#19
There is no infinity, there are no trillions of moves.
If there is no infinity a lot of mathematicians have been sorely misguided. I've heard of ultra-finitism, but the proponents I've read will usually concede trillions.
Each chess game ends in 5898.5 moves at most.
Not quite true. Every game played under FIDE competition rules ends in 8848.5 moves at most. This is a consequence of the mandatory 75 move rule. A game of this length is possible under those rules without breaking the mandatory 5 fold repetition rule. See https://wismuth.com/chess/longest-game.html. The figure 5898.5 would apply under those rules if it were not necessary to claim a draw under the 50 move rule. But it is.
Under Fide basic rules (which I explicitly mention in post #19) there are neither mandatory nor claimable rules limiting the length of a game, though with current medical technology the length of a game involving a human would in practice be limited by the expiration of one of the players and with most current theories about the evolution of the universe would in any case be limited by e.g. the heat death of the universe.
In each position there is a finite number of legal moves.
Hence there are only a finite number of possible chess games.
The number is large because of transpositions.
There are billions of ways to reach the position after 1 e4 e5.
The number of games is irrelevant, it is the number of positions 4*10^37 that counts.
Agreed. But that is of course not to say that the number of games that need to be considered is the same as the number of positions. It does give a limit on the number of positions that would need to be indexed in a full set of tablebases.
#20
Each decisive game implies an error. Increasing the time reduces the error rate.
Does that mean you've already weakly solved chess and it's a draw and also proved that AlphaZero's evaluation function doesn't suffer from minimax pathology? (http://izvolitve.ijs.si/Stacks/Articles/19805735.pdf)
At 1 second per move: 88.2% draw rate hence 1 error per 8.5 games.
At 1 minute per move 97.9% draw rate hence 1 error per 47.6 games.
Extrapolating:
At 1 hour per move 1 error per 266.6 games.
At 60 hours per move 1 error per 1493 games.
But you have no idea how the data from which you're extrapolating corresponds with objectively perfect play. (I'm assuming the answer to my previous question is "no and no".) The difference between that and Haworth's extrapolations is that his are based on statistics from the Nalimov tabebases which do correspond with perfect play under basic rules.
Also the ICCF results support that chess is a draw.
https://www.iccf.com/event?id=66745
Poppycock. They support nothing of the kind.
If you take a random game from the middle of the table in your link
you will notice that the draw reached is not a dead position. It's an agreed draw. This is irrelevant in respect of perfect play.
What has clearly happened is that the players have followed their respective opening books or that of their silicon aids for a couple of dozen moves, played on without seeing any forced mate for another dozen moves or so and then having no good ideas on how to continue in any advantageous way given up playing in a position which, while no doubt theoretically less intractable to solve than the starting position is still well beyond current resources. (They might as well have agreed a draw at the outset.)
This is exactly to be expected.
When equally matched players are out of their depth in a position the most common result is a draw. This has nothing to do with the objective evaluation of the position.
An example would be the KQKNN endgame which has for centuries been regarded as generally drawn. Nalimov says that around 75% of positions are actually winning for the queen. But in practice the general result is a draw. I've left SF playing itself in a number of times in such positions with mate depth over 45 moves and about 5 times out of 6 it draws under the 55 move rule.
For centuries humans have been weeding out lines that result in losses that they can fathom from move sequences following the starting position and for decades computers have been doing the same faster and more accurately. The result is that for equally matched players who have an enyclopaedic knowledge of opening analysis the opportunity to reach winning positions that are within their depth is shrinking and you can expect that the number of draws will correspondingly increase.
But that is not an indication that chess is objectively a draw, merely that players with limited look ahead are increasingly reaching positions where they're out of their depth.
The comparable results with stalemate = win show that stalemate = win does not increase decisiveness. This reinforces that classical chess with stalemate as an additional drawing resource is a draw indeed.
As I already remarked, without any way of comparing AlphaZero's play with accurate play when the tablebases peter out it shows nothing at all about accurate play, only about play at AlphaZero's level (neither for normal chess nor stalemate=win chess).
Engines are weak at openings without their opening books as positions with >26 men have too many legal moves.
Engines without their opening books are weak against strong human players only because the latter carry their opening books around with them on top of their shoulders.
Engines are weak at endings without their endgame table bases as positions with <7 men have too deep lines.
Not anything like as deep as positions with >7 men (or even to a lesser extent =7 men).
But its not just engines.
In 2000 Topalov (rated #1 at the time) and Karpov (i.e. Karpov) reached a KNNKP endgame in a position the same as the one I posted in #20 but the pawn and blockading knight shifted one square NW.
In the next nine moves the players blundered three half points between them.
#19
There is no infinity, there are no trillions of moves.
Each chess game ends in 5898.5 moves at most.
In each position there is a finite number of legal moves.
Hence there are only a finite number of possible chess games.
The number is large because of transpositions.
There are billions of ways to reach the position after 1 e4 e5.
The number of games is irrelevant, it is the number of positions 4*10^37 that counts.
#20
Each decisive game implies an error. Increasing the time reduces the error rate.
At 1 second per move: 88.2% draw rate hence 1 error per 8.5 games.
At 1 minute per move 97.9% draw rate hence 1 error per 47.6 games.
Extrapolating:
At 1 hour per move 1 error per 266.6 games.
At 60 hours per move 1 error per 1493 games.
Also the ICCF results support that chess is a draw.
https://www.iccf.com/event?id=66745
The comparable results with stalemate = win show that stalemate = win does not increase decisiveness. This reinforces that classical chess with stalemate as an additional drawing resource is a draw indeed.
Engines are weak at openings without their opening books as positions with >26 men have too many legal moves.
Engines are weak at endings without their endgame table bases as positions with <7 men have too deep lines.