#49
Do you know how computers were 60 years ago? Big as a room, vacuum tubes, Hollerith punch cards, magnetic tape...
As explained we do not need 10^120 positions: a 13-men table base probably suffices.
We got +1 man per past decade, there is no reason to believe it will not progress at the same rate.
#50
Checkers has stones and queens and the queens caused most of the trouble.
#47 See
en.wikipedia.org/wiki/Solving_chessWatch out for terms-of-art which have different meanings than are normally used in chess.
Also
en.wikipedia.org/wiki/Solved_game#Solved_gamesLosing chess
Weakly solved as a win for white beginning with 1. e3.
@Goldrider /#40
"To model Chess, we need to somehow make every particle in the observed universe store around 10 ^ 38 (a very huge number: 100 Trillion Trillion Trillion) Chess positions.
So, any conjectures about chess will need to be based on empirical observations."
That's not true. You can make statements and proofs of those statements of games without going through all the states of a game. For instance, it's not hard to proof that a game of Hex, with perfect play, is a win for the first player. Even if the board is large enough such that there are more possible games than there are in chess. (On a 19x19 board, there are (19 x 19)! (about 10^768) possible games -- and even a smaller 11x11 board has more legal positions than chess has).
But the proof that Hex is always a win for the first player is not constructive. It does not show how to win the game -- it does not even show what the best first move is. And that does give some hope -- there is the possibility that it is shown that with perfect player chess is a win/draw/loss for white, without giving insight on how to reach that.
#54 Perhaps people already know, but if not ...
The variant of chess where each player moves twice on their turn has been proved to be either a win for White or a draw. That is, Black has no win. The proof is like the proof for Hex; by contradiction. The proof is even easier than the Hex proof because in the Hex proof it has to be shown that extra pieces on the board for a player cannot hurt him, and it had to be proved that Hex cannot end in a draw.
Idea of the proof: Assume Black has a winning set of moves. White plays 1.Nf3 & Ng1 and for the rest of the game plays Black's winning set of moves. White will win. Contradiction.
I guess I caused a big kerfuffle by admitting the possibility that chess is winning for black. Let me clarify my position:
1. Chess could be a win for black. This is unarguable. I don't think it is likely, but I do think there are ways to disprove this and we haven't.
2. Chess probably isn't a win for black. Duh.
3. Chess could plausibly be a win for white, or a draw. People who spend their whole lives studying chess come to different opinions on this, there are good reasons for why it might be one or the other.
4. I think chess is *probably* a draw.
5. Talking about "better" or "worse" play doesn't make sense when you're also talking about things which can be forced. If we want to talk about forcing lines, that is the realm of mathematical proof. If we want to talk about good or bad or chances or likelihood, than it is fully appropriate to bring in empirical and anecdotal evidence.
TL;DR Chess might be a win for black, the fact that winning as black is harder isn't evidence against that. Chess might be a win or a draw and what happens in imperfect grandmaster games is only somewhat informative there.
"We got +1 man per past decade, there is no reason to believe it will not progress at the same rate"
An 8-piece tablebase has 38,176,306,877,748,245 nodes. The expansion factor is around 75, but let’s be optimistic and say it continues to go down. I guesstimate 42,792,750,000,000,000,000,000,000 (4 * 10^25 or 4E25) for a 13-piece tablebase, or about 2 ^ 84. A cubic centimeter has 1.8E23 atoms, and we need about 3 bytes (24 bits) to store a single node. So, one would need a cube about 16-17 centimeters on a side to have enough atoms to store this 13-piece tablebase. But that’s optimistic: I’m assuming the tablebase expansion factor contracts more than it currently does, and I’m assuming we develop a technology which can store a bit using a single atom.
Next, we need to store all the moves to go from the opening position to this 13-piece tablebase. Roughly, 20 possible moves each move, and we will just have one reply per move to save space. 40 moves to get to 13 pieces, so that’s a lot bigger than our 13-piece tablebase: 10 ^ 52, which is more than the number of atoms in the Earth.
So, yeah, we could weakly solve Chess if we make all of Jupiter a database to store the weak solution to Chess, so it’s within the rage of possibility to solve, but only barely.
— — —
In terms of there being a mathematical breakthrough to solve Chess (and, yes, the ones for Hex and double move Chess are quite interesting), Chess has been around for about 500 years and we haven’t come up with one yet. We went from giving huge numbers for the amount of computation needed to solve RSA-129 in 1977 to solving it in 1994 using mathematical breakthroughs, so, if there is a breakthrough, it’s unusual that mathematicians have not been able to come up with one in the last 500 years.
— — —
The “we can’t prove a negative, so it might exist” line of reasoning can be answered with a famous quote: “That which can be asserted without evidence, can be dismissed without evidence”. I haven’t seen any actual evidence for a Black win presented in this thread, so I will dismiss this assertion until when and if compelling evidence for it is presented. All of the evidence presented leans towards a draw, with an unlikely possibility White has a win.
Yes, sure, a Black win is possible, in the same way there might be a teapot floating in space between Earth and Mars is possible, but there isn’t a really good reason to believe this assertion (but, if you believe it, match me with the Black pieces and we can play a 7-day rated correspondence game with me playing White once I finish one of my games).
@Goldrider Mostly good analysis, but there are a few things that reduce the space needed for 13 piece tables.
Firstly, every time that you have two of the same piece in an egtb, the space needed is halved vs with different pieces. Meaning that KNNvK has roughly half as many unique positions as KNBvK. And the occurrences of duplicates go up a lot with 8+ pieces on the board.
Second you estimated that we need 3 bytes to store a single position value. With no compression and the 50 move rule, we need 8 bits per position, allowing for -100(loss in 100 ply) through +100(win in 100 ply) however compression currently brings it to less than 1 bit per position for the 7 piece tables, and with compression optimized for very large tables, it should be practical to fit 3-4 positions per bit on average.
There are also some things that would reduce the space needed for the opening lines, mostly that weak moves tend to result in shorter games, so no need to store a full 40 turns deep for most lines. Only the best few moves at each level will be good enough to need that much depth. Although it is quite possible that the best few will need far more than 40 deep, so it might balance out.
#57
This is an incredibly silly argument, and it's telling that you seem to base most of your epistemological beliefs on pithy unattributed quotes... There is meaning in the fact that the result of chess is a fact, there is exactly one well defined correct outcome and at most some finite set of perfect games that lead to that outcome. Admitting the possibility that the perfect outcome is a win for black doesn't damage you or your understanding of the game...
If we were talking empirically, I'd say without a doubt, in a second, human chess is of course a draw, with an advantage for white. But we are talking about the mathematical structure of the game of chess. There is no nuance in that. We say things about it that we know, that we can prove. One thing we have proven is that zugzwang is possible, and white does not have a strategy to "switch colors" by throwing away a tempo.
You don't gain anything by baselessly deciding which conjectures are true or false, regardless of how likely or unlikely they are to you, and then trying to ridicule people for having an open mind. That is exactly the opposite attitude that people should bring to mathematical reasoning.
