How close can we come to proving that white draws?

[Madeleine Birchfield]

I mentioned ignoring the 50 move rule in my OP. In that case, this is false. And false by a lot. Your post is about proving chess is a draw, which is not what I'm talking about.

Your post mentions building an opening book with only one choice for white. How do you prove that that one choice for white at each move of the game is the right choice? You have to search the rest of the choices for white to show that the the choice you make as white is the best choice amongst all the choices. Otherwise you prove that your opening book draws, but white might be able to choose a different move at any single stage and win and you still haven't proven that white draws. Taking all this into consideration, proving that white draws reduces the problem down to the original problem of proving chess is a draw, and my post holds.

[Dann Corbit]

We aren't close at all. So far we only have a hypothesis/theory (chess is a draw) and experiments from current engines such as Stockfish and Leela; they say that it is highly likely that chess is a draw, but do not mathematically prove anything at all.

For that to happen, one would need a database consisting of all positions reachable from the starting position and an evaluation whether the position is won, drawn, or lost, or a shortcut in the mathematical proof for chess like in the proof for solving nim. The former is physically impossible as the number of positions in chess greatly exceeds the number of subatomic particles in the universe, and the time scale to generate such a database is such that if somebody were to begin the task today, the Earth would be swallowed by the Sun becoming a red giant before the database is fully constructed. And the latter is nowhere close to being found for combinatorial games much simpler than chess (like Los Alamos Chess and Minishogi), and there is nothing to indicate that chess is anything special when it comes to finite combinatorial games.

People who say otherwise do not understand the difference between science, which hinges on experiment, and mathematics, which hinges on proof. Science doesn't actually prove anything definitive about the world, it just shows that our models/theories/hypotheses are good approximations of reality.

Even the worst possible opening move is not proven a loss.
We still only have empirical evidence.

As for the current position. The most common move sequence leading to this position is:
1.d4 d5 2.Bf4 c5 3.e3 Nc6 4.c3 Nf6 5.Nd2 *
there are of course, others. Now, 36 ^ 9 = 101,559,956,668,416
which is 101 trillion.

Now, the first move has only 20 choices, and the first response only 20 as well, but it opens up quickly after that.

[mmt]

Your post mentions building an opening book with only one choice for white. How do you prove that that one choice for white at each move of the game is the right choice?

I don't need to prove any such thing. I only need to find one line for white that draws.

[Dann Corbit]

That proves exactly that a single line draws. And what about the other trillion lines?

1.d4 d5 2.Bf4 Nf6 3.e3 c5 4.c3 Nc6 5.Nd2 * is the most popular path to the position I posed earlier.
I posted the second most probable sequence by accident.
There were 18 different paths to that position in my database.

[mmt]

Even the worst possible opening move is not proven a loss.

Your position was not after one move, so this is not relevant. And we can prove some lines after 9 plies are lost for black and even more can be drawn by white. This thread is about finding draws, not proving losses.

there are of course, others. Now, 36 ^ 9 = 101,559,956,668,416
which is 101 trillion.

Now, the first move has only 20 choices, and the first response only 20 as well, but it opens up quickly after that.

No, we only need 1 move for white at any position. We want to prove that white can draw (or win). For this problem we only need one line for white, we don't care what happens after other possible moves.

[Madeleine Birchfield]

I don't need to prove any such thing. I only need to find one line for white that draws.

One line is not enough. What you are looking at is an entire game tree, or an entire opening book. Because if you only have one line, then black could deviate from your line at any time and you would no longer be in your line. You still need to prove that every move that you make can force the game back to a draw; how do you do that? You still have to check every single move to see that your selected move in your book doesn't inadvertently cause white to lose or win the game. There are certain positions where white might be winning, but any move that white makes that doesn't continue to be winning immediately leads to a loss, and you have to make sure that you do not end up in one of those lines. And for that everything in my posts still hold.

[Madeleine Birchfield]

No, we only need 1 move for white at any position. We want to prove that white can draw (or win). For this problem we only need one line for white, we don't care what happens after other possible moves.

Proving that white can draw or win is different from proving that white could force a draw. Which one are you referring to in this topic?

[mmt]

Proving that white can draw or win is different from proving that white could force a draw. Which one are you referring to in this topic?

Maybe it's simpler from the other direction: I want to prove that black does not win. I want to prove that white at least draws. Which means that I can do this by proving that white wins or white draws. Proving that white draws is of course easier.

[Madeleine Birchfield]

Maybe it's simpler from the other direction: I want to prove that black does not win. I want to prove that white at least draws. Which means that I can do this by proving that white wins or white draws. Proving that white draws is of course easier.

If black blunders at any time then your constructed opening book should allow white to win, if you are proving that black cannot win.

[mmt]

One line is not enough. What you are looking at is an entire game tree, or an entire opening book. Because if you only have one line, then black could deviate from your line at any time and you would no longer be in your line. You still need to prove that every move that you make can force the game back to a draw; how do you do that? You still have to check every single move to see that your selected move in your book doesn't inadvertently cause white to lose or win the game. There are certain positions where white might be winning, but any move that white makes that doesn't continue to be winning immediately leads to a loss, and you have to make sure that you do not end up in one of those lines. And for that everything in my posts still hold.

Like I said, I only need one move for white for any position that can occur in the game. Of course I know that I have to respond to all possible moves by black, how is that not clear from the OP?