New Ways To Solve Chess

[syzygy]

I had this discussion with him before as well:
viewtopic.php?p=859438#p859438

From that post: "The whole point of referring to this paper and such results in general is that it shows that in general there is no quick way to solve games like chess."

That statement is plain wrong. It shows that there's no quick way to solve it by reduction - and that is the absolute limit of what it shows.

You so clearly have no clue.

[syzygy]

As I have already explained to you, the first paper just calculates a value for a position based on the number of pieces that attack and defend each other. This is what hand-crafted evaluation functions have been doing since forever. Please explain how such a concept could possibly help "solve" chess.

Maybe it won't - but betweenness centrality, as explained, is very different from an attack table, and yields more valuable information - so:

1. Maybe looking at it more closely will yield more breakthroughs

2. It shows that there are aspects of the mathematics of chess to be uncovered

No it does not. It is just a number that the author hopes says something meaningful to human chess players. Basically the paper proposes a chess GUI feature.

The second paper is not about chess at all. It is about "evolutionary games", which chess is not. Please explain how the second paper could possibly help "solve" chess.

This is all explained in the original post: persistent homology provides a way to uncover shape in noisy data of many dimensions - which is a good basic description of chess.

No, "shape in noisy data of many dimensions" in the context of evolutionary games certainly is not a good description of chess.
Come on, you're just formulating bla bla and then say "so it will solve chess if someone would just look at it". You can't seriously believe that you are making sense here.

This is getting pathological.

Your P=NP argument is false.
Let's go back first to your "in polynomial time". What do you mean by it? Do you have any actual idea of what it means in this context? Please explain,

The same as everybody else who uses the expression.

So you have no idea what it means.

[syzygy]

The word 'implication' does not mean what you imagine and there is no implicit assumption that you claim. You have zero idea about what that paper has proved or not proved. You can't even see that GMs knowing the outcome of some positions is irrelevant to the result. You think authors spend months and years figuring out that game becomes or can become longer and longer as the board size increases? Yeah, not easy to prove that. Do you think "exponential complete" means it takes exponential time to complete the game? That is the only phrase in the abstract that would explain your understanding.

Please refer to the previous threads linked earlier in this thread. I'll summarise:

1. The chess paper claims an implication that there are positions in chess in which it will take exponential time in relation to the size of the board to determine which side will win

It an implication of the theorem that the paper proves.

2. The word "imply" means "seems to" - it does not claim a proof

The word "imply" in this context clearly means logically implies. It does not mean "seems to".

The statement is simply the logical consequence of generalized NxN chess being EXPTIME-complete.

4. The good paper limits the exponential-in-board-size claim to showing that it's exponential to board side in proving by reduction only (using the exact words "by reduction")

You simply do not understand these words. You formed a sentence without meaning.

[towforce]

As I have already explained to you, the first paper just calculates a value for a position based on the number of pieces that attack and defend each other. This is what hand-crafted evaluation functions have been doing since forever. Please explain how such a concept could possibly help "solve" chess.

Maybe it won't - but betweenness centrality, as explained, is very different from an attack table, and yields more valuable information - so:

1. Maybe looking at it more closely will yield more breakthroughs

2. It shows that there are aspects of the mathematics of chess to be uncovered

No it does not. It is just a number that the author hopes says something meaningful to human chess players. Basically the paper proposes a chess GUI feature.

You are quoting me out of context left, right and centre: the above is an example. I keep explaining over and over that the paper doesn't claim to have a way to solve chess. I did this right from the very first time I cited it. I then explained the how betweenness centrality might advance chess knowledge, and how it's an example of new methods uncovering previously unknown chess knowledge. Then you reply as if I had claimed that this had come from the paper.

I will not do this to you: my track record of questioning other people is close to 100% exemplary. So for the sake of progress, let me ask you questions:

1. Is it true that many seemingly complex chess positions can actually be resolved by relatively simple heuristics?

2. Is there an upper limit to the difficulty of 8x8 chess positions that could possibly be solved by heuristics?

[syzygy]

As I have already explained to you, the first paper just calculates a value for a position based on the number of pieces that attack and defend each other. This is what hand-crafted evaluation functions have been doing since forever. Please explain how such a concept could possibly help "solve" chess.

Maybe it won't - but betweenness centrality, as explained, is very different from an attack table, and yields more valuable information - so:

1. Maybe looking at it more closely will yield more breakthroughs

2. It shows that there are aspects of the mathematics of chess to be uncovered

No it does not. It is just a number that the author hopes says something meaningful to human chess players. Basically the paper proposes a chess GUI feature.

You are quoting me out of context left, right and centre: the above is an example.

What are you talking about, I included the context.

I keep explaining over and over that the paper doesn't claim to have a way to solve chess. I did this right from the very first time I cited it. I then explained the how betweenness centrality might advance chess knowledge, and how it's an example of new methods uncovering previously unknown chess knowledge. Then you reply as if I had claimed that this had come from the paper.

No you have to read what I wrote. The paper does not "show that there are spects of the mathematics of chess to be uncovered". The paper only proposes "a number that the author hopes says something meaningful to human chess players". It really is just a number "F" assigned to each chess position as a function of how pieces attack and defend each other.

1. Is it true that many seemingly complex chess positions can actually be resolved by relatively simple heuristics?

It is irrelevant to solving chess that some tiny fraction of chess positions, e.g. all positions with a lone black king and sufficient material for white, can be resolved by relatively simple heuristics.

2. Is there an upper limit to the difficulty of 8x8 chess positions that could possibly be solved by heuristics?

8x8 chess is a finite problem so in that sense it is trivial. You can solve it using a gigantic, but finite, look-up table. It doesn't get much simpler than that.

However, as a computational problem, there is absolutely zero evidence that a significant shortcut exists which we haven't yet discovered. Your wishing there to be a shortcut is no evidence.

It is clear that almost no computational problem allows for a significant shortcut. I'm pretty sure this statement can be made more formal and proved using some kind of counting argument. (Similar to how one can prove that random data cannot be compressed. And now I half expect you to claim that if someone looks long enough, they should find a compression algorithm that works on random data.)

For solving 8x8 chess to allow for a significant shortcut, 8x8 chess would have be to very special. There is no absolutely evidence that it is computationally special. The rules of chess are all quite arbitrary. One could come up with millions or billions of similar games. Not all of them can have significant shortcuts. To think that 8x8 chess "certainly" has one, you just have to be very, very naive. (Or exceedingly unwilling to learn and improve.)

Stephen Wolfram has written about this:
https://en.wikipedia.org/wiki/Computati ... ducibility
I'm not sure how rigorously his ideas have been developed.
A well known example of "computationally irreducible" system is Game of Life. A few simple rules, much simpler than chess, but there is no general way to predict the outcome other than just running the game. (Of course Game of Life is infinite, and 8x8 chess is finite, so chess in can be solved in O(1) - constant time, but the point is we cannot expect there to be some hidden structure that is yet to be uncovered. You can always hope or wish for such a structure to be discovered, nobody can prevent you from dreaming, but it is just naive dreaming.)

[Dicaste]

I have implemented MCGS algorithm to my chess engine that use Lc0 weight. It performs quite amazing.

[pgn][Event "LAPTOP-RTA0OKO2, Blitz 2.0min+10.0sec"]
[Site "LAPTOP-RTA0OKO2"]
[Date "2026.05.03"]
[Round "2"]
[White "Stockfish 18"]
[Black "AlterEgo 0.0.1"]
[Result "1/2-1/2"]
[ECO "C54"]
[Annotator "0.21;0.01"]
[PlyCount "98"]
[TimeControl "120+10"]

{13th Gen Intel(R) Core(TM) i9-13980HX 2419 MHz W=46.8 plies; 15.078kN/s B=76.6 plies; 2kN/s} 1. e4 {0.21/41 83 Both last book move} e5 {0.01/24 13} 2. Nf3 {0.25/32 6} Nc6 {0.21/22 7} 3. Bc4 {0.25/38 22 (Bb5)} Bc5 {0.16/23 7 (Nf6)} 4. d3 {0.23/38 34 (c3)} d6 {0.16/26 13} 5. O-O {0.25/34 9 (c3)} h6 {0.30/14 13 (Nf6)} 6. c3 {0.25/26 2} Nf6 {0.40/18 13} 7. Nbd2 {0.24/36 28 (b4)} a6 {0.36/14 12 (0-0)} 8. Re1 {0.22/24 2 (a4)} Ba7 {0.37/19 12 (0-0)} 9. a4 {0.23/31 5 (Nf1)} O-O {0.32/14 12} 10. h3 {0.29/34 6 (Nf1)} Be6 {0.20/22 12 (Re8)} 11. Bxe6 {0.29/29 4} fxe6 {0.26/20 7} 12. b4 {0.31/28 5 (Nf1)} Nh5 {0.31/17 12 (Ne7)} 13. Nf1 {0.29/36 28 (Ra2)} Nf4 {0.42/18 12 (Qe8)} 14. b5 {0.33/27 3 (Ra2)} Ne7 {0.33/15 6} 15. d4 {0.30/38 21 (Bxf4)} Qe8 {0.38/15 12 (exd4)} 16. N1h2 {0.36/27 3} axb5 {0.51/21 12} 17. Bxf4 {0.00/75 12} Rxf4 {0.59/24 7} 18. axb5 {0.00/51 3} exd4 {0.62/28 12 (Qxb5)} 19. cxd4 {0.23/31 4} Bb6 {0.63/0 12} 20. Rxa8 {0.19/35 5} Qxa8 {0.62/26 7} 21. Qb3 {0.09/42 36} Qc8 {0.62/27 7} 22. g3 {0.08/41 3} Rf8 {0.60/15 7} 23. Kg2 {0.07/35 5} Qd7 {0.55/23 8} 24. Ng4 {0.04/41 5 (Re2)} Nc8 {0.51/21 13} 25. Qd3 {0.11/45 21} Qf7 {0.50/12 12} 26. Qe2 {0.06/38 7 (Re2)} Qg6 {0.52/15 12 (Na7)} 27. Rd1 {0.11/46 10 (Qd3)} Ne7 {0.48/13 12} 28. Nh4 {0.07/41 2 (Rd2)} Qe8 {0.54/19 12} 29. Qa2 {0.07/36 4 (Rd2)} d5 {0.36/24 12 (Qf7)} 30. exd5 {0.01/35 5 (Qb3)} Nxd5 {0.24/21 6} 31. Qc4 {0.04/35 9} c6 {0.22/20 6} 32. bxc6 {0.03/40 5} Qxc6 {0.16/20 7} 33. Qxc6 {0.03/61 11} bxc6 {0.16/18 7} 34. Nf3 {0.04/42 7 (Ng6)} c5 {0.12/11 10} 35. dxc5 {0.04/62 6} Bxc5 {0.12/9 6} 36. Ra1 {0.04/62 43 (Re1)} Rb8 {0.12/12 13 (Rd8)} 37. Ra2 {0.00/56 2 (Nge5)} Bd6 {0.13/18 13} 38. h4 {0.00/59 5 (Ra6)} Rb4 {0.12/12 13} 39. Nge5 {0.00/69 6} Bxe5 {0.17/11 12} 40. Nxe5 {0.00/79 4} Rb5 {0.15/0 12} 41. Re2 {0.00/48 6 (Ra8+)} Nf6 {0.09/9 12 (Kf8)} 42. Nc6 {0.11/43 7} Kf7 {0.11/9 12 (Ng4)} 43. Nd8+ {0.08/38 6} Kg8 {0.12/12 12} 44. Nxe6 {0.03/48 15} g5 {0.19/9 11} 45. Nd4 {0.00/75 7} Rd5 {0.18/0 11 (Ra5)} 46. Nf3 {0.00/67 6} Kf7 {0.24/20 11 (Ra5)} 47. Ra2 {0.00/74 6} Kg6 {0.24/12 11} 48. Ra4 {0.00/86 8 (Ra7)} Kf7 {0.29/9 11 (Kf5)} 49. Ng1 {0.00/107 9 (g4)} Kg6 {0.18/0 11 (Kg7) Draw accepted} 1/2-1/2

[/pgn]

[pgn][Event "LAPTOP-RTA0OKO2, Rapid 2.0min+20.0sec"]
[Site "LAPTOP-RTA0OKO2"]
[Date "2026.05.03"]
[Round "1"]
[White "AlterEgo 0.0.1"]
[Black "Stockfish 18"]
[Result "1/2-1/2"]
[ECO "E08"]
[Annotator "0.05;0.20"]
[PlyCount "31"]
[TimeControl "120+20"]

{13th Gen Intel(R) Core(TM) i9-13980HX 2419 MHz W=25.4 plies; 2kN/s B=41.2 plies; 12.933kN/s} 1. d4 {0.05/42 20 Both last book move} Nf6 {0.20/38 31} 2. c4 {0.26/28 20} e6 {0.21/35 13 (c6)} 3. g3 {0.40/26 20 (Nf3)} Bb4+ {0.14/42 87 (d5)} 4. Bd2 {0.40/24 20} Be7 {0.16/34 6} 5. Qc2 {0.36/24 20 (Bg2)} d5 {0.12/43 62 (c6)} 6. Bg2 {0.42/24 20 (Nf3)} c6 {0.07/32 18} 7. Nf3 {0.42/23 10} O-O {0.06/34 17} 8. O-O {0.33/26 20} b6 {0.03/32 5 (Nbd7)} 9. Bf4 {0.39/31 20 (Nc3)} Bb7 {0.05/35 10 (Nbd7)} 10. Rd1 {0.47/26 11} Nbd7 {0.05/40 10} 11. Nc3 {0.51/20 11} dxc4 {0.00/38 11} 12. Nd2 {0.42/22 11} Nd5 {0.00/50 11} 13. Nxc4 {0.40/23 11} Nxf4 {0.00/56 12} 14. gxf4 {0.36/16 13} Qc7 {0.00/73 49} 15. e3 {0.33/22 13} Rac8 {0.00/34 12} 16. Rac1 {0.26/17 22 (Ne5) Draw accepted} 1/2-1/2

[/pgn]

[pgn][Event "LAPTOP-RTA0OKO2, Rapid 2.0min+20.0sec"]
[Site "LAPTOP-RTA0OKO2"]
[Date "2026.05.04"]
[Round "1"]
[White "AlterEgo 0.0.1"]
[Black "Stockfish 18"]
[Result "1/2-1/2"]
[ECO "E59"]
[Annotator "0.03;0.20"]
[PlyCount "37"]
[TimeControl "120+20"]

{13th Gen Intel(R) Core(TM) i9-13980HX 2419 MHz W=22.7 plies; 1kN/s B=46.4 plies; 12.434kN/s} 1. d4 {0.03/20 20 Both last book move} Nf6 {0.20/37 31} 2. c4 {0.22/23 20} e6 {0.20/36 14 (c6)} 3. Nc3 {0.39/21 20 (Nf3)} Bb4 {0.12/34 16} 4. e3 {0.35/20 20 (Nf3)} O-O {0.13/35 17} 5. Bd3 {0.04/22 20 (Nf3)} d5 {0.06/38 18} 6. a3 {-0.02/29 20 (Nf3)} Bxc3+ {0.08/43 61 (dxc4)} 7. bxc3 {-0.19/23 11} dxc4 {0.07/32 9} 8. Bxc4 {-0.17/20 11} c5 {0.03/37 16} 9. Nf3 {-0.11/20 20} Qc7 {0.06/37 14} 10. Be2 {0.01/18 11} Nc6 {0.02/41 15 (b6)} 11. O-O {0.15/22 21} e5 {0.06/37 16 (b6)} 12. Bb2 {0.18/22 21 (h3)} Rd8 {0.07/41 18} 13. Qc2 {0.28/26 11 (a4)} Bg4 {0.01/38 27} 14. h3 {0.15/28 21} Bh5 {0.00/46 16} 15. dxe5 {0.09/22 21 (Rae1)} Nxe5 {0.00/72 16} 16. Nxe5 {0.04/21 21 (c4)} Bxe2 {0.00/76 17} 17. Qxe2 {0.01/17 11} Qxe5 {0.00/81 17} 18. c4 {0.02/15 21 (Rfd1)} Qe6 {0.00/72 19} 19. Bxf6 {0.00/23 21 Draw accepted} 1/2-1/2

[/pgn]

[pgn][Event "LAPTOP-RTA0OKO2, Rapid 2.0min+20.0sec"]
[Site "LAPTOP-RTA0OKO2"]
[Date "2026.05.04"]
[Round "2"]
[White "Stockfish 18"]
[Black "AlterEgo 0.0.1"]
[Result "1/2-1/2"]
[ECO "C83"]
[Annotator "0.22;0.00"]
[PlyCount "52"]
[TimeControl "120+20"]

{13th Gen Intel(R) Core(TM) i9-13980HX 2419 MHz W=44.7 plies; 12.984kN/s B=61.0 plies; 3kN/s} 1. e4 {0.22/43 116 Both last book move} e5 {0.00/19 10} 2. Nf3 {0.21/32 5} Nc6 {0.10/20 11} 3. Bb5 {0.20/41 18} a6 {0.37/36 20 (Nf6)} 4. Ba4 {0.29/28 6} Nf6 {0.46/37 11} 5. O-O {0.30/28 13} Nxe4 {0.65/35 11 (Be7)} 6. d4 {0.40/33 11} b5 {0.61/53 11} 7. Bb3 {0.32/37 24} d5 {0.58/32 13} 8. dxe5 {0.17/41 31} Be6 {0.59/30 11} 9. c3 {0.12/40 56 (Nbd2)} Be7 {0.48/41 14} 10. Nbd2 {0.12/38 34 (Be3)} Nc5 {0.56/28 11} 11. Bc2 {0.15/26 3} d4 {0.55/31 12} 12. Nb3 {0.12/30 6} d3 {0.57/43 12} 13. Bb1 {0.05/45 48} Nxb3 {0.55/45 12} 14. axb3 {0.09/25 3} Bf5 {0.53/43 12} 15. Re1 {0.11/37 40} O-O {0.52/27 17} 16. Be3 {0.08/35 20} Qd5 {0.41/23 25} 17. b4 {0.07/28 2 (Bd4)} Qd7 {0.25/18 25} 18. h3 {0.02/41 38} Rfd8 {0.18/17 14} 19. Ba2 {0.00/51 3 (Bd4)} a5 {-0.16/28 14 (Be6)} 20. bxa5 {0.19/29 5} Rxa5 {-0.10/30 14} 21. Bxf7+ {0.08/39 12 (b4)} Kxf7 {0.43/47 13} 22. e6+ {0.00/76 11} Qxe6 {0.54/54 12} 23. Rxa5 {0.00/77 12} Nxa5 {0.82/44 12} 24. Ng5+ {0.00/75 13} Bxg5 {0.38/0 13} 25. Bxg5 {0.00/65 11} Qd6 {0.24/0 26} 26. Bxd8 {0.00/111 25} d2 {0.07/0 13 Draw accepted} 1/2-1/2

[/pgn]

[heroku]

I've removed the hanging thens', and now here's some example output of gofchess:

Code: Select all

FEN: r1r4k/pp3p1p/3Qb1pP/q3p1PR/4P3/2N2P2/1PP5/2KR1B2 b - - 0 19
if has_attack_hanging { Rc6 }..
ve attack_and_through { Rc6 }..
 if evades_attack_non_confrontational { Rc6 Qd2 }..
  if checks { Rc6 Qd2 Qa1+ }..
   if blocks { Rc6 Qd2 Qa1+ Nb1 }..
    if sacrifices { Rc6 Qd2 Qa1+ Nb1 Qxb1+ Kxb1 }..
     if captures_with_checkmate { Rd8 Qb4 Qa1+ Nb1 Qxb1+ Kxb1 Rxd1# }..
 if blocks_check { Rc6 Qa3 }..
  if exchange { Rc6 Qa3 Qxa3 bxa3 }..
   if captures { Rc6 Qa3 Qxa3 bxa3 gxh5 }..

The fen is provided at the top, you can analyse the position for yourself, and see that it correctly finds the witness line that leads to checkmate.

There are some caveats like what is Rc6 doing there? But if you want to solve chess with human reasoning, I believe this is the true way to go about it.

[towforce]

Thank you. We're now making good progress!

(Similar to how one can prove that random data cannot be compressed. And now I half expect you to claim that if someone looks long enough, they should find a compression algorithm that works on random data.)

Just for fun, I'm going to take the bait: while what you say is strictly true, given such data (e.g. random points on a two dimensional piece of paper), one could construct a pattern that would encode them to "sufficient accuracy" for many purposes, and all that would need to be encoded would be the parameters for generating the pattern - which would usually require less data than directly storing all the points.

Based on what you've said I'm going to have to give a very brief overview of the entire proof the two papers contain (not claiming this summary is 100% watertight at this stage):

The chess paper shows that chess can be transformed by reduction to the game of Peek - so the complexity properties of Peek also apply to chess.

The Peek paper shows that any exponential time Turing machine computation will exist in the game of Peek.

The proof of the existence of exponential algorithms originally goes to the proof that ATMs (Alternating Turing Machines) exist. The Peek paper shows that they exist in the game of Peek. Thus it is proven that there are positions that require time to resolve that is exponential in the size of the game.

However, this proof says ABSOLUTELY NOTHING WHATSOEVER about the density of such positions within the game of chess (or, indeed, within the game of Peek: the paper says that there are "infinitely many" of them - but that's allowing the game to go to infinite size), and I can easily show ("beyond reasonable doubt" - not a mathematically watertight proof) that heuristic or approximation solutions will exist in many chess positions in which you're not aware of them.

[towforce]

I've removed the hanging thens', and now here's some example output of gofchess:

Code: Select all

FEN: r1r4k/pp3p1p/3Qb1pP/q3p1PR/4P3/2N2P2/1PP5/2KR1B2 b - - 0 19
if has_attack_hanging { Rc6 }..
ve attack_and_through { Rc6 }..
 if evades_attack_non_confrontational { Rc6 Qd2 }..
  if checks { Rc6 Qd2 Qa1+ }..
   if blocks { Rc6 Qd2 Qa1+ Nb1 }..
    if sacrifices { Rc6 Qd2 Qa1+ Nb1 Qxb1+ Kxb1 }..
     if captures_with_checkmate { Rd8 Qb4 Qa1+ Nb1 Qxb1+ Kxb1 Rxd1# }..
 if blocks_check { Rc6 Qa3 }..
  if exchange { Rc6 Qa3 Qxa3 bxa3 }..
   if captures { Rc6 Qa3 Qxa3 bxa3 gxh5 }..

The fen is provided at the top, you can analyse the position for yourself, and see that it correctly finds the witness line that leads to checkmate.

There are some caveats like what is Rc6 doing there? But if you want to solve chess with human reasoning, I believe this is the true way to go about it.

Thank you very much for this helpful clarification!

[Peter Berger]

The AI responded: Likely assisted. The prose is too "perfect" compared to his other 2,000+ posts on the forum. It lacks the "human noise" (slight repetitions, informal phrasing) present in his live discussions.

Doesn't towforce's text from 1997 sound exactly the same as his most recent one? We can safely assume the old one wasn't written by an AI.

I don't think any human can pass this 2000+ posts tests ( assuming someone +really+ wrote that amount of posts). Most will probably be random reactions to someone else or some sort of random contribution. I don't think +you+ can, if you give the AI bot one of your serious posts and let it check it against all your other board contributions .