What do you think the difference between an algorithm and mathematic formulae is?
Math as a Mathematical Formula using Matrix/Algebra?
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i believe he means rigorous closed form math formulation not numerical analysis and iterration methods like in algorithms.....i guess its possible , (it came to my mind before) but what will be its use? it will be difficult to be handed easily as u may imagine , its a big matrix and endless probabilities , unless it is made as sake of science , it wont be v much handy to use.....but shapo its a brilliant idea to think about
Sep 10, 2008
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#4
Since it is a closed system with patterns and sequence, if a matrix algebraic formula could be created, (and it's been too long since my math days but it's tempting to get into), there could be a determined calculation (s). It might have uses beyond chess. For example, since the possibilities are so large even in a closed system, it could be used in cryptography. Cryptography is not just for sending secret messages, but it is the basis for all banking which now uses a prime number formula developed by 3 MIT mathematicians years ago. Now extend the concept to an open system, that is not just 64 squares but more squares, more dimensions, or infinite dimensions. To solve such a complex equation you would need the dimensions and that would be the key, but how to trade the key would be another matter though there are ways. In code by sarah flannery an brilliant young Irish mathematician who won European and Irish young scientist awards for her cryptography work explains some of this at a level even I could understand. I'm not sure if algebraic notation could represent chess piece moves and progressive positions but there must be a mathematical interpretation to chess as opposed to move calculation and then position evaluation which is precisely what I meant by algorithm judgement.
As far as I know there is no function/formular to determine a chess move, and I can't imagine how it would look either.
There is a thing called the "Game Theory" though. An Algorithm. It's a mathematical principe used in Casinos and Ensurance Companies to ensure they will always gain profit from their games.
Computers use this system to determine a good move, with certain fixed variables such as, 2 Bishops in the endgame is better than a Bishop and a Knight etc. so it's not calculating x in a function, or isolating a variable in an equation, it's using an algorithmic system.
Problem is, if the computer were to determine the best move (The ultimate best move, as I suppose you're trying to find, or I misunderstood the topic), it would need to calculate the outcome of every single chess game playable, because it would need to know what (for example: 36. Bb4) would lead to.
If the computer were to do this though, the calculation would be backwards. It would determine the mate, and then the moves before, then before, etc.
Currently, there is a database on all possible 6 piece endgames, which is several Terrabytes!
Now, mathematicans has determined that the amount of chess game outcomes is about equal to the number of atoms in the universe, here's a headbreaker:
If the computer were to store all these games, and it would use 1 atom per game, the computer would be the size of the entire universe!
Sep 10, 2008
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#6
Interesting. Though I suspect when the computers 'calculate' their moves they use mathematics and little else (their isn't much else available to a computer except for mathematics, except where the game is at a point where all possible moves can be listed in a database and a computer can then simply play as best as the game allows - once all positions can be placed in a databse, the game is presumably solved).
my very rough understanding though, is that currently to calculate chess mathematically, the computer programs assign set values to particular situations/characteristics (e.g. a queen may be assigned nine points, and a doubled pawn may be assigned a value of 0.2 or something).
the mathematical value to assign the various positions/characteristics presumably still relies on consultation with a human who understands the game independently of the mathematics (i guess they pay and consult GMs).
but perhaps it can work the other way - with current understanding (i.e. the game is not solved and a table cannot be completed for all possible positions), if a serious expert knows the values a computer assigns to each, and those values cannot be altered during a game, the player can manipulate that insider knowledge to win (regardless of the computers starting assigned value assumptions).
either way, the project of designing a mathematical formulat to 'win' chess, is not dissimilar to designing the unbeatable computer program.
Sep 10, 2008
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#7
Not to win or for best move, but a matrix with a formula that represents the moves of chess pieces in that matrix. the formula would then generate the various positions without using a single atom of space beyond the written formula. However, I'm not sure if such a thing is doable using variables and functions but it's something to think about. You'd have to look into matrix algebra and game theory and see what exists, but that's not my field. Still, I'm curious.
Sep 10, 2008
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#8
DaBear wrote:
...
Currently, there is a database on all possible 6 piece endgames, which is several Terrabytes!
Now, mathematicans has determined that the amount of chess game outcomes is about equal to the number of atoms in the universe, here's a headbreaker:
If the computer were to store all these games, and it would use 1 atom per game, the computer would be the size of the entire universe!
btw, falling back on my limited high school physics, given we have identified objects smaller than atoms (neutrons, electrons, etc) why is it assumed (or known/believed) that science could only ever reach a point where a game must be stored on a full atom and only one game can be stored on/in each atom?
i accept the answer may be obvious to those in the know (and that i clearly demonstrate a nerdishness in asking).
joly wrote:
btw, falling back on my limited high school physics, given we have identified objects smaller than atoms (neutrons, electrons, etc) why is it assumed (or known/believed) that science could only ever reach a point where a game must be stored on a full atom and only one game can be stored on/in each atom?
i accept the answer may be obvious to those in the know (and that i clearly demonstrate a nerdishness in asking).
Subatomic particles are extremely unstable and unless restrained by a force, such as the strong neuclear force (holds together the neucleus) or the electromagnetic force (holds the electrons to the neucleus), they go flying off randomly, colliding soon and causing a voilent, although miniture, physical reaction. These are not the building blocks of a stable machine.
In stars, fusion burns hydrogen (the electrons and protons being in like a soup all mixed up- electrons disassociated from their neuclii) in a series of physical reactions somehow creating neutrons along the way) into helium neuclii (two protons and two neutrons, a stable unit, something to do with the strong force) and as the hydrogen gets used up, the helium is fused once into heavier particles, neuclii of heavier elements, with the neutron count roughly equal to the proton count (the atomic number) once again because of stability points in the strong atomic force, the force that overcomes the tendency of protons to repel. As the process continues, the neuclii of just about every element is created. Some of these stars end up exploding, and their dust can collect to make new stars, called second generation stars, as well as other celestial bodies including planets such as Earth.
In cooler temperatures these neuclii must gain an equal number of electrons as protons in order to form an electrically balanced atom. This is very significant, because without this equality of charge the material cannot be physically stable. It would be an ion, and would react with the nearest atom or molecule to release energy and form a new compound. Even most electrically neutral atoms are chemically unstable and rarely appear in elemental form in nature. In order to build a reliable computer or anything for that matter, one must start with materials that are physically stable.
You just called people nerds by posting on a social networking site on the internet about chess.
Just putting things into perspective.
You're not writing a paper on Maths, now are you? ;D
I doubt chess play can be formulated based on some equation or algorithm. The math involved is only a representation between the pieces on the board, not the abstract strategies behind the pieces. Only what is concrete can be evaluated as a number or a symbol.
Chess engines play the numbers game even though what they are mainly doing is searching the position, to crunch out the best move it "plays chess" even though it really doesn’t play chess.
I dont' see any reason why there can't be a mathematical formula for chess. on any given move there is a determined number of possibilities with a determined number of outcomes. why couldn't there be a formula for determining the best possible move given all the possible outcomes (not by using a database but by having all the possible outcomes qualified in the formula)
on another note, for a computer to determine the best possible move from a database of games would not require the computer to hold in memory ALL possible chess positions. for instance, a possible chess position is white has 9 queens and black has 1 King.
that is a possible position but not one that a computer would need to memorize because well before the game got to that position, white could have won.
the computer would only need to hold a database of winning positions and as it discovered more winning positions it could also eliminate the need to store previously held positions. A forced mate in 3 position could be replaced by a mate in 4 position where all the possible moves that could be made to defend the mate in 4 position will be beaten in 4 or less moves (2 different mate in 3's, 1 mate in 2, a mate in 1, etc.).
regardless, the memory needed would not be for every possible positino, just for all positions that lead up to a win. or a loss.
This subject is very interesting but the key to solving this may be beyond human knowledge. or it can be right in our face but we just cant see it or we over look it. I think "chess" can be from another world or even another universe that was passed down to humans.
johnny263 wrote:
I dont' see any reason why there can't be a mathematical formula for chess. on any given move there is a determined number of possibilities with a determined number of outcomes. why couldn't there be a formula for determining the best possible move given all the possible outcomes (not by using a database but by having all the possible outcomes qualified in the formula)
Just because a system follows a well defined set of rules does not at all mean it can be predicted by formula. An excellent example of this is the prime numbers. A prime number can only be divided evenly by one and itself. So a simple test for prime would be: For all x from 2 to itself-1 if remainder (itself/x) =0 then prime=false. Yet there is no known formula that generates only prime numbers, let alone predict them in order of succession.
johnny263 wrote:
the computer would only need to hold a database of winning positions and as it discovered more winning positions it could also eliminate the need to store previously held positions. A forced mate in 3 position could be replaced by a mate in 4 position where all the possible moves that could be made to defend the mate in 4 position will be beaten in 4 or less moves (2 different mate in 3's, 1 mate in 2, a mate in 1, etc.).
regardless, the memory needed would not be for every possible positino, just for all positions that lead up to a win. or a loss.
For a computer relying soley on the database to make its move, it does need to store every position because it must be able to look up the winning move in every position.
Sep 10, 2008
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#18
Yes no formula for solving Rieman's prime numbers--Marcus du Sautoy wrote a great book on this. Perhaps this would be a good math challenge. A formula that allows the computation of moves in a 64 square matrix. We need some great mathematicians for this.
bowanza wrote: Just because a system follows a well defined set of rules does not at all mean it can be predicted by formula. An excellent example of this is the prime numbers. A prime number can only be divided evenly by one and itself. So a simple test for prime would be: For all x from 2 to itself-1 if remainder (itself/x) =0 then prime=false. Yet there is no known formula that generates only prime numbers, let alone predict them in order of succession.
Bowanza, do you have another example of what it is you're trying to say? your prime number example doesn't exactly work because there are infinite prime numbers and a finite number of chess positions. further, i'm not looking for a formula that will give out EVERY winning chess move, i'm talking about a formula that will give out a winning chess move given an exact position.
you gave an example of a formula for determining IF a number is prime
Bowanza: "For all x from 2 to itself-1 if remainder (itself/x) =0 then prime=false"
Similarly, i'm talking about a formula for determining IF a move is winning. So, again, i don't think the prime number example works especially since a formula DOES exist for determining IF a number is prime. do you have another example?
bowanza wrote:
johnny263 wrote:
the computer would only need to hold a database of winning positions and as it discovered more winning positions it could also eliminate the need to store previously held positions. A forced mate in 3 position could be replaced by a mate in 4 position where all the possible moves that could be made to defend the mate in 4 position will be beaten in 4 or less moves (2 different mate in 3's, 1 mate in 2, a mate in 1, etc.).
regardless, the memory needed would not be for every possible positino, just for all positions that lead up to a win. or a loss.
For a computer relying soley on the database to make its move, it does need to store every position because it must be able to look up the winning move in every position.
my point was that there actually ARE positions from which the database does not need to be able to look up winning positions.
my example was a position where white has nine queens and black only has 1 king. well before the game got to this point, white could have won. therefore, the database does not need to know the winning move from this position because (if the database has been playing the winning move the whole time) this positino will NEVER be reached.
Here's a thought--can you make a mathematical representation of chess? Since the moves are within a given field that seems like a matrix, and the patterns are predetermined with some rules, can a formula be created that recreates the game in mathematical terms that have some functional use. If so, we can use our math to beat computers. Not being a good mathematician (I can add and subtract but division, hm...) it's beyond me, but I wonder if someone in the field could figure something out. All computers do is calculate moves and then use an algorithm to judge a position. Mathematic formulae may be superior. Again, beyond me.