[teiferer]

I agree it's hard and non-obvious. If it wasn't then chess would have long been solved by now.

Let's start from the other end. Just a pawn and two kings. It's possible to describe some quite succinct rules for when that's a draw versus a win for the side with the pawn. Agreed? Club players know these by heart. You could write that doen as invariants. As long as the side with the pawn stays inside the "green zone" of the state space, there is nothing the other side can do to void mate. And vice versa, if the game is in the red zone and the other player manages to stay inside that red zone, there is nothing the side with the pawn can do to win. Those areas of the state space, green and red zones, can be described as invariants, in contrast to just enumerating them. It's very compact and can easily be checked by a machine that it's correct.

Now let's add a pawn. And another. And a rook perhaps. The more you add, the harder the condition is to describe, but we live in the age of billion-node-sized neural nets, we have the resources. Eventually you get all pieces on the board, and if the starting position satisfies the draw invariant, that's it. And likely the 960 freestyle chess positions too.

[ctchocula]

> Let's start from the other end. Just a pawn and two kings. It's possible to describe some quite succinct rules for when that's a draw versus a win for the side with the pawn. Agreed? Club players know these by heart. You could write that doen as invariants. As long as the side with the pawn stays inside the "green zone" of the state space, there is nothing the other side can do to void mate.

The flaw in this line of reasoning is that it's easy to come up with a theorem that works for KP vs K. However as the number of pieces increases, it becomes impossible to distill all the branches of possible moves into a simple theorem like that. If what you said were possible, endgame would be a simple flowchart, but look at how much time even GM players use in endgame and how often mistakes are made, and you'll recognize Chess endgame is not distillable into a simple flowchart when there are even as few as 7 pieces on the board.

Given the above, the only option is enumeration, if you want to prove that in all cases the outcome is White win or draw.

[halfcat]

People have done this. They call it tablebases [1]. They’ve computed 7-piece tablebases, but not 8 pieces.

”After the completion of the 7-man, some people were curious about the feasibility of building the 8-man. Ronald de Man estimated that without modifying much the generator, the task requires computers with 64 TB RAM and 2000 TB hard disks[10] (cost about $640K and $40K respectively in 2020). The generator can be modified to work on much cheaper computers with 64 GB RAM but that may need a few thousand years of computing”

And to solve chess you’d need to calculate the 32-piece tablebases.

[1] https://www.chessprogramming.org/Syzygy_Bases

[teiferer]

Like your sibling comment, you seem to be missing my point. Tablebases are fundamentally enumerating positions explicitly even though they use lots of tricks to make that more efficient. I'm talking about foregoing that entirely.

[necklesspen]

You may want to take a look at the Shannon number, we would need quite a large neural net to solve chess in this way.

[teiferer]

The Shannon number is about the size of the state space. What I am dreaming up here is to avoid explicit enumeration by succinct descriptions in the form of logical formulas which have the potential of cutting down the representation size dramatically.