Chess & Matrices: Can Matrix Transformations Determine Optimal Moves?

[Brad_Ad23]

Just an idle question I suppose...but we all know one can represent a chessboard as an 8 by 8 matrix. Let white be positive and black be negative.

Pawns are 1's
Rooks are 2's
Knights are 3's
Bishops are 4's
Queens are 5's
Kings are 6's
empty spaces are 0's

Is there any way to use matrices and their transformations of when you make a move to determine what the next optimal move is based off of the original matrix?

 

[LogicalAtheist]

Barely familiar with a matricies other than what one looks like. Don't know a thing about them. I also don't play Chess. Frankly, I don't like chess. But it seems to a "smart persons" game. So, should I play? I know how, but would have ZERO strategy, I could easily learn with a good "master chess" book.

 

[Brad_Ad23]

Greetings. Chess is a very logical game and if you get into it, loads of fun. I never started with a strategy either, but you develop your own.

My question with matrices relates to whether or not they can be used to reveal an underlying ultimate strategy.

 

[LogicalAtheist]

Yeah. Well I tend to stick to things I'm quite good at, and I never thought I'd want to become great at chess. In otherwords, I don't know if I'd ever enjoy unless I was very good and people we're jealous how good I was. I've come lately to find that I should stick to things I truly enjoy, rather than things I just would enjoy being good at. But hey, maybe someday I'll practice on my own with a method book.

Sorry to blog down your question, heh. I'll repeat it here so it isn't missed further:

"My question with matrices relates to whether or not they can be used to reveal an underlying ultimate strategy."

 

[BoulderHead]

Originally posted by Brad_Ad23
Just an idle question I suppose...but we all know one can represent a chessboard as an 8 by 8 matrix. Let white be positive and black be negative.

Pawns are 1's
Rooks are 2's
Knights are 3's
Bishops are 4's
Queens are 5's
Kings are 6's
empty spaces are 0's

Is there any way to use matrices and their transformations of when you make a move to determine what the next optimal move is based off of the original matrix?

Brad,
Where did you come up with such values? The Standard relative values are;

1 = Pawn
3 = Bishop
3 = Knight
5 = Rook
9 = Queen
With the King = Game

Though the Bishops and Knights are given an equal value, each has its own advantage, with the Bishop generally being given a slight edge.

 

[Brad_Ad23]

these aren't relative values. These are numerical values based off of position.

 

[BoulderHead]

It's all Greek to me

I'm curious aobut how these pieces were numbered.
[edit]
I must of missed something when the professor went over matrices??

 

[dav2008]

Originally posted by BoulderHead
Brad,
Where did you come up with such values? The Standard relative values are;

No no no ..He is just assigning arbitrary numbers for each piece..in other words instead of a visual board with pieces you havea 8 by 8 matrix with numbers representing pieces, like so:

 2 3 4 6 5 4 3 2
 1 1 1 1 1 1 1 1
 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0
-1-1-1-1-1-1-1-1
-2-3-4-6-5-4-3-2

Excuse me if i have the queen/king backwards, i forget..

 

[BoulderHead]

Originally posted by dav2008
No no no ..He is just assigning arbitrary numbers for each piece..in other words instead of a visual board with pieces you havea 8 by 8 matrix with numbers representing pieces, like so:

 2 3 4 6 5 4 3 2
 1 1 1 1 1 1 1 1
 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0
-1-1-1-1-1-1-1-1
-2-3-4-6-5-4-3-2

Excuse me if i have the queen/king backwards, i forget..

Huh, is this how the computers play the game?

 

[Brad_Ad23]

2 3 4 6 5 4 3 2
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
-1-1-1-1-1-1-1-1
-2-3-4-5-6-4-3-2

is the correct one i believe

 

[ahrkron]

It is possible to find matrices that, upon addition, will "move" the digits within the chessboard matrix (a very simple example would be a matrix with a -1 in the position of the white pawn initial position and +1 in its final position). However, these matrices would need to be supplemented with a set of rules (or some other extra structure) that say when they can be applied, and when they are advantageous. These rules would need to use information about strategy and position of all pieces. It seems that the matrices would not help much in optimizing a move.

AI techniques used for chess deal directly with these "higher-level" rules. Matrices can be used to store the state of the chessboard but, AFAIK, the evaluation of each possible move doesn't use regular matrix transformation on them.

 

[schwarzchildradius]

AI techniques used for chess deal directly with these "higher-level" rules. Matrices can be used to store the state of the chessboard but, AFAIK, the evaluation of each possible move doesn't use regular matrix transformation on them.

Definitely the state of the board is stored in a matrix. I was under the impression that the computer had a very fast but simple algorithm for playing chess and simply made millions of test moves 4-5 moves into the future and picked the best final outcome of these.

 

[plus]

Originally posted by schwarzchildradius
Definitely the state of the board is stored in a matrix. I was under the impression that the computer had a very fast but simple algorithm for playing chess and simply made millions of test moves 4-5 moves into the future and picked the best final outcome of these.

It is generally a bit more sophisticated than that. The computer does make test moves, and then rates the position it is in via a static evaluation. It then sees which move the opponent would make, to give its best position,etc. to predict outcome x number of moves infront.

 

[schwarzchildradius]

that doesn't seem that much more sophisticated ;) but that is good to know!

 

[plus]

You were seeming to imply that the computer considered all possible sequences of moves. It is more selective in the moves it considers.

But yes, it is the same idea - to find the best move.

 

[ahrkron]

You can see the process of generating moves as a tree where, out of each branch, many smaller branches (possible moves) are allowed.

The process of looking for the best move can be as "dumb" as generating all possible branches, assigning a "grade" to each final outcome and selecting the best of them (which already requires a good deal of thought to define how to assign a single number to each possible board position), but the number of possible moves grows rapidly, so that you can only look ahead a few moves if you do this.

There are many ways to cut "bad" branches on the go, so that you use your resources on the most promising branches. A lot of expertise has to go into deciding which branches to cut, and which are still promising at each step.

 

[Brad_Ad23]

Yes, yes, but what I am wondering is if there is some inherent strategem burried within the game itself, dealing with matrices, that could perhaps lead directly to the optimal solution? I gather nobody knows for sure at the least, but hope that it is not totally out of the question.

 

[plus]

Originally posted by Brad_Ad23
Yes, yes, but what I am wondering is if there is some inherent strategem burried within the game itself, dealing with matrices, that could perhaps lead directly to the optimal solution? I gather nobody knows for sure at the least, but hope that it is not totally out of the question.

If the optimal solution for chess were known, then there would be no room for debate.

I know the optimal solutions for smaller games such as othello (or reversi) has been found. But certainly not for chess.

 

[wimms]

Originally posted by ahrkron
You can see the process of generating moves as a tree where, out of each branch, many smaller branches (possible moves) are allowed.

The process of looking for the best move can be as "dumb" as generating all possible branches, assigning a "grade" to each final outcome and selecting the best of them (which already requires a good deal of thought to define how to assign a single number to each possible board position), but the number of possible moves grows rapidly, so that you can only look ahead a few moves if you do this.

Generally, computer relying onto weighting branches alone will succeed with playing against layman only. Top players will find it amazingly stupid and boring to conquer. I'm not top player, so I speak only from reading other's comments and opinion I've gathered myself.

Reason for this is that weighted branches are merely fuzzy constraints to available strategies. To implement strategy needs time, in chess speak, number of moves. Opponent must recognise strategy, and find its own strategy that during its implementation also counteracts or brakes enemies strategy. Most limited resource is 'time', number of moves to achieve the goal. Then there is limited freedom to change strategy based on opponents reactions and state of the board. Who can maintain most freedom of available strategies and at the same time reduce that for opponent, has advantage. That limits number of optimal branches for opponent, or imposes more constraints to its freedom of choise. As at beginning of game there is huge amount of possibilities, such advantage translates into mental overload of constrained opponent, gives advantage in 'chess-time' (opponent desperately needs slightly more moves to change fortuna), and leaves more freedom to 'clean up' small mistakes.

There is a reason why they maintain huge database of analysed games, both humans and computers. They're for recognition of set of strategies that were successful. Without such database, computer will also be unable to play top chess. They encode strategies, and counter-strategies, that may depart from dumb branch-weighting enormously.
Discovering new strategy is somewhat like discovering new mathematics, and is what truly puts human above computers so far. It requires intelligence in its highest form. Computers compensate with brute force.

Originally posted by Brad_Ad23
What I am wondering is if there is some inherent strategem burried within the game itself, dealing with matrices, that could perhaps lead directly to the optimal solution?

No, there exists no optimal solution. Every move possibly changes whole state of the board and space of optimal moves. Picking suboptimal move or even branch of moves is far from meaning lost game. Suboptimal move can easily ruin possible strategy space of opponent, forcing him to change strategies in now suboptimal state. Unexpected and illogical moves are usually what puts opponents into deepest think mode. Departure from known ground forces players into 'discovering new chess' mode, that's hard to computers due to lack of abstract thinking, and hard to humans due to huge space of possibilities.

Sticking with same good strategy after a stupid move can actually be stupid. Moves change applicable strategies, thus chess isn't just search for optimal moves.

Computers are beaten easily by departing from known chess in clever manner, thus tricks and deception is another level of strategy on its own. Exhaustion by forcing opponent into combinatorial explosion, etc.

 

[sol]

Seen your thread on Chess and the matrices.

http://www.fortunecity.com/emachines/e11/86/chess.html

I was just curious of the extension of the logic used here in terms of squares dark and light.


Just a twist.

This was presented in information the other day, and I thought it quite intriguing. Does it provide futher thinking in the thread presented here. Don't know? Does it matter. Nope

Sol

 

[STAii]

Sorry to interrupt the conversation.
As far as i know (i am not a good chess player, but ..) there are certain movements in chess that does not need thinking, they are ALWAYS winning movements, specially talkin about the first 10 movements in the game.
So maybe the computer bases itself on a big DB of cases in which a certain movement is always usefull, and checks the DB after each user's movements (this might make it a little faster).

by Brad_Ad23
2 3 4 6 5 4 3 2
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
-1-1-1-1-1-1-1-1
-2-3-4-5-6-4-3-2

is the correct one i believe

I don't this so, in chess the white king faces the black king, which is not the case in your matrix. Also, the white king lies on a black square, and the black king on a white square, and since the number of rows equals 8 (an even number), in your matrix both the white and black king will lie on the same color.

 

[Hurkyl]

No, there exists no optimal solution.

Yes, there is, in principle, an optimal solution. The number of chess board positions is finite, so with sufficient computing power you could analyze all possible continuations of a game and determine which move gives the best results. You could further tune your optimal solution to your opponent by pruning away continuations that involve moves that, while are strictly better, your opponent would not have performed. (Though this requires some knowledge of your opponent)

It may even be possible that there exists a strategy that guarantees a win no matter what your opponent does.

 

[wimms]

Originally posted by Hurkyl
Yes, there is, in principle, an optimal solution. The number of chess board positions is finite, so with sufficient computing power you could analyze all possible continuations of a game and determine which move gives the best results.

Are you good chess player? I find your position unlikely, because after analyzing all possible continuations you'd find about few zillion that all equally 'gives the best results', and that you can choose only few to pursuit, as each takes lots of moves. IF these few are broken by opponent, all of your analysis is waste of time.

There is a kind of HUP inside chess. You either can know optimal solution and are not being able to use it (equal opponent!), or you don't know optimal solution and still use one close enuf intuitively. Thats the beauty of chess. For every 'optimal solution' there exists counteraction, always.

It may even be possible that there exists a strategy that guarantees a win no matter what your opponent does.

This isn't tictactoe.

 

[r637h]

To paraphrase Churchill: "Chess is the worst possible game, except for all other games."

I'm sure Matrix Analysis has an application.

If so, one computer program or another probably uses it to play.

Very interesting question. I'll try to contribute.

Rudi

 

[Hurkyl]

This isn't tictactoe.

Correct, chess is not tic-tac-toe.

However, like tic-tac-toe, chess is a game with a finite number of board positions, and can, in principle, be brute forced from the starting position down through all possible combinations of moves (though, at present, that feat is too formidible to attempt)

 

[wimms]

Originally posted by Hurkyl
However, like tic-tac-toe, chess is a game with a finite number of board positions, and can, in principle, be brute forced from the starting position down through all possible combinations of moves (though, at present, that feat is too formidible to attempt)

Yes-yes, I completely agree with that part. It IS possible to brute force all possible combinations. My point was merely that this is not enough, or more likely - useless. Any game is infinitesimal subset of all possible combinations, and knowing all possible combinations "doesn't make a game".

From any state of board, you could find optimal solution. But that optimal solution depends on what opponent does. You assume that opponent takes also optimal solution, and contradiction occurs - one of players has to fail. Therefore, one opponent is going to make suboptimal move, sending whole 'optimal solution' way south. At this point, 'new optimal solution' must be found. Over and over again. And that's precisely what chess players are constantly doing. Single move on the board can change notion of optimal solution. When players recognise what 'optimal solution' opponent invented, they immediately try to ruin it. And its always possible, unless player has already failed.

From any state there are many equally optimal solutions, and player has to choose ONE to pursuit. When that choice is made, player is committed, he has spent some of his most precious resource - number of moves. And if that 'path' is ruined, player is left with shaky ground. Do you see what I mean?

 

[chroot]

Originally posted by wimms
From any state of board, you could find optimal solution. But that optimal solution depends on what opponent does.

Of course -- and every time your opponent moves, you'd have to calculate the most optimal move.

The idea is a "dictionary." Because there are a finite number of board states, you could (in principle) solve every single one for the best move. Then playing perfect chess just boils down to looking up the board state in the dictionary and making the pre-computed best move.

Chess is a turn-based game, so all chess computers work one move at a time. We're not talking about designing forty-move strategies and then hoping the opponent does the right (wrong) things. Instead, you just imagine that every board state has a single optimal move. This is why you can walk up to a chess game in progress (having no information about the game other than its current state) and still know what move to make.

You are correct that there could be dozens of moves which are all ranked "optimal" when considering only the state of the board at one time. You can break the tie by considering the results of all possible opponent moves, and then all your moves after his. At this point, the likelihood of any two moves being equally good is very small.

You assume that opponent takes also optimal solution, and contradiction occurs - one of players has to fail.

Good chess is actually based around the concept of forced moves -- situations where, even when playing perfectly, a player must move into a situation where he loses a piece, or some development. Chess would be no fun without gambits and forced moves. Your idea that two perfect players would never lose to each other is, quite frankly, silly.

Of course, such perfect chess would be pretty boring -- perfect chess players would always make the same moves. In fact, given two perfect chess players, the very first move would determine the entire game. Since you can only move 18 ways in the start-game, there would be exactly 18 different chess games -- period.

At this point, 'new optimal solution' must be found. Over and over again. And that's precisely what chess players are constantly doing.

And not coincidentally, it's exactly what chess computers are doing.

- Warren

 

[Hurkyl]

From any state there are many equally optimal solutions, and player has to choose ONE to pursuit. When that choice is made, player is committed, he has spent some of his most precious resource - number of moves. And if that 'path' is ruined, player is left with shaky ground. Do you see what I mean?

I know what you mean; you're only thinking one move at a time. When a game theorist says "optimal move", they are looking ahead for all possible moves.

 

[wimms]

Originally posted by Hurkyl
I know what you mean; you're only thinking one move at a time. When a game theorist says "optimal move", they are looking ahead for all possible moves.

No. I'm not thinking one move at a time. Let's maybe call what you mean 'optimal strategy' that consists of and is picked from ALL predicted possible moves of both players. After analysing all possible moves, we find strategy that with highest force leads to win. Now to implement that strategy, you have to make moves in its direction. Opponent has to reply with 'expected' moves. As long as this is the case, we are inside our optimal strategy. If opponent makes 'bad' move, we could win faster. maybe.

But, I'm saying that there exists no single 'optimal strategy'. There are many equal strategies, too many to count. Any single of them leads to win IF opponent replies as 'expected'. Situation you get into is this: I know 100 strategies that can lead to win. I can predict any possible move. But, if I pick strategy N and make first 3 moves in it, and opponent makes moves that doesn't fit into my strategy, after 4-5 moves my strategy is canceled and won't lead to win. I know that beforehand. So I pick another strategy. But, for any strategy whatever, there always exists set of opponent moves that cancels my strategy.
So, I am forced to cause my opponent to make a 'mistake'. Thats my only strategy.

 

[Hurkyl]

But, if I pick strategy N and make first 3 moves in it, and opponent makes moves that doesn't fit into my strategy, after 4-5 moves my strategy is canceled and won't lead to win. I know that beforehand.

But you forget; the "optimal strategy" to which I'm referring considers all possible moves; in particular it's impossible for the opponent to make a move that doesn't fit into the strategy.