Is Chess solvable? [Archive] - Two Plus Two Newer Archives

thylacine

01-31-2007, 01:48 AM

[ QUOTE ]
There are a finite amount of moves so my initial reaction was yes, but I don’t know enough about the game to know how much uncertainty over your opponents’ future moves affects things. So is Chess (theoretically at least) solvable?

[Not sure if this is the right forum. Feel free to move it.]

[/ QUOTE ]

A solution (and a proof that it is a solution) might collapse under its own weight into a black hole. Seriously.

ChrisV

01-31-2007, 02:02 AM

Chess is a completely deterministic game, so in principle it is solvable. How could it not be? In practice, the number of legal positions is around 10 to the power of 42. To put that in perspective, assuming the universe is 15 billion years old, if 2 million billion billion positions were looked at by a computer every second since the beginning of the universe, it still wouldn't be done looking.

valenzuela

01-31-2007, 02:24 AM

If the PC doesnt analyze clearly lost situation it can be done, btw this is just instinct

MusashiStyle

01-31-2007, 05:13 AM

eh... I think I can answer this. There is really no need to solve chess. Anyone who has alot of knowledge and experience in the game will tell you: the result of perfect play is a draw:

check out the draw Percentages at this years Corus:

Group A (best players in the world): 61 % of games drawn

Group B (strong GM's): 40% games drawn

Group C (GM's IM's): 30 % games drawn

at the bottom of this page it shows the stats

http://www.chessbase.com/newsdetail.asp?newsid=3638

ChrisV

01-31-2007, 08:26 AM

I agree that chess played perfectly is almost certainly a draw. That doesn't tell us how to draw versus any given opening line, though, which is the point of solving the game.

chezlaw

01-31-2007, 08:30 AM

[ QUOTE ]
I agree that chess played perfectly is almost certainly a draw. That doesn't tell us how to draw versus any given opening line, though, which is the point of solving the game.

[/ QUOTE ]
Hopefully it will be solved one-day and proven that black wins. Ultimate Zugzwang.

chez

Magic_Man

01-31-2007, 10:49 AM

I believe all endgames with 5 pieces have been solved, and most or all of the 6 piece solutions. My far-better-at-chess-than-I friend says that a few interesting 7 piece configurations have been solved as well. Kings don't count as pieces here. Presumably, with enough computing power, these databases could continue to be expanded, but they grow at an amazing rate. You'd need enormous data storage and a very powerful computer.

Al68

01-31-2007, 03:52 PM

Although there is a limited number of legal moves, you have to consider the total number of combinations of all those legal moves. This would give you the total number of all possible endgames. This number is greater than the total number of atoms in the universe.

No, I'm not joking and I'm not making it up.
This is why IBM cannot just have their chess computer analize all possible endgames and always win. There is not enough computing power in the world to do that.
They have to try to teach the computor how to play, analize the strength of their position, anticipate future moves, etc., similar to the way a human plays.

soon2bepro

01-31-2007, 04:52 PM

Computing power advances at an amazing rate. Chess is not that complicated. I think it wont be so long before it can be solved. A couple centuries maybe. Have in mind we didn't even get to the point where we can solve a game as simple as checkers.

Now talk about solving Go, there you have a challenge.

Siegmund

01-31-2007, 06:02 PM

I personally would not be THAT amazed to see chess solved in my lifetime, though I am not holding my breath.

The database holding all the solved positions is of course huge and will keep getting huger. The database necessary to make a meaningful contribution is not so huge -- a 7-piece ending either leads to a draw or to one of seven families of 6-piece endings - so all you need access to to solve that position are those seven families of 6-piece endings.

ChrisV

01-31-2007, 10:51 PM

[ QUOTE ]
Although there is a limited number of legal moves, you have to consider the total number of combinations of all those legal moves. This would give you the total number of all possible endgames. This number is greater than the total number of atoms in the universe.

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The game-tree complexity of chess (that is, the number of possible games of chess that can be played) is much, MUCH higher than the number of atoms in the universe. In fact, if you took the number of atoms in the universe and squared it, that would be closer to the mark. However, the number of legal positions in chess is far less than the number of atoms in the universe. In fact, the Earth alone contains at a minimum 10,000 times more atoms than the number of legal chess positions. But even so, you can imagine that this makes a database storing all possible positions completely infeasible - it would take up a good chunk of the Earth.

Siegmund

02-01-2007, 04:46 AM

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In fact, the Earth alone contains at a minimum 10,000 times more atoms than the number of legal chess positions. But even so, you can imagine that this makes a database storing all possible positions completely infeasible

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No, it doesn't. A simple list of positions is infeasible. But what matters is how much *information* is contained in that database. It can be compressed, greatly - recall, for instance, a king and a bishop cannot mate a king, but a king and rook can. That one sentence covers just short of a million positions. (2*64*63*62 arrangements of three distinct pieces, only a tiny handful of which need handled individually - for instance those where the singleton king moves next and can capture the threatening rook and cause a draw)

The simple endings have been known for years, of course -- but there are many more officially unexplored areas which won't take up much storage at all. Say, white has lost no pieces and black has 3 randomly selected pieces left: the list of such positions from which black can force a draw is gonna be realllll short.

bocablkr

02-01-2007, 12:49 PM

[ QUOTE ]
Although there is a limited number of legal moves, you have to consider the total number of combinations of all those legal moves. This would give you the total number of all possible endgames. This number is greater than the total number of atoms in the universe.

No, I'm not joking and I'm not making it up.
This is why IBM cannot just have their chess computer analize all possible endgames and always win. There is not enough computing power in the world to do that.
They have to try to teach the computor how to play, analize the strength of their position, anticipate future moves, etc., similar to the way a human plays.

[/ QUOTE ]

Can you show me the math behind your statement.

disjunction

02-01-2007, 01:02 PM

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No, it doesn't. A simple list of positions is infeasible. But what matters is how much *information* is contained in that database. It can be compressed, greatly - recall, for instance, a king and a bishop cannot mate a king, but a king and rook can. That one sentence covers just short of a million positions. (2*64*63*62 arrangements of three distinct pieces, only a tiny handful of which need handled individually - for instance those where the singleton king moves next and can capture the threatening rook and cause a draw)

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This is an interesting point. But I don't think anybody looks at this type of compression. I assuming that chess programs are just doing a heuristic search. They take a move, and the resulting position, and they have a really good metric for evaluating whether the resulting position is good. The technology direction would then be to (1) Do better evaluations (2) Find new, interesting ways to explore the search space. I wouldn't think there's much current effort to say "Well, this position really reduces the same as this other position", in a provable way, beyond what is already there (i.e., the endgames you mentioned).

Maybe there's some automated way to do these reductions (to AI people (or you if you're one of them), it seems like you are proposing finding interesting relations that can prune the search space ). I dunno, it's an interesting thought, but personally I'd be a little surprised the compression is effective enough.

Joerii

02-01-2007, 01:40 PM

kurto

02-01-2007, 01:48 PM

Can they even reasonable estimate the matter in the Universe? If its infinite and we cannot see the 'end of it'... how can they reasonable assume they know the amount of matter?

Joerii

02-01-2007, 02:00 PM

Exactly. With the galaxy we could make an estimate but it would be a pain in the ass to write it down :-)

bisonbison

02-01-2007, 02:42 PM

Okay, so I've read before about endgame tablebases, and clearly there's a mind-numbing gap between the current coverage (wikipedia says all 6-piece endgames have been solved: 4 pieces, 2 kings) and the complete 32-piece endgame tablebase that would solve that pesky "chess" issue.

Are there any Folding-at-home type distributed efforts to advance this? Or is the effort pretty much proprietary?

And someone must have done the math somewhere to estimate the total storage space necessary for storing that data. Petabytes? Exabytes? Zettabytes?

disjunction

02-01-2007, 03:23 PM

[ QUOTE ]
Okay, so I've read before about endgame tablebases, and clearly there's a mind-numbing gap between the current coverage (wikipedia says all 6-piece endgames have been solved: 4 pieces, 2 kings) and the complete 32-piece endgame tablebase that would solve that pesky "chess" issue.

Are there any Folding-at-home type distributed efforts to advance this? Or is the effort pretty much proprietary?

And someone must have done the math somewhere to estimate the total storage space necessary for storing that data. Petabytes? Exabytes? Zettabytes?

[/ QUOTE ]

Bison I'm not quite sure what you're asking. If you want to explore all the possible moves with 32 pieces, the problem grows exponentially the further you look forward. In one step we can make 10 moves, change this horizon to 2 and there are 10*10 possible moves, etc. It is growing exponentially. (When you grow things exponentially, you very quickly exceed the number of atoms in the universe, it's no big deal)

By distributing the computing, you are increasing your computing power and storage arithmetically. It doesn't really help and won't even bring you close.

Al68

02-01-2007, 03:30 PM

[ QUOTE ]
[ QUOTE ]
Although there is a limited number of legal moves, you have to consider the total number of combinations of all those legal moves. This would give you the total number of all possible endgames. This number is greater than the total number of atoms in the universe.

No, I'm not joking and I'm not making it up.
This is why IBM cannot just have their chess computer analize all possible endgames and always win. There is not enough computing power in the world to do that.
They have to try to teach the computor how to play, analize the strength of their position, anticipate future moves, etc., similar to the way a human plays.

[/ QUOTE ]

Can you show me the math behind your statement.

[/ QUOTE ]
Offhand, no, I can't. I may try to do a little research when I get the chance.
As far as where this claim came from, it was in a documentary about the IBM computer built to challenge the world chess champion, and the challenges they faced in building and programming the computer. They were asked why couldn't they just have the computer memorize every possible combination of legal chess moves, like you can do with a simpler game like checkers or tic tac toe, then it will always be a draw unless a mistake is made. That's when they said that the number of different combinations of legal chess moves is greater than the total number of atoms in the universe. So that was not an option.

bisonbison

02-01-2007, 03:43 PM

Bison I'm not quite sure what you're asking. If you want to explore all the possible moves with 32 pieces, the problem grows exponentially the further you look forward. In one step we can make 10 moves, change this horizon to 2 and there are 10*10 possible moves, etc. It is growing exponentially. (When you grow things exponentially, you very quickly exceed the number of atoms in the universe, it's no big deal)

By distributing the computing, you are increasing your computing power and storage arithmetically. It doesn't really help and won't even bring you close.

Dis,

Oh, I'm not saying that I think this is a small issue, and I understand the implications of the exponential growth of possibilities. I'm just curious how the current effort to solve the 7-piece endgames is proceeding practically, and wondering what the theoretical processing time and required storage space is for that step, for the 8-piece, for the 9-piece, etc.

The followup question then is, who's generating these tablebases, and are they relying on proprietary technology to advance it (so their efforts are siloed) or are teams interested in solving the current frontier (7-piece games) using software that could marshall distributed computing to (at least in a minor way) augment their efforts?

Al68

02-01-2007, 04:24 PM

According to Wikipedia, the number of atoms in the observable universe is estimated to be 10^80. And the game-tree complexity (the total number of possible games that can be played) of chess is 10^123.

I don't know how many bytes would be required to store each single "possible game", but no matter how small you make the "byte storage device", it wouldn't be able to store more bytes than the number of atoms it contains. So there would never be enough atoms in the universe to make the storage device. All of this tells me that chess cannot be "solved" completely, ever.

disjunction

02-01-2007, 05:12 PM

[ QUOTE ]

Dis,

I'm just curious how the current effort to solve the 7-piece endgames is proceeding practically, and wondering what the theoretical processing time and required storage space is for that step, for the 8-piece, for the 9-piece, etc.

The followup question then is, who's generating these tablebases, and are they relying on proprietary technology to advance it (so their efforts are siloed) or are teams interested in solving the current frontier (7-piece games) using software that could marshall distributed computing to (at least in a minor way) augment their efforts?

[/ QUOTE ]

Ah, I misunderstood your question. I just read the wiki article and I had never heard of endgame tablebases before. This could be because I'm a dumb ass who avoids going to department seminars, or maybe it's more of a chess problem than a search problem. You'd have to be a theory geek as well as a chess geek to be up on this stuff. Maybe someone else can answer, I'm in awe right now because apparently a lot of people here are both. /images/graemlins/laugh.gif

valenzuela

02-01-2007, 05:17 PM

Al but many of those games would have missed mates.

Philo

02-01-2007, 05:58 PM

And don't forget they have to program the computer to play specifically against Kasparov!

chezlaw

02-01-2007, 06:14 PM

[ QUOTE ]
According to Wikipedia, the number of atoms in the observable universe is estimated to be 10^80. And the game-tree complexity (the total number of possible games that can be played) of chess is 10^123.

I don't know how many bytes would be required to store each single "possible game", but no matter how small you make the "byte storage device", it wouldn't be able to store more bytes than the number of atoms it contains. So there would never be enough atoms in the universe to make the storage device. All of this tells me that chess cannot be "solved" completely, ever.

[/ QUOTE ]
Can't we solve it with a quantumn computer. Searching every node on the tree simultaneously should only take a few minutes.

chez

ChrisV

02-01-2007, 08:41 PM

[ QUOTE ]
Can't we solve it with a quantumn computer. Searching every node on the tree simultaneously should only take a few minutes.

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I don't really know what I'm talking about, but I'm not sure a quantum algorithm could be designed that could do this. The quantum algorithms I've seen are designed to find one specific "correct" result, for example encryption cracking.

In any case, a quantum computer of sufficient complexity to do this is a couple of centuries off, I would estimate.

arahant

02-01-2007, 08:59 PM

[ QUOTE ]
[ QUOTE ]
Can't we solve it with a quantumn computer. Searching every node on the tree simultaneously should only take a few minutes.

[/ QUOTE ]

I don't really know what I'm talking about, but I'm not sure a quantum algorithm could be designed that could do this. The quantum algorithms I've seen are designed to find one specific "correct" result, for example encryption cracking.

In any case, a quantum computer of sufficient complexity to do this is a couple of centuries off, I would estimate.

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CENTURIES?!?! c'mon man....if it's doable, that is WAY too long. Look at the pace of advancement lately.

I'm actually willing to bet that chess is 'solved' within 100 years. Sadly, I only expect to live another 80 tops...

Al68

02-02-2007, 03:49 PM

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Al but many of those games would have missed mates.

[/ QUOTE ]
I would guess that most of those games would never be played as a practical matter. Most of them would certainly never be played by competent players.

But the question was, could chess be "solved"? I assumed this meant all possible endgames.

Siegmund

02-02-2007, 06:11 PM

"Solving chess," to me, implied calculating the value of the game when two rational players oppose each other, and showing how each can achieve that value: not giving a list of the best move in every legal position, but rather, a list only of those moves which (assuming chess is a draw) leave you still able to force a draw after moving them... or even one particular sequence of moves that leads to a draw, accompanied by a proof that no alternative sequence allows either player to force a win, only to win if his opponent makes an error.

thylacine

02-02-2007, 06:47 PM

Al68

02-02-2007, 07:40 PM

[ QUOTE ]
"Solving chess," to me, implied calculating the value of the game when two rational players oppose each other, and showing how each can achieve that value: not giving a list of the best move in every legal position, but rather, a list only of those moves which (assuming chess is a draw) leave you still able to force a draw after moving them... or even one particular sequence of moves that leads to a draw, accompanied by a proof that no alternative sequence allows either player to force a win, only to win if his opponent makes an error.

[/ QUOTE ]
This second proof, that no alternative sequence would allow either player to force a win, would have to account for all alternative sequences possible, not just rational. Otherwise, you could beat a "perfect" player by making "irrational" sequences of moves. So a "perfect" player would have to be able to counter any possible sequence of moves, not just the rational ones.

Siegmund

02-02-2007, 08:59 PM

A perfect player has to counter any possible opposing move, yes.
It is often quite easy, however, to show that a particular move may cost but cannot gain, without evaluating a complete game tree.
My basic point is that some parts of solving chess will be done by exhaustive search, and others by thinking clearly.

On a slightly related note.... does anyone have an example of a situation where a player's position would be improved by removing one of his own pieces from the board without compensation, that does not involve a discovered attack?

chezlaw

02-02-2007, 09:45 PM

Al68

02-02-2007, 10:14 PM

[ QUOTE ]
A perfect player has to counter any possible opposing move, yes.
It is often quite easy, however, to show that a particular move may cost but cannot gain, without evaluating a complete game tree.
My basic point is that some parts of solving chess will be done by exhaustive search, and others by thinking clearly.

On a slightly related note.... does anyone have an example of a situation where a player's position would be improved by removing one of his own pieces from the board without compensation, that does not involve a discovered attack?

[/ QUOTE ]
Well, you may have a point there. Maybe some possible endgames could be eliminated using logic. However, even if you eliminated 99.99999999999999999999999999999999999999 % of all possible endgames through logic, you would still have more endgames left than there are atoms in the universe.
And that would only be a partial solution, not a total solution.
As a practical matter, there is no reason to completely "solve" chess, but that was not the original question.

"Solving" chess would involve all endgames, not just the small fraction (still a large number) that would conceivably be played or the small fraction where neither player makes a clearly "costly" move.

And even in your example where a move may cost but cannot gain, if you exclude those game trees from the "solution", a player could make such a move early (at cost), and then take advantage of the fact that the resulting game tree is not a part of the "solution". So a partial solution cannot be said to ensure a win or draw against any opponent, only those opponents who would never make such moves.

Mickey Brausch

02-03-2007, 04:07 AM

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[ QUOTE ]
I agree that chess played perfectly is almost certainly a draw.

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Hopefully it will be solved one-day and proven that black wins. Ultimate Zugzwang.

[/ QUOTE ]"With 1. e4, White's game is in its last throes".

Who said that?

Mickey Brausch

02-03-2007, 04:12 AM

From a game-theory approach, "solving chess" can only mean finding a specific series of moves which enable each side to claim at least a draw.

In other words, every move by either W or B should, first and foremost, not provide the other side with an advantage (or, in the case of a move by B, with an increase in advantage), no matter what the other side plays.

Mickey Brausch

FortunaMaximus

02-03-2007, 04:40 AM

[ QUOTE ]
[ QUOTE ]
According to Wikipedia, the number of atoms in the observable universe is estimated to be 10^80. And the game-tree complexity (the total number of possible games that can be played) of chess is 10^123.

I don't know how many bytes would be required to store each single "possible game", but no matter how small you make the "byte storage device", it wouldn't be able to store more bytes than the number of atoms it contains. So there would never be enough atoms in the universe to make the storage device. All of this tells me that chess cannot be "solved" completely, ever.

[/ QUOTE ]
Can't we solve it with a quantumn computer. Searching every node on the tree simultaneously should only take a few minutes.

chez

[/ QUOTE ]

Problem: This would result in having a solution set that compromises of every possible chess game that can be played.

It's less of an algorithm problem in that instance as implied from ChrisV's post as it is a problem of storing and accessing the whole set. At minimum, you have to be able to store vast sums of data temporarily, locally. Doing that with something with such dense complexity as chess would be a ridiculous waste of resources.

And you would probably want to filter for the games that basically consist of zero strategy and opposing pieces move at seemingly random impulses irregardless of what the other does.

Maybe that part is more problematic to devise an algorithm for, indeed, as you'd need a fairly large, consistent database of world-class games to be able to design a reasonably accurate filtering algorithm.

(Typed more about q-computing, but that's for another discussion probably. Anything finite is solvable though, it's a matter of efficiency and purpose.)

MusashiStyle

02-03-2007, 05:46 AM

remember that both sides may choose to maneuvre their pieces back and forth ad infinitum ( this is if you exclude the 50 move rule or the 3 fold repition rule)

So then you want to solve chess with the 50 move rule and the treefold repition rule.

It seems to me that chess would be solvable then, but only because of these rules. Basically, these rules state that one side must move a pawn forward or make a piece exchange at least once every 50 moves or else the game may be declared "drawn".

So this rule forces all the pawns to queen or be exchanged. This would leave a situation with just pieces. But then the other side of the 50 move rule kicks in which forces all these pieces to be exchanged or else the game will be declared drawn etc.
So my point is that once you have a computer that can calculate all the way to the end of these lines, the 50 move rule enforces the non-infinity of moves.

Hope that makes sense.

FortunaMaximus

02-03-2007, 09:59 AM

[ QUOTE ]
So my point is that once you have a computer that can calculate all the way to the end of these lines, the 50 move rule enforces the non-infinity of moves.

[/ QUOTE ]

You'd be able to keep the set within a finite bracket, sure.

That hardly lessens the scope of the problem. It's still a pretty huge solution set.

The issue probably is that it wouldn't provide a complete set.

And to devise an algorithm that takes into account solvable games while avoiding irrelevant loops, which the 3-fold repetition rule can filter for does not remove the problem of the one-fold and two-fold repetition issues. If those two other issues are accounted for, sure, then the algorithm clarifies a bit.

The halting probability of such a program would be nonzero, however. Even if the parameters involved are finite. Simply because of the space the data would occupy.

In linear processing systems, it is solvable, but it'd take too long. Memory can be reused as you would be able to discard the trash data after checking it. You would have to be extremely careful with the trashing process though. So the carrying capacity never needs to be astronomical.

In systems that use the compressed amounts of time that q-computing would permit, you run into an immediate problem that comes from being able to temporarily store vast sums of data. But it's possible, but not probable.

It leaves me wondering if such a calculation has a nonzero halting probability. It would seem this is not so given the seemingly simple exponential scope of the problem. The bothersome aspect is such an incorrect algorithm can run for eons without being certain of this.

If number or solution sets can be said to have substance or a form in three-dimensional space, what one is able to intuitively visualize is the solution, or the pieces, are far too heavy for the chessboard.

(1. a3) o_O

madnak

02-03-2007, 12:22 PM

[ QUOTE ]
remember that both sides may choose to maneuvre their pieces back and forth ad infinitum ( this is if you exclude the 50 move rule or the 3 fold repition rule)

So then you want to solve chess with the 50 move rule and the treefold repition rule.

[/ QUOTE ]

Nah, you would only need to record each position. The positions themselves are independent of the specific game tree (that is, there is no difference between repeating the moves 50 times and 5,000,000 times, in game terms, so the computer doesn't have to distinguish between the two cases).

Daisydog

02-03-2007, 06:13 PM

I believe it was either Ernst Zermelo or Harold Kuhn who proved that every finite game of perfect information has a pure strategy Nash equilibrium.

Hercules

02-04-2007, 04:09 AM

To everyone, who says that the perfect ending in a chessgame is a draw, you might wanna read this:

1) The fact that many games end in a draw does not proof this fact at any point. Most players play prejudiced like 1.h4 is bad and 1.e4 is good. We (chessplayers:) all know, that like 50 years ago everybody would laugh at you, if you`d have played the Sveshnikov defense in the sicilian. Or what about isolated pawns?
There have been so many shifts in paradigmata, that it is impossible for us today to say that the current top players are playing optimal. Modern chess is more about dynamic factors and its more about exceptions than rules.
So if the GM`s don`t play optimal,then the results of them do not show anything.

2) However i believe that chess is a solved game, but i don`t know anybody, who has the key. Although computer programs are amazingly strong, they are still only "perfect humans". All their "knowledge" is knowledge given by humans. The main difference between top players and computers are, that the computers simple compute better and don`t do devastating mistakes (see the match Fritz vs Kramnik)
So i think we are still far away from "solving" chess. Every month there are some creative new ideas in the opening out there, which no one has ever thought of before.
So i can`t see anybody drawing conclusions on what the "perfect score" must be.
3) I have a friend and he ran some simulations with a computer and his output was, that BLACK wins most games theoretically. This is however a contradiction to reality, because white has obv a decent advantage in today`s thinking and playing.
But his simulation didn`t use brain. It just ran every legal move order (randomly) and spit out the fact that black won most games. That doesn`t prove anything, but it is something to consider before saying "a perfect game ends draw" etc...

ChrisV

02-04-2007, 04:41 AM

[ QUOTE ]
2) However i believe that chess is a solved game, but i don`t know anybody, who has the key. Although computer programs are amazingly strong, they are still only "perfect humans". All their "knowledge" is knowledge given by humans.

[/ QUOTE ]

This is not true. Computers don't have any "knowledge" of how to play chess, really. Their main advantage is brute force evaluation of millions of moves, which is completely different to how humans play chess. In fact, against the lesser computers of yesteryear, and even to some extent today, the tactic used by humans against computers was to play "anti-computer" chess, manouevering the computer into a situation it would play suboptimally. If computers were simply "perfect humans" this would not be possible.

[ QUOTE ]
I have a friend and he ran some simulations with a computer and his output was, that BLACK wins most games theoretically. This is however a contradiction to reality, because white has obv a decent advantage in today`s thinking and playing.
But his simulation didn`t use brain. It just ran every legal move order (randomly) and spit out the fact that black won most games.

[/ QUOTE ]

I would be surprised if this were not a sample size issue. I wouldn't expect to see any bias to White in this situation, I'd expect it to be 50/50.