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 Post subject: What is the theoretical value of the first move of a game? #1 Posted: Mon Aug 24, 2020 1:35 pm
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During yose or in a tsume-go problem the value of a move may be quite simple.

`[go]\$\$W\$\$ -------------------\$\$ | . . . X O . . . .\$\$ | . . . X O . . . .\$\$ | X X X X O . . . .\$\$ | O O O O O . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .[/go]`

In the above diagram the value of a move in the upper left corner is 22 points.

From a theoretical point of view is it possible to give the value of the very first move of a game?

My calculation is the following: assuming a parfait game will end with an advantage of exactly 7 points for black the value of the first move must be 4 x 7 = 28 points.
Do you agree with such evaluation ?

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 Post subject: Re: What is the theoritical value of the first move of a gam #2 Posted: Mon Aug 24, 2020 2:49 pm
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Value of first move in an even game between evenly matched has been debated for many centuries. Did you not find the answer?

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 Post subject: Re: What is the theoritical value of the first move of a gam #3 Posted: Mon Aug 24, 2020 2:56 pm
 Judan

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Why did you multiply by 4?

If you model the game as each player taking a gote move of some clear number of points in a non interacting way (like the cards of environmental Go) then there's many ways you can end up with black having 7 more at the end (I'm using a short game to illustrate the point):

28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15
100, 100, 100, 100, 100, 93
7, 7, 7, 7, 7, 7, 7
24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11
12, 12, 12, 12, 11, 10, 45, 43, 30, 28, 2, 2, 2, 1, 1, 1, 1
28, 21, 12, 12, 10, 10, 5, 5, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1

Does it make sense to talk about the value of the first move as the gote number of points it grabs (which could be e.g 7 or 100 in the examples), or is "the value of sente" more useful, which is something closer to the difference between the first 2 numbers. Of course Go isn't so simple as grabbing a bunch of non interacting gote moves in decreasing order, but it'll do for starters.

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 Post subject: Re: What is the theoritical value of the first move of a gam #4 Posted: Mon Aug 24, 2020 4:18 pm
 Honinbo

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A reasonable estimate for how much an opening move, if gote, gains is twice komi. So with a 7 pt. komi we can estimate the gain for the first play as 14 pts. There is a great deal of uncertainty about that estimate, however.

KataGo provides an estimate for the value of each position, or position with a certain player to play, so we can use that as a way of estimating how much a play gains. I expect that it will be close to 14 pts. (Years ago I estimated it as 13.75 pts. )

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Last edited by Bill Spight on Mon Aug 24, 2020 4:24 pm, edited 1 time in total.
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 Post subject: Re: What is the theoritical value of the first move of a gam #5 Posted: Mon Aug 24, 2020 4:23 pm
 Oza

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Gérard TAILLE wrote:
...

From a theorical point of view is it possible to give the value of the very first move of a game?

My calculation is the following: assuming a parfait game will end with an advantage of exactly 7 points for black the value of the first move must be 4 x 7 = 28 points.
Do you agree with such evaluation ?

Is this a "theoretical" value? What does that even mean? Notice that if we change 7 to 11, the first move value changes to 44. If we change the 4 to 3, it changes to 21. Is there a "theory" behind either the 7 or the 4?

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 Post subject: Re: What is the theoretical value of the first move of a gam #6 Posted: Mon Aug 24, 2020 6:50 pm
 Honinbo

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Quote:
In the above diagram the value of a move in the upper left corner is 22 points.

Just a side note... I agree with the assessment of 22 points for playing a move in that corner. But the position made me wonder about ko threats. Specifically, is a 22 point corner that has fewer ko threats that can be used against it different in "value" from a 22 point corner with many ko threats that can be used against it?

I don't think so in terms of the way we count value of a position. But I wonder if it'd ever be useful to distinguish when counting. It's hard to predict how many kos might arise in the course of a game anyway, so maybe it's not really that practical to differentiate.

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 Post subject: Re: What is the theoretical value of the first move of a gam #7 Posted: Mon Aug 24, 2020 8:16 pm
 Honinbo

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Kirby wrote:
Quote:
In the above diagram the value of a move in the upper left corner is 22 points.

Just a side note... I agree with the assessment of 22 points for playing a move in that corner. But the position made me wonder about ko threats. Specifically, is a 22 point corner that has fewer ko threats that can be used against it different in "value" from a 22 point corner with many ko threats that can be used against it?

One way of dealing with the value of a ko threat when there is only a potential ko of unknown type and size on the board is to assign the threat a small positive value, ε, negative if it is your opponent's threat. Back in the '90s Yonghoan Kim, one of Berlekamp's students, tackled the idea of the value of ko threats in his dissertation, in what he called perturbation theory. OC, if you know the specifics you can work things out.

As for the value of 22 pts. for a move in the corner, I know you guys understand these things, but for the readers, the value of 22 pts. does not mean what a lot of people think it means.

Assuming that the White stones are immortal, the average value of the corner is 6 pts. for White. If Black plays first, Black lives and gets 5 pts., an average gain of 11 pts. If White plays first, White kills and gets 17 pts., also an average gain of 11 pts. 22 pts. is the sum of the two average gains. It is equal to the difference between 5 pts. for Black and 17 pts. for White, which is a little easier to calculate than the average gain.

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 Post subject: Re: What is the theoritical value of the first move of a gam #8 Posted: Mon Aug 24, 2020 8:40 pm
 Honinbo

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Uberdude wrote:
Why did you multiply by 4?

If you model the game as each player taking a gote move of some clear number of points in a non interacting way (like the cards of environmental Go) then there's many ways you can end up with black having 7 more at the end (I'm using a short game to illustrate the point):

28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15
100, 100, 100, 100, 100, 93
7, 7, 7, 7, 7, 7, 7
24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11
12, 12, 12, 12, 11, 10, 45, 43, 30, 28, 2, 2, 2, 1, 1, 1, 1
28, 21, 12, 12, 10, 10, 5, 5, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1

Does it make sense to talk about the value of the first move as the gote number of points it grabs (which could be e.g 7 or 100 in the examples), or is "the value of sente" more useful, which is something closer to the difference between the first 2 numbers. Of course Go isn't so simple as grabbing a bunch of non interacting gote moves in decreasing order, but it'll do for starters.

Suppose that the stack of cards model is appropriate for go in general when you don't know the specifics of the position, but you do know how much the most valuable card (the top card) gains. Also suppose that at each turn each player takes the top card. Also, as in your example, there could be any number of cards of the same size, and suppose that there is a 50:50 chance that the number of cards of a given size is odd or even.

Then the expected result of the stack of cards with a top card that gains T pts. is T/2 pts.

Let S(v) be the expected value of having the move when the top card gains v pts.

Let the gain of the top card be T and the gain of the second to top card be R.

Half the time the number of cards that gain T pts. will be odd and the player with the move (the first player) will take the last one for a gain of T pts. Then the second player will take one of the second to top cards, for an expected gain of S(R). The expected result will be T - S(R).

Half the time the number of top cards will be even and the players will split them, for net 0 pts. Then the first player will have an expected gain of S(R).

The expected result will then be ((T - S(R)) + S(R))/2 = T/2. QED.

Edit: T/2 will also be the expected komi because for N cards you will have 2N-1 pairs of equiprobable results that add to T. Half of the addends will be less than or equal to T/2 and half of them will be greater than or equal to T/2.

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 Post subject: Re: What is the theoretical value of the first move of a gam #9 Posted: Mon Aug 24, 2020 11:17 pm
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Echoing what I think Bill has tried to teach on these forums.

The "swing value" - the total swing in points between white playing first and black playing first - of the upper left is 22 points.

But it's weird to call that the "value of the move". For example, the phrase "value of the move" suggests some other possible intuitions: that maybe 22 should be the amount that you'd have to make a card worth in a model like Uberdude's model to be indifferent between moving there and taking the car. Or that if you were forced to pass rather than to make a move in the upper left, 22 points would be the amount of extra komi you'd need in compensation to make it fair. Or that if you were to blunder and to play a useless dame move somewhere else when the upper left was the biggest move, the amount you'd lose on average would be 22 points.

But none of these is the case, in all cases 11 is generally closer to the right answer, not 22. We can see 22 points is too high as follows:

1. Suppose it's white's turn, and white plays in the upper left. White has 17 points, with black to move next.
2. Suppose white is instead forced to pass and is awarded 22 points in compensation for this forced pass. Black plays in the upper left and has 5 points. So net, white has 17 points matching situation 1 in points. But now *white* is to move next.

So generally, situation 2 is better for white than situation 1. Therefore 22 points is too large, white preferred to receive the 22 points and forego the move than to play the move.

3. Suppose white is instead forced to pass and is awarded 11 points in compensation for this forced pass. Black plays in the upper left and has 5 points. But now white is to move next instead of black, so to match things up we need to force white to pass again. Let's award another 11 points for this pass. Now white has 22 points, while black has 5 points, so net white has 17 points, and black is to move next.

In situation 3, we've achieved equivalence with situation 1. We had to force white to pass twice though. But there's still a clear sense in which we can say that the average value of each move here was 11 points. Per move we forced white to give up, we needed to give 11 points of compensation to white to make it equivalent to if white was never forced.

If you still find "swing values" easier to think about, just call them "swing values" or some other phrasing less ambiguous than "value of a move" when you talk about them.

 This post by lightvector was liked by 2 people: Bill Spight, Uberdude
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 Post subject: Re: What is the theoretical value of the first move of a gam #10 Posted: Mon Aug 24, 2020 11:33 pm
 Tengen

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 Post subject: Re: What is the theoretical value of the first move of a gam #11 Posted: Tue Aug 25, 2020 12:09 am
 Gosei

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 Post subject: Re: What is the theoretical value of the first move of a gam #12 Posted: Tue Aug 25, 2020 2:49 am
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We have to distinguish beetween
1) the swing value in a given area of the board
2) the value of a move
Taking my example with 22 points. It is clearly a swing value but the value of a move here is only 11 points : whitout knowing who will play first one can estimate that, on average, white will have 6 points in this area (17 points if he plays first and -5 points if black plays first) and a player can add 11 points to this value if plays first.
One sees that a swing value is twice the value of the corresponding move.

In other words:
the "value of a move" consists of comparing the move to a pass move
and the swing value consists of comparing the move to an opponent move.

With this in mind I understand that the value of the first move is 14 points.
If now you want to compare such move with a swing value (e.g. 22 points) then you have to consider that such move is equivalent to a move in an area of 28 points (swing value).

It seems to me that computers use only "value of move" but I am not sure what strong human players use. It seems to me that they can switch easily from one count (value of moves) to the other (swing values) depending of the situation (fuseki, middle game, yose) and that is a little upsetting.

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 Post subject: Re: What is the theoretical value of the first move of a gam #13 Posted: Tue Aug 25, 2020 4:02 am
 Tengen

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Computers do not use move values.

Per-move value and swing value are both move values, although the latter has restricted scope of application.

That players can use both and other values should come to no surprise. They are just values, which can be calculated:)

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 Post subject: Re: What is the theoretical value of the first move of a gam #14 Posted: Tue Aug 25, 2020 6:31 am
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Oops, I have to clarify my point concerning computer vs human play.
Taking aside the deep analysis made by a computer or a human through a vast tree of variants, my understanding is the following.
Let's suppose it is black to play.
A computer evaluates all the positions reached by a black move and chooses the best move according to these evaluations. If you substract to these evluations the evaluation of the currtent position you can see that the choice made by the computer can be seen as based on the value of each black move (but of course the computer has not to do explicitly these substractions).
Humans may act strictly as a computer but in order to choose their move they also can take into account another approach to gather some more information: they imagine they pass and they find the best white move. Using such information they can return to the computer method in order to definitely chosose their move. In that case, beside the move evaluation made by the computer the human add something like a swing evaluation in order to have a better understanding of the position.
To my knowledge I don't think a computer improve its analysis by finding the best white move when the pass move looks completetly stupid.
So my conclusion is the following:
Computer use basically "move values" though humans may add some swing values to improve their analysis.

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 Post subject: Re: What is the theoretical value of the first move of a gam #15 Posted: Tue Aug 25, 2020 7:14 am
 Honinbo

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Gérard TAILLE wrote:
We have to distinguish beetween
1) the swing value in a given area of the board
2) the value of a move
Taking my example with 22 points. It is clearly a swing value but the value of a move here is only 11 points : whitout knowing who will play first one can estimate that, on average, white will have 6 points in this area (17 points if he plays first and -5 points if black plays first) and a player can add 11 points to this value if plays first.
One sees that a swing value is twice the value of the corresponding move.

In other words:
the "value of a move" consists of comparing the move to a pass move
and the swing value consists of comparing the move to an opponent move.

That's one way to think about it. Another way is to think of the value of a move as like the distance between two points on a map. We start at point A and go to point B. How far have we gone? In go we start at position A and make a move to position B. How much have we gained?

Quote:
With this in mind I understand that the value of the first move is 14 points.
If now you want to compare such move with a swing value (e.g. 22 points) then you have to consider that such move is equivalent to a move in an area of 28 points (swing value).

I see what you are saying, but it seems to me to unnecessarily complicated. What does this hypothetical 28 point area look like? Swing values aren't really made for this kind of thing.

Quote:
It seems to me that computers use only "value of move" but I am not sure what strong human players use.

As Robert pointed out, modern computer go players do not use the value of a move as described here. They could be trained to do so, and before the virus hit that was something I was thinking about working on. Aaron Siegel has written a program that does calculate such values, called CGSuite. See http://cgsuite.sourceforge.net/ . KataGo does make use of evaluations in terms of points on the board, which the author, lightvector, can explain.

Quote:
It seems to me that they can switch easily from one count (value of moves) to the other (swing values) depending of the situation (fuseki, middle game, yose) and that is a little upsetting.

Strong players of previous generations were mostly confused. Swing values were devised to compare plays, as you indicate above. Good enough. But swing values don't work for comparing sente and gote. Despite that fact, well into the late 20th century the Nihon Kiin's Small Yose Dictionary included several pages of plays ordered by move values which lumped sente and gote together. Finally they stopped doing so, but instead of correcting the move values, simply dropped that feature. The rule of thumb for comparison was to double the value of sente. That is already misleading, as what were called the value of sente moves were actually the average gains of reverse sente moves. Early in this century O Meien wrote a popular book in Japanese about endgame evaluation that explained that. His book does not refer to the value of sente at all. And he does not use swing values.

When you try to do more sophisticated things than just compare two plays, swing values usually make things more difficult and confusing than average gains. Best, IMO, to ground your understanding on the value of positions; starting, OC, with scores, and then the average value of positions. Then think of the average value of a move as the difference between the average value of the position before the move and the average value of the position after the move.

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 Post subject: Re: What is the theoretical value of the first move of a gam #16 Posted: Tue Aug 25, 2020 7:28 am
 Tengen

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Computers do not study all positions but a small sample of positions.

There might or might not be a correlation between computer evaluation of moves and human move values.

Humans can calculate or estimate move values for endgame moves, endgame-like moves, the first move and maybe other moves with particular circustances. Humans, however, still cannot only assess all moves by move values because some aspects, such as influence, cannot always be converted to parts of move values. Therefore, we still cannot relate move values to computer values for all moves.

Once more: computers do not use move values.

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 Post subject: Re: What is the theoretical value of the first move of a gam #17 Posted: Tue Aug 25, 2020 8:34 am
 Honinbo

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Gérard TAILLE wrote:
Oops, I have to clarify my point concerning computer vs human play.
Taking aside the deep analysis made by a computer or a human through a vast tree of variants, my understanding is the following.
Let's suppose it is black to play.
A computer evaluates all the positions reached by a black move and chooses the best move according to these evaluations. If you substract to these evluations the evaluation of the currtent position you can see that the choice made by the computer can be seen as based on the value of each black move (but of course the computer has not to do explicitly these substractions).
Humans may act strictly as a computer but in order to choose their move they also can take into account another approach to gather some more information: they imagine they pass and they find the best white move. Using such information they can return to the computer method in order to definitely chosose their move. In that case, beside the move evaluation made by the computer the human add something like a swing evaluation in order to have a better understanding of the position.
To my knowledge I don't think a computer improve its analysis by finding the best white move when the pass move looks completetly stupid.
So my conclusion is the following:
Computer use basically "move values" though humans may add some swing values to improve their analysis.

Instead of talking about humans and computers let me talk about two different human theories of evaluating games and nodes in a game tree. Suppose that we have a game in which the players take turns. One player, say, Black, makes a move and then the other player, White, makes a move, then Black, then White, and so on. At each node of the game tree only one player can move. Each player tries to maximize the result of her play. From the standpoint of Black, White is trying to minimize Black's result, and vice versa, so the term, minimax, is used to describe the evaluation of these nodes. At the end of the search tree we have leaf nodes, which have a well defined value, or score. In go a win by ½ pt. is as good as a win by 35½ pts., so we can just evaluate a win as +1, a loss as -1, and a tie, if we have those, as 0. Now let's back up one move from the leaf nodes. Suppose that Black is to play from that node. Then she will move to one of the leaf nodes that maximizes her result, and we evaluate that node as the maximum value of its children. Similarly, if White is to play from a node, we back up the minimum value for Black of the children of that node as its value. Note that by minimax evaluation if you start at node A and move to node B, if your move is correct the difference between the values of A and B is zero. If you make a mistake, then the value of node B is less than that of node A. A move cannot gain anything, it can only lose. Economists call such a loss an opportunity cost.

In the go endgame, however, the board breaks up into independent regions (except, perhaps, in a ko fight), each of which may be considered as a game. Although the players take turns on the whole board, they do not have to take turns in each game. This leads to a different kind of game tree, where each player can potentially play from each node of the tree — except leaf nodes, OC. We cannot simply evaluate a node by backing up the value of one of its children. In fact, it is not exactly obvious how to evaluate it. But go players did figure out a way, two centuries or more ago. Obviously, if a node (position) has a local score, that is its value. Furthermore, if a position is a simple gote, where Black can move to a position with a score and White can also move to a position with a score, it makes sense to evaluate the original position as the average of those scores. It has an average value. Voila! Sente positions are less obvious, but a kind of backup was used, not from the immediate position after the sente play, but from the position after the (correct) reply. Hence the saying, Sente gains nothing. Double sente positions are even less obvious, and go players never figured out how evaluate them, with the exception of a few players like myself, back in the 1970s. Even so, I could not get Go World to publish my article about double sente. {shrug} Even today, O Meien does not talk about double sente. Back to the story. Now that go players could evaluate local positions by averaging gote values or by backing up sente values, plays normally make gains. At least, gote and reverse sente plays do. Also, the difference between moving to a local score of 30 is better than moving to a local score of 28. That is because no local score determines by itself the result of the game. A local score of 28 may be enough to win the game, but a local score of 30 may be necessary, and gives in general a better chance of doing so.

So we have two different types of games, two different types of game trees, and two different types of evaluation.

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 Post subject: Re: What is the theoretical value of the first move of a gam #18 Posted: Tue Aug 25, 2020 9:51 am
 Lives in gote

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That's very clear for me Bill (I worked for lot of years on a draughts program and it's not difficult for me to understand what you mean concerning tree exploration).
Dividing the goban into independant regions each one considered as a small game is a very interesting idea but only in the very final stage of the game.
Typically, when it remains area where different possbilities may exist then we cannot handled this area as an independant region.
For example:
`[go]\$\$B\$\$ -------------------\$\$ | . . . . . . . . .\$\$ | . a b . . . . . .\$\$ | . O X X . . . . .\$\$ | . . O X X X X X X\$\$ | . O O . . . . . .\$\$ | . O . . . . . . .\$\$ | . O . . . . . . .\$\$ | . O . . . . . . .\$\$ | . O . . . . . . .\$\$ | . O . . . . . . .[/go]`

In the diagram above, when you analyse a white move, how can you choose the correct move (a ot b) whithout knowing what will happen in the other regions of the goban.

Anyway I effectively can see the interest of using different evaluations depending on the circumstancies.
BTW do you think that computers may use also different evaluation techniques depending of the circumstancies (for instance in the example you mentionnened when independant areas can be identified)?

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 Post subject: Re: What is the theoretical value of the first move of a gam #19 Posted: Tue Aug 25, 2020 10:09 am
 Oza

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Quote:
O Meien does not talk about double sente.

Eh? He may not rabbit on about it but he does mention it, so it's in his mind. Example 3 Diagram 5: Black 1 is double sente and must not be overlooked. Example 4 Diagram 5: White can play the double sente of 1 and 3, which I felt very smug about.

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 Post subject: Re: What is the theoretical value of the first move of a gam #20 Posted: Tue Aug 25, 2020 11:28 am
 Tengen

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"computers may use also different evaluation techniques depending of the circumstancies"

Since when might programmers not be free?:)

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