Chess is a Draw

[jefk]

Uri B,
not yet responding to your suggestions for rule modifications (to reduce the drawing margin),
such ideas are highly worthwile for ICCF chess and should at least be investigated for an additional chess
variant (before possibly later also adopting for eg engine-engine games or even Otb chess). Robert Fischer
already knew that such a draw problem would come, and for this purpose invented chess960. Nowadays we already
have chess324 and the field still is evolving (despite some people not understanding it or being overly
resistant to change, in other words being negative).

This forum nowadays, especially the thread about 'is there a project to solve chess' has
become is full with toxic (and indeed negative) people hiding behind aliases and meanwhile
making aggressive remarks only clearly displaying there lack of understanding in this topic.
Some skeptical thinking in science is normal, but insisting on rigorous mathematical proofs for
things which don't require such proofs is ridiculous; like i wrote, there also is not rigorous
proof that checkers has been weakly solved. For a game as Nim there is a rigorous proof,
for a game as chess this might be impossible, this doesn't mean we can't determine
that it's a draw (which it is, although the reason behind this is beyond the understanding
of apparently most people in this forum (except for a tiny few in this thread).
In the future i expect people will simply accept the fact that chess is a draw with perfect play.
While not weakly solved, it imo (depending on possible reaction by vd Herik or Allis) is ultraweakly
solved because it can be determined that from the initial position that is a draw. The move 2.g4 may
even be a loss but that's another topic. The fact that the ultraweak solution for Hex indeed contained
a mathematical (nonconstructive) proof doesn't mean that for other games an ultraweak soluctions
needs to be based on a rigorous proof. And if it would, then it wouldn't change
anything, it only would mean that chess is closer to being weakly solved than some think
and we already know the outcome (quoting vd Herik (years ago)) , a draw.

[jefk]

So, in expectancy of my (draft) article about chess (next year), the preliminary claim can be announced, which is;

Chess has been ultraweakly solved and the result is a draw.

The following reasoning (to determine it's a draw beyond reasonable doubt) was applied:

Lemma (*):
1) *if* there is a winning strategy then Mcts will find it (like it will find a possible winning strategy
for Black against 1.g4 (attention towforce(!)
Emperical finding:
2) it has been determined such a winning strategy cannot be found (neither can the Chin database ergo (proof by contradiction), ergo
There is No such winning strategy,
(conclusion, proof by contradiction):
3) with application of Zermelo's Theorem: Chess is a Draw
QED

happy 2024
(with hopefully a better forum)

(*) it's a (sufficiently although maybe not yet rigorously) solid Lemma because it has been proven that Mcts is (at least as good)
as alfabeta/minimax (but better, deeper). But minimax fundamentally is the best you can get according to an
important game theorem proven by von Neumann; the latter being a logical proof beyond any doubt.

[jefk]

according to mr Kobra, the 'Lemma' isn't correct. Ok, I'll do some more research on that, and for the time being, replace
the word Lemma by proposition(*) (and possibly *temporarily* replace the term ultraweakly solved by 'essentially solved').
No big deal is it.

Meanwhile, an interesting coincidence, today starts the(n) (online) ICGA 2023 conference, hosted in Edingburgh see schedule here :
https://jstorage.app.box.com/s/t35l7wmn ... cqae3vgb1t
The first morning is the most interesting for us (about chess) but i don't know
if i'll be able to see it on zoom (got a confirmation but with date Nov 29 which is tomorrow).

(*) sure some people may also dispute my usage of the word proposition here, but i'm Not going
further into such details regarding nomenclature ; cumulative evidence at least leads to 'essentially solved'.

[jefk]

From this recent ICGA conference it became clear that game theory is not a subset of pure math, but has extensions beyond math,
being a broader discipline than pure theoretical math. This imo also has consequences for the methodology in 'solving' a game as chess.

For chess for example, we see that after 1.g4 there may be a win for Black (in all situations); while the Chinese database is running behind in
backsolving, it appears that the initial score of approx -1.4 is declining when we scroll further in the tree, after which it becomes -1.9 or so.
Example of such an endgame (FEN)
6k1/1pb2p2/p5p1/1q3p2/4P3/3pQ3/1Pr3RP/3R3K w - - 6 49
Probably won for Black. MCTS analysis, or otherwise running some complete engine vs games from the deepest endnodes should give an indication
of these endgames are won for Black So-far, haven't looked at it exhaustively like for normal openings, I think it's quite plausible that after 1.g4??
it's a win for Black). Vice versa, with initial scores of 0.0 -instead of eg. 1.40 or so if it would be an opposite situation as
for 1.g4- (and staying low when scrolling further down the tree(s) for moves as 1.e4 and 1.d4, it's clear that there simply is no chance for
White to have a win in all situations (but then i'm repeating myself). Yet when combining evidence like from many engine vs engine
at standard or rapid time control, and the results of the Live book of Chessbase, we again can confirm the 2023 statement i.e.
chess is a draw. Which we imo can say with 100 pct certainty with a methodology beyond pure, math, namely,
using cumulative evidence, logic, and proof by contradiction.

Which would mean that a more (fundamental?) comprensive approach as weakly solving would be futile number crunching, still a
useful exercise maybe in hardware and software building and testing, yet with no additional nor userful meaning for the game of chess itself.
As for the possibility of 'weakly solving' confirming the current result that chess is a draw, some have said this will not be possible
within our lifetimes. Again, something with which I as physicist simply disagree based on some expectations about new technologies.

Consider the rapidly developing technolgoy of photonics:
Weak solving refers to using computational methods to exhaustively evaluate all possible game lines from start to finish, without relying on human
expertise or proof-based techniques. The minimum computing power estimated to 'weakly' solve chess may be estimated to around 10^50 operations,
but in practice this is probably less because of the many transpositions in the chess tree. Even with major improvements in parallel tree search algorithms,
this remains an immense computational challenge. However, photonic computing promises exponentially more powerful systems than today's fastest
supercomputers. With projected developments over the next decade: By the late 2020s/early 2030s, photonic chips may achieve exascale
(10^18 operations/second) performance levels and beyond. Just a modest 1000x speedup over current machines would equate to a zettascale
system (10^21 ops/sec). A system delivering yottascale performance (10^24 ops/sec) or greater might emerge before 2035 based on technology
roadmaps. Such unprecedented parallelism provided by photonic chips, combined with algorithmic and hardware optimizations,
could potentially lower the computational requirements for chess solving below 10^50 operations.
Therefore, if photonic and related exponential technologies continue advancing as expected, there is a realistic possibility that chess may
be weakly solved, solely through computational power, somewhere between 2030-2035 once systems with capabilities in the
zetta-yottaflops range can be realized. Of course, there remain many uncertainties, but the time frame reflects photonic computing's
potential to solve problems once considered intractable through conventional supercomputing alone.

[lkaufman]

To my thinking, "chess is a draw" is in the same category as something like "there are no humans living permanently on the moon". This can't be proven; it is theoretically possible that when we landed on the moon, one or two astronauts secretly remained, drilled deep below the surface, brought enough supplies to last half a century, and have remained hidden there ever since. Hard to disprove this rigorously, but I think almost everyone would agree that the probability of the statement being true is at least 99.9999999%. Similarly we can use statistical arguments using engines to show that chess is a draw with over 99.9999999% probability, we just can't prove it 100%. Some may enjoy the challenge of a 100% proof, but for me I'm willing to accept 99.9999999% as good enough to make all decisions in life. I wouldn't call a claim false just because there might be one chance in a billion or so that it is.

[towforce]

The minimum computing power estimated to 'weakly' solve chess may be estimated to around 10^50 operations

This is not a trivial number: from link, the upper limit of computer operations from 1 joule is 10^33 (assuming no quantum computing, and understanding that this is not quite what the article section actually says - but I'm taking it as a "reasonable guide"). The mass-energy of the observable universe is 4 × 10^69 joules (link). If the universe was an optimal computer instead of a bunch of galaxies, there would thus be the following number of computing operations available:

(10^33) * (4 * 10^69) = 4 * 10^102

So - I think the green parties might object to using the resource needed to solve chess strong strongly (meaning the game tree crunched out in full).

However, as I've said before, I believe it to be possible to prove that chess is drawn with today's technology: one way to do it would be to recursively generate a list of conditions to exist for one side to be able to force material gain from the other side, and then show that these conditions don't exist in the standard starting position (that wouldn't be a complete proof, but variations of it could be).

[Uri Blass]

From this recent ICGA conference it became clear that game theory is not a subset of pure math, but has extensions beyond math,
being a broader discipline than pure theoretical math. This imo also has consequences for the methodology in 'solving' a game as chess.

For chess for example, we see that after 1.g4 there may be a win for Black (in all situations); while the Chinese database is running behind in
backsolving, it appears that the initial score of approx -1.4 is declining when we scroll further in the tree, after which it becomes -1.9 or so.
Example of such an endgame (FEN)
6k1/1pb2p2/p5p1/1q3p2/4P3/3pQ3/1Pr3RP/3R3K w - - 6 49
Probably won for Black. MCTS analysis, or otherwise running some complete engine vs games from the deepest endnodes should give an indication
of these endgames are won for Black So-far, haven't looked at it exhaustively like for normal openings, I think it's quite plausible that after 1.g4??
it's a win for Black). Vice versa, with initial scores of 0.0 -instead of eg. 1.40 or so if it would be an opposite situation as
for 1.g4- (and staying low when scrolling further down the tree(s) for moves as 1.e4 and 1.d4, it's clear that there simply is no chance for
White to have a win in all situations (but then i'm repeating myself). Yet when combining evidence like from many engine vs engine
at standard or rapid time control, and the results of the Live book of Chessbase, we again can confirm the 2023 statement i.e.
chess is a draw. Which we imo can say with 100 pct certainty with a methodology beyond pure, math, namely,
using cumulative evidence, logic, and proof by contradiction.

Which would mean that a more (fundamental?) comprensive approach as weakly solving would be futile number crunching, still a
useful exercise maybe in hardware and software building and testing, yet with no additional nor userful meaning for the game of chess itself.
As for the possibility of 'weakly solving' confirming the current result that chess is a draw, some have said this will not be possible
within our lifetimes. Again, something with which I as physicist simply disagree based on some expectations about new technologies.

Consider the rapidly developing technolgoy of photonics:
Weak solving refers to using computational methods to exhaustively evaluate all possible game lines from start to finish, without relying on human
expertise or proof-based techniques. The minimum computing power estimated to 'weakly' solve chess may be estimated to around 10^50 operations,
but in practice this is probably less because of the many transpositions in the chess tree. Even with major improvements in parallel tree search algorithms,
this remains an immense computational challenge. However, photonic computing promises exponentially more powerful systems than today's fastest
supercomputers. With projected developments over the next decade: By the late 2020s/early 2030s, photonic chips may achieve exascale
(10^18 operations/second) performance levels and beyond. Just a modest 1000x speedup over current machines would equate to a zettascale
system (10^21 ops/sec). A system delivering yottascale performance (10^24 ops/sec) or greater might emerge before 2035 based on technology
roadmaps. Such unprecedented parallelism provided by photonic chips, combined with algorithmic and hardware optimizations,
could potentially lower the computational requirements for chess solving below 10^50 operations.
Therefore, if photonic and related exponential technologies continue advancing as expected, there is a realistic possibility that chess may
be weakly solved, solely through computational power, somewhere between 2030-2035 once systems with capabilities in the
zetta-yottaflops range can be realized. Of course, there remain many uncertainties, but the time frame reflects photonic computing's
potential to solve problems once considered intractable through conventional supercomputing alone.

I do not see the reason for the estimate of 10^50

Number of legal positions in chess is smaller than 5*10^44 at least if you do not consider the fifty move rule and the history of the game

see
https://github.com/tromp/ChessPositionRanking

I strongly believe that chess is a draw even without the 50 move rule and it may be easier to strongly solve chess without repetition rule and 50 move rule with a rule that no win after 9999 moves of both sides is a draw(so result is dependent only on the move number and we can easily translate mate in 100 to draw or win based on the move number).

Note that I think 10^50 may not be enough to strongly solve chess or even to strongly solve the 7 piece tablebases because every different history of the game is different position and it may be a draw or win based on the history not only because of the 50 move rule but also because of repetition rules.

[syzygy]

Note that I think 10^50 may not be enough to strongly solve chess or even to strongly solve the 7 piece tablebases because every different history of the game is different position and it may be a draw or win based on the history not only because of the 50 move rule but also because of repetition rules.

Chess with 7 or fewer pieces has been strongly solved if you trust that the generated tablebases are correct. Even if you do not trust that the current tablebases are correct, it is clearly within our abilities to generate them correctly.

Generating the full 8-piece tables needs a lot of hardware but is probably less expensive than, say, training a large LLM. It does not seem to be a computational task that is bigger than anything humanity has accomplished so far.

Generating the full 9-piece tables might start to become problematic. I suppose it is still possible with some years of effort and a huge investment in hardware.

Generating the full 10-piece tables probably does not make any sense at this time.

[syzygy]

To my thinking, "chess is a draw" is in the same category as something like "there are no humans living permanently on the moon". This can't be proven; it is theoretically possible that when we landed on the moon, one or two astronauts secretly remained, drilled deep below the surface, brought enough supplies to last half a century, and have remained hidden there ever since. Hard to disprove this rigorously, but I think almost everyone would agree that the probability of the statement being true is at least 99.9999999%. Similarly we can use statistical arguments using engines to show that chess is a draw with over 99.9999999% probability, we just can't prove it 100%. Some may enjoy the challenge of a 100% proof, but for me I'm willing to accept 99.9999999% as good enough to make all decisions in life. I wouldn't call a claim false just because there might be one chance in a billion or so that it is.

There is no evidence that the claim is false.
What would be false is the claim that it has been deductively established as true. Most of science does not care about such a claim in the first place (and daily life even less).

[jefk]

The estimate of 10^50 positions or more needed for solving chess indeed was inaccurate. In fact, that number was copied from an AI without verification. Apparently , 10^44 or less positions may be required for brute force (strong) solving, with weak solving needing far fewer still.
The main point is that photonic computing promises dramatically more powerful systems within the next decade. Just a modest 1000x speedup over classical machines could enable zettascale computation (10^21 ops/sec) by the late 2020s or early 2030s. This level of parallelism, combined with algorithm optimizations, may be enough to weakly solve chess computationally before 2035 if the technology continues advancing.
It also will require much less energy.
Games theory also encompasses diverse fields beyond just mathematics through studying strategic interaction across social and natural sciences.
In terms of determinations like games being draws, sometimes conclusions are reached based on available evidence rather than strict mathematical proof. A judge may convict a murderer based on DNA evidence meeting a "beyond reasonable doubt" standard. While not 100% certain, it establishes a conclusion sufficient for decisions in the real world.
Similarly, if there is no known forced win for one player in chess, it follows logically that the game is a draw. Relevant evidence should be presented to decision makers (as eg. within the ICCF) rather than unclear statements
suggesting we don't know because of a lack of rigorous mathematical proofs. In summary, photonic computing shows promise for solving problems like chess that were previously almost intractable. Whereby in some situations determinations of games as draws can be sufficiently established through cumulative evidence meeting practical standards, even without strictest forms of proof.