Chess will never be solved, here's why

First page has the most amount of downvotes i've ever seen...

Nearly 100 thumbs down!

Janko is getting close

He's almost halfway there

he actually came up with it himself.

1. He takes a starting number of chess positions (which he draws from an authority),

2. and reduces that number to the amount without what are deemed 'unnecessary' positions, taking a number from another authority. (although tygxc doesnt sufficiently demonstrate that the type of positions removed by his authority is the same type of positions 'unnecessary' to factor into calculations as he claims)

3. tygxc takes the square root of that number as an heuristic to simulate considering only one side's variations. (IE, the game tree splits into every possible black response, but continues through only ONE white response, or vice versa)

the approach as a whole is sound, however the individual steps and interpretation of the resulting number are very flawed

a) insufficient justification/misinterpretation of the authority for the first reduction. The current upper bound for the number of chess positions without promotion is 10^37. (i cite an authority here that you can look up, however the number itself in the article is locked behind a paywall that I got through with a college account). tygxc's number at this stage is 10^34. it is very likely possible to reach this number (or even lower) through calculations of simplification to the 7 man table base, although I would like to reiterate that the level of proof for this would be VERY loose. tygxc provides no such justification.

b) using a square root to consider only one response from white means that tygxc's heuristic calculates only an estimate of the size of the solution tree itself, NOT the computational power required to FIND that tree.

For example, the total possible checkers positions is 10^20.

the number of positions calculated in order to solve checkers was 10^14, excluding the endgame table bases.

( https://www.cs.mcgill.ca/~dprecup/courses/AI/Materials/checkers_is_solved.pdf )

per tygxc's claims, only 10^10 positions would have needed to be calculated.

this is an obvious contradiction

the "how is the white move found" is key. there is no room provided for finding the move in tygxc's calculation.

(from tygxc, typos corrected and equivalent numbers rewritten)

"2.07e37 legal sensible positions starting from 26 men
in analogy to the checkers proof: square root of this: 4.55e18 relevant positions 
cloud engine 10^9 nodes/s
4.55e9 s = 144 years on 1 cloud engine for the whole of chess"

tygxc assigns only 1 node to each position. the ICCF games that tygxc offers as 'perfect' chess required over a billion nodes for each position.

https://www.chess.com/forum/view/general/chess-will-never-be-solved-heres-why?page=5#comment-66899947

It's simply wrong that you get a 1000-fold shrinking of the number of positions by using a "small" tablebase (like 7 or even the future 8 piece), These tablebases are just a tiny subset of the positions because there are way fewer classes (like KkQ) and typically way fewer positions for each combo, compared with with bigger tablebases. The set of positions is in fact a set of 30 tablebases for 3, 4, 5, ... 32 pieces. They keep getting bigger most of the way, but at the end it is less clear, as there are fewer classes. Only one class for the 32 piece tablebase, since you can't promote without a capture. Someone has done analysis to adequately estimate the sizes.

Maybe - but I'm thinking that tygxc is going by his indoctrination as a chess player.
Quite a strong player as players go.
He's made himself a good player. Over a period of years and years.
But like so many things - there can be flip sides.
Or are flip sides.
--------------------------
He wants to 'dismiss' e4 e5 Ba6.
For rigorous solving - such dismissal isn't legitimate.
But for 'approximations to solving' such dismissal looks quite legitimate.
The computer wouldn't have to 'Dismiss' immediately - it can take a few more billionths of seconds or whatever to make sure there's no tactical retaliations for the side that's making the blundering move and only then 'Dismiss'. Or label as 'approximately solved.'
-----------------------------------
'Material sacrifices 'to get position' are much harder than more purely tactical ones.
Although the sacrifice itself is tactical.
Its much rarer that they're sound.
Sac a knight for an f7 pawn. Say Kxf7 reply.
No immediate tactical recovery of the material.
No immediate mate.
Maybe not even immediate followup checks.
No immediate followup mate threats.
'positional' compensation.
Difficult. With its own spreading tree of game possibilities.
Might be 'sound' or 'unsound'.

@12343

"linking 10^17 to arguing from authority. However, where did he get it from?"
++ I have explained many times.
(1) There are 10^44 legal positions, but the vast majority of those makes no sense with multiple underpromotions to pieces not previously captured. https://github.com/tromp/ChessPositionRanking 
The 3 positions shown are legal, but cannot result from optimal play by both sides because of multiple underpromotions by both sides. Underpromotion to rook or bishop only makes sense to avoid a draw by stalemate, and that makes sense for 1 side only, not for both sides.

(2) A better number is 10^37 positions without promotions to pieces not previously captured.
https://arxiv.org/pdf/2112.09386 
However, that is too restrictive, as positions with 3 or 4 queens do occur in perfect games with optimal play from both sides. So multiply by 10 to include such positions.
That gives 10^38 positions without underpromotions to pieces not previously captured.

(3) Now note that even these positions cannot result from optimal play by both sides.
I inspected a random sample of 10,000 positions. None can result from optimal play by both sides. Convince yourself: take one and try to contruct a reasonable game that leads to it. E.g.

That leads to 10^38 / 10,000 = 10^34 positions.
This is a conservative estimate: Tromp conjectured only 1 in 10^6 to be sensible.
That would lead to 10^32 positions.

(4) Now assume perfect alpha-beta pruning. For N white moves we only need to look at 1 black move, hence not N * N = N², but N * 1 = N = Sqrt (N²) positions.
Hence Sqrt (10^34) = 10^17 positions relevant to weakly solving Chess.

That is also what Schaeffer wrote: perfect alpha-beta pruning halves the exponent.
Schaeffer needed 10^14 positions of his 5*10^20 legal positions, i.e. exponent 0.67.
However, chess programs have evolved more then Chinook for checkers and Chess is easier to prune than Checkers. Checkers is more like a pawn endgame in chess.

In the most optimistic case, we would end up with Sqrt(10^32) = 10^16 positions.
In the most pessimistic case, we would end up with (10^34)^0.67 = 6*10^22 positions.
10^17 relevant positions is a fair estimate of the effort needed to weakly solve Chess.

@12297

"For N = {1, 2, 3, 4, 5 ...} define the sequence of questions Q(N) like this:
Q(N) = If N games were played between two world class centaurs (engine assisted human) using current engines and it was a draw, would that be a proof that chess is a draw?
which is the least N such that the answer to Q(N) is Yes?"

++ If asked this way, then the answer is N = 0. Proof that Chess is a draw already is the deductive argument that the initiative is a white advantage of +1 tempo = +0.33 pawn = not enough to win, especially since each further move dilutes the advantage. The consecutive 110 draws out of 110 ICCF World Championship Finals games only confirm that in a convincing way. Even the results of previous ICCF World Championship Finals, which had sparse decisive games, confirmed that. To me for all practical purpose Chess is ultra-weakly solved and the game-theoretic value of the initial position is a draw.

Ultra-weakly solved to a draw does not indicate how to draw against all opposition.
That is weakly solving Chess.
The present 110 ICCF WC Finals draws are at least part of the weak solution of Chess.
The fact that they are redundant makes it fail safe.
You could object that they are not yet complete.
I would consider 272 ICCF WC Finals draws as a weak solution of Chess: both complete and redundant.

@12347

"necessity to explore games of say 150 to 200 moves "
++ The ICCF World Championship Finals games end in draws in average 39 moves.
Games of 150 to 200 moves are a myth, they only occur between two weak players that hop aroud aimlessly.
Besides 10^38 = 2^126 = 2^(63*2)
Thus after 63 moves with 2 non transposing choices per ply you have the whole of Chess.
Any game over 63 moves must contain strings of forced moves.
There are no more chess positions than there are chess positions.
That is the pigeonhole principle

@12349

"Someone has done analysis to adequately estimate the sizes."
++ Yes: Table 3

@12336

"the square root formula is based on assumptions of constant branching factor, games of equal length and no transpositions"
++ No. The square root is only based on the assumption of perfect alpha-beta pruning.
So it depends on the efficiency of the engine: 0.67 for Chinook of Schaeffer for Checkers and close to 0.5 for modern Chess engines, which have evolved more over years.

Variable branching factor, games of unequal length and transpositions do not affect that.
To cope with transpositions it is enough to consider an equivalent branching factor of non-transposing moves. That is also how an engine works: if a branch reaches a transposition, then that branch is not calculated, but looked up in the transposition table.

"Transpositions are ubiquitous in chess analysis." ++ Yes.
As 10^38 = 3^80 a game of average 40 moves has only 3 non transposing choices per ply.

"all positions with promotions or multiple promotions"
++ No. Underpromotions are rare. Promotions to pieces not previously captured are rare. Underpromotions to pieces not previously captured do not occur in games with optimal play from both sides and can be ignored indeed.

"states can be reached in an ENORMOUS number of ways" ++ Yes, that is why positions, not games count. That is why a way to handle transpositions is necessary i.e. a transposition table.

"The square root is only based on the assumption of perfect alpha-beta pruning.
So it depends on the efficiency of the engine: 0.67 for Chinook of Schaeffer for Checkers and close to 0.5 for modern Chess engines, which have evolved more over years."

no justification given for the 0.5. in fact, most modern chess engines are actually far less efficient.

in addition, if you actually read the solution paper to checkers, you would have noticed how schaeffer points out that an alpha beta search does not prevent double counting positions.

tygxc is going to respond to this so im going to give you a heads up

tygxc's end claims here are not supported by any area of research, and ive actually shown tygxc's "logic" to dozens of math majors and math professors, and they all found the exact same errors that have been pointed out to tygxc many times.

"states can be reached in an ENORMOUS number of ways" ++ Yes, that is why positions, not games count. That is why a way to handle transpositions is necessary i.e. a transposition table." completely ignores the computing costs of using such a transposition table

its funny, i just discovered several new ways that tygxc makes errors.

"An alternative would be to use only computing—i.e., build a search tree using the alpha-beta algorithm. Consider the following unreasonably optimistic assumptions: number of moves to consider is eight in noncapture positions, a game lasts 70 ply, all captures are of a single piece (23 capture moves), and the alphabeta search does the least possible work. The assumptions result in a search tree of 8^(70–23) = 8^47 states. The perfect alpha-beta search will halve the exponent, leading to a search of roughly 8^47/2 ≈ 10^24"

alpha beta pruning is calculated from the game tree complexity, not from a stored positions table.

and i bet tygxc wont figure out why schaeffer did that calculation. but i figured it out.

this is why I'm partially convinced there's a sort of language misunderstanding on tygxc's end. nobody with a math education beyond middle school would call that a "proof", unless "proof" translates to something weaker in the different language that tygxc is native in.

I think I see what tygxc is now trying to say though.
He's apparently claiming that in 'optimal play' the advantage for either side will never approach 1.0 ...
but he's totally blind to the fact that that's today's engines against today's engines.
Has he yet responded to the fact that today's engines would beat yesteryear's engines?
Has he responded to the argument that engines of the future will be stronger than today's?
He's not looking at 'the big picture'.
And probably isn't going to.
'No cheddar' there for him.
But that's OK.
Its more like 'intellectual tunnel vision' than 'intellectual dishonesty'. 
Its the other guy that's dishonest.
Like claiming he's the only one discussing the forum subject.
Blatant dishonesty.

@12362

"in 'optimal play' the advantage for either side will never approach 1.0" ++ Correct.

"today's engines would beat yesteryear's engines?"
++ Today's ICCF World Championship Finalists with each 2 engines of 90 million positions per second at average 5 days per move have now reached 0 error, 110 draws out of 110 games.
In previous years there were sparse decisive games, thus some errors.

"engines of the future will be stronger than today's" ++ Yes, engines and humans of the future might reach the same 0 error in a shorter time than average 5 days per move.