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 Post subject: Generalized thermography question #1 Posted: Mon Apr 11, 2022 12:44 pm
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Any thermography experts in the room?

Suppose the following game, where I list each state as state: [list of left options] , [list of right options]

k1: [-9] , [9, k2]
k2: [k1] , [k3]
k3: [k2, -9] , [9]

The scores here are in the CGT sense of numbers, that is playing a move = 1 point. When both players pass and the game is in a non-terminal position, that's 0 points for both. This CGT form of scoring in the context of go is basically stone counting but that's a tangent.

Let's assume we start at k2 and positional super ko (the standard in CGT) Obviously this is a "simple" game, i.e. every subposition has at most one left ko option and one right ko option. Therefore it should be solvable with Berlekamp, Spight, Muller's generalized thermography. Observe that the game will end up either in a seki at k1 or a seki at k3, the difference between these two that left (resp. right) has played one more move in the game hence they have one more point. Recall that in this context, passing does not lift ko bans but playing an environment play does.

Let's first analyse it in isolation. If left goes first in k1 then they get a move, and right cannot play to k2 because the ko ban so they pass, left passes, score is +1. Analogous with right playing first.

Now let's assume we add a -1 temperature switch: [-1] , [1]. In other words a move that both can play that lifts the ko bans and is non committal to end the game should the next player pass. The -1 temperature means this will not affect the score. If left goes first they could play the environment play right away but then Right plays the aforementioned sequence and wins an extra point. Then Left plays to k1, now right could pass but then they lose a point. Instead, they play the environment play. Now left has no good moves so they pass, and right plays to k2, then right passes, then k3. Now left can only pass, right passes, the final score is that right played one more move so it's -1. Analogous with right playing first, but the score is then +1.

The same argument holds if you put arbitrary switches ranging from temperature -1 to 1. Then the outcome of the local position depends on the parity of environment plays and there is no way the thermograph is well defined and converges to anything.

Observe this looks a bit like molasses ko. The optimal play between t=-1 to t=1 is to alternate the local sequence from k1 to k3 with environment plays, like LEELELEELE.

I believe that there is a problem in the BSM paper about generalized thermography because of the forcing pass lemma. Usually when a player plays in a ko, then the opponent passes/plays in the environment, the pass forces the first player to continue playing locally. However at t=-1, in this example this is not the case. When we reach k3, then left passes, right has no good moves from k3 either.

It would be great to make this example in go. For that we would need a capturing race between two groups with a double ko in the middle, such that no player can fill in the kos, and the status of the groups is seki independently of who "wins" the ko. "wins" here because there is no good move after winning the two stage ko except the stone that you just captured.

I'm not sure where was my mistake, either it is in my understanding on the definition of simple loopy game, or the definition of thermography, or the position, or... well the proof of generalized thermography might be wrong, and stronger assumptions on the game are required.

tl;dr: Assuming I didn't make a mistake (strong assumption) the position above is simple yet admits no thermography under PSK and no threat environment, hence it has no temperature. The optimal sequence at certain temperature range seems to involve a loop with environment plays which afaik only is known in go for molasses ko. If we can realize this in the go board it would be a particularly nasty endgame beast.

What do you think?

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 Post subject: Re: Generalized thermography question #2 Posted: Fri May 13, 2022 3:26 pm
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I have some interest, but very little understanding of the terms and theory involved. Do you have a link to the paper?

I do think that depending on wording, passes often make classical results more confusing. For example, even in Benson's paper, I doubted for some time that his proof that you shouldn't sacrifice a living group was correct. IIRC, I think it is correct but it says a lot less than I thought it did, because of the way his set up his definitions.

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 Post subject: Re: Generalized thermography question #3 Posted: Fri May 13, 2022 7:35 pm
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This seems to be continued in this thread, and the above post also seem to shed some light on where the logical problem is.

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 Post subject: Re: Generalized thermography question #4 Posted: Thu May 19, 2022 4:37 am
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dhu163 wrote:
I have some interest, but very little understanding of the terms and theory involved. Do you have a link to the paper?

Of course, here is part 1 http://webdocs.cs.ualberta.ca/~mmueller ... 6-030a.pdf and part 2 http://webdocs.cs.ualberta.ca/~mmueller ... 6-030b.pdf .

For the formalization of their fuzzy claims I went to Aaron Siegel's excellent book "combinatorial game theory". However Mueller confirmed me that Siegel is claiming a slightly stronger thing than what they had in mind when they wrote that article. Mueller emphasized to me that at the time they could only solve one ko at a time, whereas Siegel claims the BMS algorithm generalizes to "simple" loopy positions. There's also the elephant in the room that, with the construction they present, go is technically not a finite game, but a transfinite one. Mueller's program cleverly reuses the information in a position to account for different numbers of captures. Recall that in the mathematical rules they use, a capture gives you the right to pass so the postion is not technically the same with two captures as it is with three. In general, the CGT model of go has at the moment a big margin of improvement I believe.

I was thinking that a more realistic formal framework for temperature theory would require to forget about a lot of CGT stuff and do things from the beginning again. In particular I was thinking a way to model various ko rules and what affects the ko bans (passes, environment plays, etc) would be to model that information as a finite state automaton. In that way we can develop a formal understanding of go thermography that is more rule-agnostic. Regardless of your particular dialect of rules, one can not expect thermographs are always defined. The other element for a new formalization is labelling graph edges with the number of captures it produces to avoid the transfiniteness of the game, and perhaps also labelling somehow the nodes. Also the formalization I have in mind breaks for some japanese rules quirks, particularly the infamous torazu sanmoku I'm afraid. Hopefully I will be able to write it soon with the care it deserves.

As a side note and progress update, I found a loopy combinatorial game whose thermograph is well defined under Siegel's koban setting for which the thermograph has a segment with slope 1/2. As far as I know under Berlekamp-Mueller-Spight's construction every thermograph they could analyse always has integer slope. It is not contradicting any of their claims in that sense but I still believe it's cool.

In any case, the example presented in the other post still admits no thermography independently of accepting passes as lifting ko bans or not. Environment plays should definitely lift ko bans as they represent moving somewhere else in the board.

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 Post subject: Re: Generalized thermography question #5 Posted: Thu May 19, 2022 5:49 am
 Judan

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"I was thinking that a more realistic formal framework for temperature theory would require to forget about a lot of CGT stuff and do things from the beginning again."

I am delighted that you consider using my style!

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 Post subject: Re: Generalized thermography question #6 Posted: Mon May 23, 2022 10:32 am
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RobertJasiek wrote:
I am delighted that you consider using my style!

I'm not sure I'd call that following your style, as the formalization I have in mind is also quite different to yours. However one thing is certain: if I ever write this stuff down seriously, your method of hypotheses and the result we found proving its necessity are definitely going in there (credited of course). I think it's just such an useful tool for determining when things are sente. If someone is going to write "Mathematical go endgames 2", then I think those conditions should be there.

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 Post subject: Re: Generalized thermography question #7 Posted: Mon May 23, 2022 9:36 pm
 Judan

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With my style, I have meant something more fundamental:

- For rules and scoring theory, I start afresh with axioms to build definitions.

- For ko definition, I start afresh with the most basic model rules and definitions.

- For most of my endgame theory, I forget about combinatorial game theory but start afresh and simply compare numbers to define types of local endgames and decide correct move orders.

- For applied go theory, I start afresh with basic definitions to derive principles and methods.

My style is to start afresh and build everything bottom-up.

Current AI derived from Alphago Zero thinks similarly when ignoring human experience but learning afresh.

When you will model endgame ko evaluation, you plan to start afresh and build buttom-up.

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