There's SL's definition by Bill, closely related to CGT: https://senseis.xmp.net/?Temperature
I have added my own understanding: https://senseis.xmp.net/?Dieter%2FTemperature

I prefer move value and only use [ambient] temperature as "largest move value on the rest of the board". I speak of move value (and - different concept - gain) instead of local temperature and - CGT concept - incentive. I speak of the largest move value on the board instead of the global temperature. Simple! No need to follow CGT when one does not use CGT.

In your article you have written under late endgame:
"The global temperature is still more complicated, as we need to compare two sequences of endgame moves, with either player going first. That's why we estimate the global temperature in the endgame to be half of the biggest local temperature."
Two comments :
1) for me the global temperature is in general equal (very near) to the biggest local temperature and not half of this local temperature.
2) In your sentence it is difficult to see what means each occurence of the word "tenmperature". Is it the "new" temperature calculated by the difference between position after the good move and after the pass, or is it the temperature defined in CGT?

No my magic function cannot lead to a negative value but if you use katago instead of the magic function, then yes in very rare situations I can imagine you might get a negative value.
BTW do you have a better defintion of temperature when you are not in the endgame phase?

A minority occurrence but not rare!

Definition of global temperature: If you want to do it CGT style, ko thermography must be taken into account and this is far beyond of what I have completely understood of CGT. So, no, I cannot provide an even more thorough definition. As Francesco Criado points out, ko thermography is incomplete. So before creating somthing better, one should consider first completing CGT definitions... Will my ko definition be needed?

Interesting. In order to avoid any misunderstanding can you show us at least one example?

I do not have an example at hand because such AI mistakes I quickly discard / avoid / dissolve by deeper or follow-up position search. Typically, the occurrences are of ordinay kind when choice between close competitors depend on a) tactics of medium complexity or b) conquering vast empty space when dozens of candidates make finding the best hard but eventually understanding emerges that one candidate is significantly better and the reasons are only revealed by search of at least medium depth. Similar to what humans can experience in their games when good follow-up sequences are found early enough in the game sequence to outplay the opponent.

Typical choices are between 0.1% or 0.1 points off candidates. Above cases, however, can change about 2% or 0.3 to 2 points.

Maybe there is a misunderstanding Robert.
I perfectly understand that katago can prefer a move that is not the best one and, as a consequence I understand that evaluating the result of the real best move may lead to some contraction but this point has nothing to do with my defintion of move value.
For me the move value is defined by comparing the local move to the PASS move (which in general is not a close competitor!). I do not compare the local move to the best move on the board.

Remove the magic and AI reference and create perfect play. This avoid misunderstandings.

Ok let's remove the magic and AI reference. What remains on the table? The defintion of "the value move" disappears and and the definition of the "temperature" disappears also. Your are not very constructive Robert.

Oops its a wrong to say that "the magic function cannot lead to a negative value".
Let'me clarify this point:
1) the magic function magic(P) is the result of the game starting from position P. The result can be of course any number, for example +7 or -7
2) the value of move M = magic(P + M) - magic(P + pass) can also be a negative value: if the move M is a terrible mistake (like playing in its own eye) then the pass can lead to a better result than a move M.
3) concerning the temperature of a local area (defined as the max value of all possible moves in this area) this temperature cannot be negative because a pass move is always a part of all possible moves in this area.

I do not do all your work:) Hint: you might do what others did in the past - imagine the perfect play automaton. Katago assumed to play perfectly so to say or brute force playing perfectly instantly by assumption. We do not know how it decides but we assume it does play perfectly instantly.

You begin to understand. Now, if you did not remove AI, it could also pass incorrectly.

Obviously you did not understand my point Robert and I have to explain again.
in my post https://lifein19x19.com/viewtopic.php?p=280075#p280075 I wrote:
"Assume you have a magic fonction (katago?) which gives you the result of a game starting from a given position P : result = magic(P)."
In this sentence the magic function looks like a God function. The parenthesis suggests only to take katago as a good estimation of this function. As you see, stricly speaking the magic function has nothing to do with AI.
This being, AI appears to me to be good tool to give a good estimation of the magic function. OC katago may be not accurate to evaluate magic(P) or magic(P+M) or magic(P+pass) and it will never be perfect. I said magic(P) can be positive or negative, I said also that the value of move M = magic(P + M) - magic(P + pass) can be positive or negative (in this case the pass is better that the move M) and finally I said that the temperature of an area cannot be negative.
Where is your problem?

Actually, it is very much simpler than automatons, magic or AI! We apply my Definition 2 in [22], which repeats this kind of CGT definition:

"Definitions 2 [score of a position]
If Black or White starts, B(P) or W(P) is the resulting score of P, respectively.

Remarks

The resulting score occurs by applying the presuppositions, that is, letting the players alternate to a settled position while Black maximises and White minimises. Combinatorial game theory assumes a game G, the players Left and Right, and calls L(G) or R(G) the left or right stop, respectively, which reveals either score. Compare [31]"

Now, I think what you want is for either player to move. Therefore, let me define while generalising:

S(P) is the resulting score of P with its player having the turn.

References:

https://www.lifein19x19.com/viewtopic.p ... 45#p143245

The problem in your definition is the magic instead of constructing the result! Compare my new definition in my previous message and note that it does construct the result because it relies on earlier CGT definitions effectively prescribing alternate play to a game end position (called "stop" and such in CGT) having some particular result. Therefore, you should be writing your definition more like this:

"The _score_ of a position P is the score of a final position Z reached by min-max alternation abiding by the [non-cyclic] rules."

Be serious Robert. I agree that your definition allows to construct this result but it's only a purely theoritical approach without any interest for a fuseki position.

Click Here To Show Diagram Code

[go]$$Bc Start with a one-space high pincer...
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . X . . . . . . . . . . . . . . |
$$ | . . . . . O . . . , . . . . . X . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

What is the result of this position, white to play?
Are you able to give a better result than AI?

I'll try to respond to the OP.

I think this quote quite sufficient to understand temperature.



Later the idea, which I doubt is true, was spread around that Go players talked a lot about temperature. I think that is only true when Go players talk about theories such as those presented in the quoted monograph.

Like many concepts it has various realizations, depending on the intention.

For example

Let G be a game G := { L | R }, L and R also games.
Then define an operator, that we call cooling,

cool(G, t) = { cool(L, t) - t | cool(R, t) + t }

unless for some T < t, { cool(L, t) - t | cool(R, t) + t } is a number x (i.e. L - T <= R + T), then

cool(G, t) = x

Now G is said to have temperature T, mean(G) x and to freeze to cool(G, T).

Many other definition that allows for the following properties could be called temperature, even if they are not exactly the same, and I think you could be justified to call something temperature even if these properties are only usually correct. If one wished, then could be very precise about what is meant in each case.

Linearity: cool(G, t) + cool(H, t) = cool(G + H, t)
Order preserving: G >= H implies cool(G, t) >= cool(H, t)
mean(G + H) = mean(G) + mean(H)
temperature(G + H) <= max(temperature(G), temperature(H))

I doubt that these hold if we define

G := { f(L) | f(R) }

where f(x) is katago's score evaluation function, L and R as before. The most obvious violation is that f(x) isn't exact and will violate equalities and inequalities for that simple reason. Another problem is that while statements like G + H may work in form you usually can't actually add two 19x19 game together on a 19x19 board.

But it sure is similar in many ways, especially in form, to be useful.

I like that it is similar on form, I think that can be useful sometimes. If I were to suggest a less problematic definition that doesn't need to be similar in form then that would probably be

mean(G) := (f(L) + f(R)) / 2
temperature(G) := (f(L) - f(R)) / 2

Gérard, you can do either maths or informal opening theory but a combination of both requires research from scratch. My related maths has been as weak as yours so far, except for specific strategic concepts, such as influence stone difference or neutral stone difference. Endgame maths is decades ahead.

In the OP I was trying to explain temperature in layman terms, at least for myself. Now let me apply the concepts to a particular joseki.

Click Here To Show Diagram Code

[go]$$B Joseki
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . b . . . . . , . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . 1 . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . a 9 7 B . . . |
$$ | . . . . . . . . . . . . 0 W 5 2 3 . . |
$$ | . . . . . . . . . . . . . . 8 6 4 . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

For the sake of brevity, let's introduce a few abbreviations and functions

S = the score
L = local temperature
T = ambient temperature
P = pass

I'm now evaluating this position, with 7.5 komi, on an otherwise empty board, using KataGo's pointwise evaluation function. This is our substitute for Gerard's function magic(). As Robert has argued sufficiently, it needs many playouts to stabilize at a score which we can rely on. I have not done that.

Assume A the best local move (can be verified) and B as the most valuable move elsewhere, which is often the 4-4 diagonally opposite the corner played out.

if Black A, S = W+1
if Black P, White A, S = W+11.7
L = |S(A) - S(P, A)| = 10.7

if Black B, as KG recommends, S = 0
if Black P, White B, S = W+13.4
T = |S(B) - S(P, B)| = 13.4
T > L

Since the ambient temperature is higher than the local temperature, this position can be considered finished
or in other words, wasn't sente and Black can/should play elsewhere

We can apply the same reasoning along the whole joseki. I refer to the sgf below for all the computations.

Results:

The marked moves are not sente, neither is
and are all sente
is somewhat ambiguous: L and T are close and with the few playouts done the precision of the evaluation might surpass their difference. Also ...
is sente again, so Black has all reasons to play it

Another observation is that and both increase the local temperature by a lot and also decrease the ambient temperature. I interpret this as the influence the obtained strength after a corner capture has on the whole board.

Edit: I'm now doing the same but on a board where the other corners are occupied by one or two stones.