in area than 50*50, and even just barely (31*79 = 2449 = 2500 - 51)

Technically, it doesn't actually fit in a 50*50 box, though.

Either way, the part in italics should now be updated IMO.
`x = 81, y = 96, rule = LifeHistory58.2A\$58.2A3\$59.2A17.2A\$59.2A17.2A3\$79.2A\$79.2A2\$57.A\$56.A\$56.3A4\$27.A\$27.A.A\$27.2A21\$3.2A\$3.2A2.2A\$7.2A18\$7.2A\$7.2A2.2A\$11.2A11\$2A\$2A2.2A\$4.2A18\$4.2A\$4.2A2.2A\$8.2A!`
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### Re: Unproven conjectures

dvgrn wrote:Conjecture: No glider eater can be constructed with a recovery time of three ticks or less.
(All known glider eaters take at least four ticks to recover to their original state after eating a glider.)

I think this one is somewhat easier to tackle than others; Is it proved?

The definition of "recovery time" of a catalyst isn't very well-defined, I think...
1) Traditional catalyst searchers / gencols etc. define them as "having a common neighbor".
e.g. two blinkers in a row are "reacting" every 2 generations in this sense.
2) Personally I think it's appropriate to define as the following:
`Pattern p, c; // Pattern and Catalystreaction_gen(p, c) = min{i | i∈ℕ, p[i] ∪ c[i] != (p∪c)[i]};recovery_gen(p, c) = min{i | i∈ℕ, i >= first_reaction_gen(p, c), c[i] ⊆ (p∪c)[i]};recovery_time(p, c) = recovery_gen(p, c) - reaction_gen(p, c);`

That makes the eater1's recovery time 3 ticks, which smakes sense in a catalyst-program's perspective, but not so much in Real Life(TM), I guess.
Best wishes to you, Scorbie

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### Re: Unproven conjectures

Is the smallest possible Garden of Eden known?
Life is hard. Deal with it.
My favorite oscillator of all time:
`x = 7, y = 5, rule = B3/S2-i3-y4i4b3o\$6bo\$o3b3o\$2o\$bo!`

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### Re: Unproven conjectures

Hdjensofjfnen wrote:Is the smallest possible Garden of Eden known?

Probably not. There are no Gardens of Eden inside 6x6, as proved by exhaustive search. Extending that search to 6x7 or 7x7 will be quite a computational feat -- roughly 30 times or 8000 times as much work, respectively, as the 6x6 case.

My guess would be that all 7x7 patterns have at least one predecessor, and quite possibly all 8x8 patterns also. There might be a Garden of Eden hiding inside 9x9. Steven Eker has almost made it down to 9x10, but it appears to be very difficult to get rid of those last few cells sticking out into 9x11.

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### Re: Unproven conjectures

What is the smallest pattern that has exactly one father?
Life is hard. Deal with it.
My favorite oscillator of all time:
`x = 7, y = 5, rule = B3/S2-i3-y4i4b3o\$6bo\$o3b3o\$2o\$bo!`

Hdjensofjfnen

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### Re: Unproven conjectures

dvgrn wrote:
simsim314 wrote:
Tom Mazanec wrote:How about things where we don't even have a conjecture as to the answer, much less a theorem?

I've demonstrated the concept of constructible ship of any speed and direction - but the period of the constructed speed is extremely high. It's completely unknown and we don't have any clue whether there exists ships with relatively low period and very high speed...

A huge number of existence questions are like this. The answer is "yes" or "no", but there isn't enough information to come up with even an educated guess.

Is there a true period-14 glider gun inside a 50x50 box?
Is there a glider collision that produces a 4x4 array of blocks?
Is there a 16x16 methuselah that takes more than a billion ticks to stabilize?
Is there a stable reflector smaller than a Snark?
Is there a 2c/3 signal elbow with a repeat time less than 20?
Is there a two-engine Cordership using a lucky clean c/12 debris-burning reaction?

... I'd guess yes, yes, no, yes, yes, no, but what do I know really? These aren't particularly long-standing or important questions, by the way -- there are hundreds more just like them.

I'd guess no, yes, no, no, yes, no.
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### Re: Unproven conjectures

There is a known 2-engine Cordership.
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calcyman

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### Re: Unproven conjectures

calcyman wrote:There is a known 2-engine Cordership.

Oh, I forget that.
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### Re: Unproven conjectures

A billion ticks is a lot.

KittyTac

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### Re: Unproven conjectures

A classic time-traveler paradox involves accidentally becoming one's own parent.

Are there any Still Life objects where the only father is themselves?
EdPeggJr

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### Re: Unproven conjectures

EdPeggJr wrote:A classic time-traveler paradox involves accidentally becoming one's own parent.

Are there any Still Life objects where the only father is themselves?

This question showed up on the previous page of this thread, in a couple of variants:

Conjecture: No still life can be constructed such that every possible predecessor pattern contains the same set of ON cells.
(The existence of one of these would disprove the "All still lifes can be synthesized by colliding salvos of gliders" conjecture, among other things.)

Conjecture: No oscillator or spaceship can be constructed such that the only predecessors include the same oscillator or spaceship (some fading junk around the edges not being counted).
(The LifeWiki Did-You-Know form is as follows: It is currently an open question whether there exists a periodic pattern whose only predecessors are its own evolutionary sequence.)

It's also in the Open Problems list on the LifeWiki, listed as having been posed in LifeLine Volume 6 by John Conway Himself -- the "unique father problem". Not sure if the \$50 prize is still available, or if there was some kind of statute of limitations on it... but in either case the problem is definitely still open.

dvgrn
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### Re: Unproven conjectures

Oh, of course. The Unique father problem.
Are there any handy father-finding programs? I'd like to see other fathers for this pattern.

`x = 9, y = 9, rule = B3/S234bo\$3bobo\$2bobobo\$bo2bo2bo\$ob2ob2obo\$bo2bo2bo\$2bobobo\$3bobo\$4bo!`
EdPeggJr

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### Re: Unproven conjectures

Wait...ARE there parents for that? My mind says it's really unlikely that that...thing has any nontrivial parents whatsoever But of course it's probably wrong of me to assume that.

EDIT: Oh, here's one from catagolue:
`x = 31, y = 31, rule = B3/S23oobobooobbbbbbbooboooboobobbbbo\$oobbboboboooboobbbbboooboooooob\$bbbbbobboobbboboobbbobobobbobob\$obboobooobobooboobboooooobbooob\$bbboooobooooobboboobboooboobbob\$ooobooboobbooobobbboobboooobboo\$obboobbobbbobbobooboobobbobooob\$ooboboooobobobooobboobbobooobbo\$bbooooboobobbooobbooboobobooooo\$boobobbbbboobobobbbbobobbboobob\$boboobbooooobobbobobooboooboooo\$bobbooobboobobobbbobbboooobobbo\$bbbooobobbboboboooboobobbbobbbo\$booobobbooobooboboobbbbbobobbbb\$bobbbbooobbobbbobbobobboobboobo\$oboooobooobbooooooobboooboooobo\$oboobboobbobobbobbbobbooobbbbob\$bbbbobobbbbbooboboobooobbobooob\$obbbobbboboobooobobobbbobooobbb\$obboboooobbbobbboboboobbooobbob\$ooooboooboobobobbobooooobboobob\$boboobbbobobbbboboboobbbbboboob\$ooooobobooboobbooobbobooboooobb\$obbooobobboobbooobobobooooboboo\$booobobbobooboobobbobbbobboobbo\$oobboooobboobbbobooobbooboobooo\$bobboobooobboobobboooooboooobbb\$booobboooooobboobooboboooboobbo\$bobobbobobobbboobobbboobbobbbbb\$boooooobooobbbbboobooobobobbboo\$obbbbobooboooboobbbbbbboooboboo!`

That's my most frequently used 'parent finder', but I think you can get WLS/JLS to do it. Idk
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danny

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### Re: Unproven conjectures

danny wrote:[Catagolue]'s my most frequently used 'parent finder', but I think you can get WLS/JLS to do it.

Yup. In JLS, if you just want to find a simple predecessor, you choose a search size under Edit > Properties, and also set the number of generations to 2 and in the Tiling/translation tab set the "After the last generation is..." setting to Unknown.

Then in generation 1, draw all OFF cells (select all, then right-click), then draw the ON cells you want. In generation 0, draw all OFF cells also, then hollow out an area of empty cells where you want to allow either ON or off.

For a challenging search, it's often important to set the search options correctly -- Search > Options > Sorting and Search > Options > Constraints. Maybe the most important detail is to try to start the search from the most difficult point and work outwards from there. If you start the search in an easy corner somewhere with a lot of possibilities, JLS will take enormously longer, or sometimes roughly forever, to get through the entire search. If you start the search in the right place, it will be able to eliminate huge swaths of the search space without having to fill in those easy corners in all possible ways.

Here's a .jdf file that shows a sample setup. Run it and in a second or two it will report 11,902 father patterns for the pattern in question, inside the arbitrary space I defined for Generation 0.

sample-parent-search.jdf.txt
Sample setup with unnecessarily large area (but you should leave a couple of blank cells around the edges, like the top and bottom here)

If you want to actually see the solutions it finds, change the "Pause search after each solution" and/or "Append solutions to file" settings in Search > Options > Processing.

dvgrn
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### Re: Unproven conjectures

Does there exist a synthesis for every billiard table?
Does there exist a glider synthesis for every stable object?
Life is hard. Deal with it.
My favorite oscillator of all time:
`x = 7, y = 5, rule = B3/S2-i3-y4i4b3o\$6bo\$o3b3o\$2o\$bo!`

Hdjensofjfnen

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### Re: Unproven conjectures

dvgrn wrote:
danny wrote:[Catagolue]'s my most frequently used 'parent finder', but I think you can get WLS/JLS to do it.

Yup. In JLS, if you just want to find a simple predecessor, you choose a search size under Edit > Properties, and also set the number of generations to 2 and in the Tiling/translation tab set the "After the last generation is..." setting to Unknown.

Then in generation 1, draw all OFF cells (select all, then right-click), then draw the ON cells you want. In generation 0, draw all OFF cells also, then hollow out an area of empty cells where you want to allow either ON or off.

For a challenging search, it's often important to set the search options correctly -- Search > Options > Sorting and Search > Options > Constraints. Maybe the most important detail is to try to start the search from the most difficult point and work outwards from there. If you start the search in an easy corner somewhere with a lot of possibilities, JLS will take enormously longer, or sometimes roughly forever, to get through the entire search. If you start the search in the right place, it will be able to eliminate huge swaths of the search space without having to fill in those easy corners in all possible ways.

Here's a .jdf file that shows a sample setup. Run it and in a second or two it will report 11,902 father patterns for the pattern in question, inside the arbitrary space I defined for Generation 0.

sample-parent-search.jdf.txt

If you want to actually see the solutions it finds, change the "Pause search after each solution" and/or "Append solutions to file" settings in Search > Options > Processing.

or you can use LLS, it's simpler

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### Re: Unproven conjectures

Hdjensofjfnen wrote:Does there exist a synthesis for every billiard table?
Does there exist a glider synthesis for every stable object?

The answer to both those questions is the same: It is unknown whether there exists a glider synthesis for every still life or for every oscillator. I'm not sure if it would be worthwhile making a distinction between billiard tables and other oscillators.

For the second of those about stable objects you can find some discussion in this very thread, as well as elsewhere on this board. From the top of the first page:
dvgrn wrote:Conjecture: All still lifes can be synthesized by colliding salvos of gliders.
(All still lifes up to 18 bits have a known glider synthesis, but it is still not known whether all still lifes are synthesizable.)
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wildmyron

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### Re: Unproven conjectures

How would I go about searching for a still life with only one parent?
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danny

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### Re: Unproven conjectures

danny wrote:How would I go about searching for a still life with only one parent?

Well, I guess that can't exist. Even though there are many still life They are composed by simple components.
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### Re: Unproven conjectures

danny wrote:How would I go about searching for a still life with only one parent?

You'd want to take a program that searches for GoEs and adapt it. You'd have to change two things, one is to make it only check still lives, and the other is to allow the pattern to have itself as a predecessor. I think you'd probably want to use the program in this paper. For each potential GoE it creates a SAT instance that asks for a predecessor of the pattern. The instance is satisfiable if and only if the pattern is not a GoE. So you would want to add clauses to the SAT instance that ban the predecessor from being the pattern itself. You'd then want to find some quick way to find still lives to feed into this program. I think Nathaniel has been working on some still-life generating projects so he might have such a program.

Macbi

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### Re: Unproven conjectures

Macbi wrote:
danny wrote:How would I go about searching for a still life with only one parent?

You'd want to take a program that searches for GoEs and adapt it. You'd have to change two things, one is to make it only check still lives, and the other is to allow the pattern to have itself as a predecessor.

There's a little more defining and decision-making that might have to be done. Any still life is guaranteed to have multiple predecessors by normal definitions: just pick any edge and add a single cell or a domino spark or whatever, to make a new larger cluster, but the spark fades away harmlessly. At worst you can always duplicate the entire outermost edge of the still life two cells away. Still life edges can't have more than two cells next to each other, so the line of dots and dominos on the other side of the temporary gutter will always fade in one tick.

If for some reason you want the parent to be connected in the same way as the still life, we should be able to enumerate all possible width-2 strips of edges of valid still lifes -- say for a 12x3 strip -- and then I bet we can show for each one that there's some modification that adds one or more cells toward the center of the adjacent row, such that the ends of the strip aren't affected but the strip converges to the same still life after one tick.

In other words, Dvgrn's Incautious Conjecture is that there are multiple parents of every 10x4 strip of still life edge with one blank row, even if you only alter the centered 8x3 section of the strip that includes the blank row.

It might not be an impossible task these days to enumerate all of those strips, and then run each one through LLS and find a nontrivial predecessor. But maybe just doing the enumeration would suggest a line of attack that would simplify the proof of nonexistence. Start with smaller strips and see if a centered 4x3 or 5x3 or 6x3 section is enough, maybe?

-- I should probably have said sooner that this is all something of a wild herring, or a red goose chase, or whatever you call it. Conway's statement of the problem from LIFELINE 6:1 is "is there a stable configuration whose only father is itself (with some fading junk some distance away not being counted)?" Slightly extending that, I think it's okay if the edges of a still life can have multiple parents, as long as there's an MxN chunk in the middle that is common to all possible ancestors.

It seems intuitively unlikely to me that any such MxN chunk is possible. But do we have enough computing power these days to enumerate all possible MxN chunks consistent with still-life-ish-ness, and then find a way to alter each center (M-2)*(N-2) area in a way that converges back to the same MxN chunk? ... Maybe we do, actually. Would that be not good enough for some reason? What did I miss?

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### Re: Unproven conjectures

This is all strongly related to the question of whether every still life is synthesizable. In order to demonstrate that not every still life is synthesizable it would suffice to produce a arrangement of alive and dead cells which could be extended to a still life and such that the parent of any pattern containing that arrangement also contained that arrangement in the same place.

If we wanted to disprove that such an arrangement existed then one way to do it would be to find by computer search some M and N such that for every N×M pattern that can extend to a still life there are some cells in the interior (N-2)×(M-2) rectangle that can be changed to produce a parent.

Macbi

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### Re: Unproven conjectures

Macbi wrote:This is all strongly related to the question of whether every still life is synthesizable. In order to demonstrate that not every still life is synthesizable it would suffice to produce a arrangement of alive and dead cells which could be extended to a still life and such that the parent of any pattern containing that arrangement also contained that arrangement in the same place.

That seems like a good short modern restatement of Conway's unique-father problem. I guess I never noticed before that that conjecture isn't very well named. Seems like "unique father" ought to mean "find a pattern that has only one parent", but it actually means "find a stable pattern where some part in the middle of it has only one parent".

Anyway, yeah, I'd definitely put my research time on the "disprove that such an arrangement exists" side of that idea. I think that these days we might actually be able to exhaustively search MxN rectangles until we find parents for every possible (M-2)x(N-2) still-life-compatible configuration.

That doesn't mean that I'd bet on all still lifes being synthesizable, though. I still vaguely suspect there's an unsynthesizable still life out there somewhere. But I'm glad I don't have to prove it. My real bet would be that neither a proof nor a disproof will be coming along in my lifetime.

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### Re: Unproven conjectures

Maybe unrelated, but does anyone have a proof that any speed greater than c/4 diagonal is unachievable in Life? After all, there's a 2c/7 in a rule one transition from Life:
`x = 5, y = 5, rule = B3/S234w3o\$b2o\$2b3o\$3b2o\$4bo!`
Life is hard. Deal with it.
My favorite oscillator of all time:
`x = 7, y = 5, rule = B3/S2-i3-y4i4b3o\$6bo\$o3b3o\$2o\$bo!`

Hdjensofjfnen

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### Re: Unproven conjectures

Hdjensofjfnen wrote:Maybe unrelated, but does anyone have a proof that any speed greater than c/4 diagonal is unachievable in Life? After all, there's a 2c/7 in a rule one transition from Life:
`x = 5, y = 5, rule = B3/S234w3o\$b2o\$2b3o\$3b2o\$4bo!`

Here: http://www.njohnston.ca/2009/10/spacesh ... -automata/
Still drifting.
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