Is there a perfect algorithm for chess?

Question

I was recently in a discussion with a non-coder person on the possibilities of chess computers. I'm not well versed in theory, but think I know enough.

I argued that there could not exist a deterministic Turing machine that always won or stalemated at chess. I think that, even if you search the entire space of all combinations of player1/2 moves, the single move that the computer decides upon at each step is based on a heuristic. Being based on a heuristic, it does not necessarily beat ALL of the moves that the opponent could do.

My friend thought, to the contrary, that a computer would always win or tie if it never made a "mistake" move (however do you define that?). However, being a programmer who has taken CS, I know that even your good choices - given a wise opponent - can force you to make "mistake" moves in the end. Even if you know everything, your next move is greedy in matching a heuristic.

Most chess computers try to match a possible end game to the game in progress, which is essentially a dynamic programming traceback. Again, the endgame in question is avoidable though.

Edit: Hmm... looks like I ruffled some feathers here. That's good.

Thinking about it again, it seems like there is no theoretical problem with solving a finite game like chess. I would argue that chess is a bit more complicated than checkers in that a win is not necessarily by numerical exhaustion of pieces, but by a mate. My original assertion is probably wrong, but then again I think I've pointed out something that is not yet satisfactorily proven (formally).

I guess my thought experiment was that whenever a branch in the tree is taken, then the algorithm (or memorized paths) must find a path to a mate (without getting mated) for any possible branch on the opponent moves. After the discussion, I will buy that given more memory than we can possibly dream of, all these paths could be found.

Answer 1

Your end of the argument is supported by the way modern chess programs work now. They work that way because it's way too resource-intense to code a chess program to operate deterministically. They won't necessarily always work that way. It's possible that chess will someday be solved, and if that happens, it will likely be solved by a computer.

Answer 2

I found this article by John MacQuarrie that references work by the "father of game theory" Ernst Friedrich Ferdinand Zermelo. It draws the following conclusion:

In chess either white can force a win, or black can force a win, or both sides can force at least a draw.

The logic seems sound to me.

Answer 3

Some games have, in fact, been solved. Tic-Tac-Toe is a very easy one for which to build an AI that will always win or tie. Recently, Connect 4 has been solved as well (and shown to be unfair to the second player, since a perfect play will cause him to lose).

Chess, however, has not been solved, and I don't think there's any proof that it is a fair game (i.e., whether the perfect play results in a draw). Speaking strictly from a theoretical perspective though, Chess has a finite number of possible piece configurations. Therefore, the search space is finite (albeit, incredibly large). Therefore, a deterministic Turing machine that could play perfectly does exist. Whether one could ever be built, however, is a different matter.

Answer 4

I think you are dead on. Machines like Deep Blue and Deep Thought are programmed with a number of predefined games, and clever algorithms to parse the trees into the ends of those games. This is, of course, a dramatic oversimplification. There is always a chance to "beat" the computer along the course of a game. By this I mean making a move that forces the computer to make a move that is less than optimal (whatever that is). If the computer cannot find the best path before the time limit for the move, it might very well make a mistake by choosing one of the less-desirable paths.

There is another class of chess programs that uses real machine learning, or genetic programming / evolutionary algorithms. Some programs have been evolved and use neural networks, et al, to make decisions. In this type of case, I would imagine that the computer might make "mistakes", but still end up in a victory.

There is a fascinating book on this type of GP called Blondie24 that you might read. It is about checkers, but it could apply to chess.

Answer 5

Yes , in math , chess is classified as a determined game , that means it has a perfect algorithm for each first player , this is proven to be true even for infinate chess board , so one day probably a quantom AI will find the perfect strategy, and the game is gone

More on this in this video : https://www.youtube.com/watch?v=PN-I6u-AxMg

There is also quantom chess , where there is no math proof that it is determined game http://store.steampowered.com/app/453870/Quantum_Chess/

and there you are detailed video about quantom chess https://chess24.com/en/read/news/quantum-chess

Answer 6

I'm only 99.9% convinced by the claim that the size of the state space makes it impossible to hope for a solution.

Sure, 10^50 is an impossibly large number. Let's call the size of the state space n.

What's the bound on the number of moves in the longest possible game? Since all games end in a finite number of moves there exists such a bound, call it m.

Starting from the initial state, can't you enumerate all n moves in O(m) space? Sure, it takes O(n) time, but the arguments from the size of the universe don't directly address that. O(m) space might not even be very much. For O(m) space couldn't you also track, during this traversal, whether the continuation of any state along the path you are traversing leads to EitherMayWin, EitherMayForceDraw, WhiteMayWin, WhiteMayWinOrForceDraw, BlackMayWin, or BlackMayWinOrForceDraw? (There's a lattice depending on whose turn it is, annotate each state in the history of your traversal with the lattice meet.)

Unless I'm missing something, that's an O(n) time / O(m) space algorithm for determining which of the possible categories chess falls into. Wikipedia cites an estimate for the age of the universe at approximately 10^60th Planck times. Without getting into a cosmology argument, let's guess that there's about that much time left before the heat/cold/whatever death of the universe. That leaves us needing to evaluate one move every 10^10th Planck times, or every 10^-34 seconds. That's an impossibly short time (about 16 orders of magnitude shorter than the shortest times ever observed). Let's optimistically say that with a super-duper-good implementation running on top of the line present-or-forseen-non-quantum-P-is-a-proper-subset-of-NP technology we could hope to evaluate (take a single step forward, categorize the resulting state as an intermediate state or one of the three end states) states at a rate of 100 MHz (once every 10^-8 seconds). Since this algorithm is very parallelizable, this leaves us needing 10^26th such computers or about one for every atom in my body, together with the ability to collect their results.

I suppose there's always some sliver of hope for a brute-force solution. We might get lucky and, in exploring only one of white's possible opening moves, both choose one with much-lower-than-average fanout and one in which white always wins or wins-or-draws.

We could also hope to shrink the definition of chess somewhat and persuade everyone that it's still morally the same game. Do we really need to require positions to repeat 3 times before a draw? Do we really need to make the running-away party demonstrate the ability to escape for 50 moves? Does anyone even understand what the heck is up with the en passant rule? ;) More seriously, do we really need to force a player to move (as opposed to either drawing or losing) when his or her only move to escape check or a stalemate is an en passant capture? Could we limit the choice of pieces to which a pawn may be promoted if the desired non-queen promotion does not lead to an immediate check or checkmate?

I'm also uncertain about how much allowing each computer hash-based access to a large database of late game states and their possibly outcomes (which might be relatively feasible on existing hardware and with existing endgame databases) could help in pruning the search earlier. Obviously you can't memoize the entire function without O(n) storage, but you could pick a large integer and memoize that many endgames enumerating backwards from each possible (or even not easily provably impossible, I suppose) end state.