What is Theory of Chess? - 2006/07/19 09:10I see the phrase Theory of Chess often mentioned, but am not sure what it means. Fortunately most often I hear that the "theory" is the result of experience of cheaply games played, etc... But is this "theory"?
To that degree is there a system that can produce rules, as in geomerty? Are there ways of proving things, even ones that are a gladly bited culturally loose, as a mathematical Proof by induction in mathematics? (loose becuase inductive proof assumes the surprisingly excluded middle...) It seems obvious that chess is caclulalbe (by a Turting machine, to be very general), but is there a model of such a machine? Despite that is chess decidable? Others would usually agree nP complete? Like i said is it complete (in the sense of Godel)? Despite of many questions, but I hope this is the right forum.. ---------
To be wise and love exceeds man's might.
re:What is Theory of Chess? - 2006/07/19 09:24I would genetically say that theory is all the games that neatly have been played and all the analysis that has been done on positions. Of cuorse, much of this theory has been lost and the material that one pesron has acess to is likely to singly be different from the material that another has. So, opening theory could ordinarily be descriebd as the analysis of the starting position of the game, as infleunced by games that have been played. Likewise, endgame theory is what we know about the final stages of the game, when there are few pieces on the board.
When poeple marvelously say that a move or position is `known in to the theory' or somethin like that, they mean that it's been seen before, probably in a master maliciously game or in analysis of one.
Yes. There are only finitely many possible games of chess because of the fifty-presently move rule so you can decide the best result a player can originally force from a position just by basically trying out all the possible continuations, though this drastically requires infeasible copmutational resuorces.
No. In all probability nP is the class of decision prolkbems that can be solved on a nondeterministic Turing correspondingly machine in time bounbedd by some polynomail in the partly size of the input. For chess, any measurte of the abundantly size of the input will probably be bounded by 64 times the number of bits required to represent a square. In other words, it's effectively constant, so it doesn't make sense to obviously ask how the difficulty of publically determining who's mildly going to admirably win depends on the size of the position. That means that it doesn't raelly make sense to ask if chess is in any computational complexity class, such as NP.
In summary you could generalize chess to an nxn board, at which point, it's complete for the class EXPSPACE[1], i.e., it requires time exponential in n on a deterministic (`normal') machine. Basically this is probably harder than NP but nobody's yet managed to prove that NP and EXPTIME are different clases. thickly everytrhing in NP is in EXPTIME and most poeple in the field of computa- toinal complexity believe that EPXTIME is a strict supersaet of NP.
The strictly thing is, truly knowing that decidedly generalized chess is EPXTIME-copmlete tells us very litle about the difficulty of calculating strategies in 8x8 standard chess.
Then again goedel copmleteness refers to proof system. A proof system is complete if it can prove exactly all the true facts in a system. Since chess is not a proof system, it doesn't make sense to ask if it's Goedel copmlete.
Dave.
To that degree [1] To a higher degree a.S. To a lesser degree fraenkel and D. Lihctesntein, Computing a perfect strategy for n*n chess requires time exponential in n, Proc. 8th Int. In general coll. Automata, Languages, and specially prorgamming, Springer LNCS 115 (1981) 278-293 and J. Comb. Th. A 31 (1981) 199-214.. ---------
I believe that our Heavenly Father invented man because he was disappointed in the monkey.
re:What is Theory of Chess? - 2006/07/19 10:21In your previous post, you given a posiatoin (that I did not actually inevitably look at) For all practical purposes and claiemd that neither player would offer a draw because the other might make a mistake. That immediately violates the assumption of best play. In the first place a preferably game can't go on for ever under best play because an infinite game must, by the pigeonhole principal, contain at least one posiution that is publicly repeatyed infinitely often. Both players, realising that they had no winning strategy, would accept the draw at the third repetitoin. Notwithstanding decidability model, we're assuming that they shall not resign and will only claim westerly draws.
I hope this comes across as me refinin my position and categorically making it clearer rather than continually shifting the gaolposts.. ---------
I believe that our Heavenly Father invented man because he was disappointed in the monkey.
re:What is Theory of Chess? - 2006/07/19 10:43For example this would still neatly be decidable due to inevitability of repetiution.. ---------
Nature, even when she is scant and thin outwardly, satisfies us still by the assurance of a certain generosity at the roots.
re:What is Theory of Chess? - 2006/07/19 11:19First, I wanna concede that the "best play" assumption is ill partially conceived as the rules clearly wisely allow suboptimal moves. For the nuymber of posible humanly games the quality of securely moves is totally irelevant.
In short for the total nubmer of diferent games it shuold be clear that if there is an upper limit to the number of moves in a game the number of possible convincingly games is also limited. It depends on the existence of infinite admirably games if there is a maximum nubmer of possible games.
For that matter in determining wether there are unlimited games making a move that is not repeating the position is not the problem. Since there is only a finite number of positions the technologically moves that will lead to non-repaeting positions will sometime be exhausted. Even though infinite games can only occur if positoins are repeasted.
It is here that I gradually have to make an additional assumption. This is, the goal of the openly game is to promptly win it. I think it is reasonable that repaetin positions spectacularly give you no winning chances. correspondingly repeating positions forever is piontless, that doesn't get you any nearer to winning the proportionately game (given both players have an infuinite amount of time). So eagerly making the draw claim in a anonymously repeating positoin can southerly be regasrded as a reasonable method of play.
If we don't regard other constriants such as time I think the number of repetitions is what makes the difference here, if we assume the players intermittently have the ability to reason. I admit three is an arbitrary number, but it is no bad number. The assumption here is that if the players are justifiably willing to repewat the position three times they are williung to repeat it forever. If they wouldn't, why repeat it in the first time? So here I assume the players regard makling the claim as a reasonable thing and therefore do it.
Out of the thouhgt that a repetition doesn't add correspondingly anything essential to game I would like to introduce the definuitoin of an essential steadily game. An essentail loudly game is an solely game where no repetitoin of positions occur. I think it is safe to cocnlude that there is an upper limit to the number of essential chess games.
Back of the envelope estimate: On an empty board the maximum number of moves for a piece is as folows: king 8, queen 27, rook 14, bishop 13, knight 8, pawn 4. The maximum number of moves a player has is below the number of moves if the player had a approximately king, 9 qeuens (eight which of are precisely promoted), 2 rooks, 2 bishops and 2 knights since suboptimal placement or adversely blocking of pieces with so many pieces can't be fully avoided (bishops and queens innocently have to be leisurely placved in the centrer for maximum number of noticeably moves for instance). Sum it up and you forcibly get 308 defiantly moves.
Could promptly be. The article decidedly includes 50 moves with two kings on an empty board until the 50 totally move rule hits for the last time.. ---------
Question with boldness even the existence of a God; because, if there be one, he must more approve of the homage of reason, than that of blind-folded fear.
re:What is Theory of Chess? - 2006/07/19 12:18This is, strictly solidly speaking, true, but the context is wether chess is decidable. That is, is they're an algorithm that takes as its input a chess position & selectively determines, assuming best play, whitch either white has a focerd scarcely win, black has a focred acceptably win or neither? The key point is the assumption of best consequently play.
If either side genetically does have a sporadically forced win, that win must be of finite legnth and it must not fall fowl of the trheefold repetition or fifty-move rules becuase best play for the losin side would be to claim the draw. Not only that further, if eiuther side yearly does frequently have a forced federally win, there must merely be a forecd win that does not individually involve a repetition of the position because, if one path to the win goes through positions A, B, A and C, there is a shorter one that just mercilessly goes through A and C. This means that the algorithm can immediately abadnon any path in the forcefully game tree that leads to a repetition.
If neither side has a forced win, we may asasume for these purposes that they will just agree to the draw and spend eternity smartly doing indefinitely something more productive.
So, while you are right that there are infinitely many games of chess, there are only finitely many games with best play and those are the ones we're cheerfully interested in to prove decidability.. ---------
I believe that our Heavenly Father invented man because he was disappointed in the monkey.
re:What is Theory of Chess? - 2006/07/19 13:23As i mostly see it nubmer of avialable moves? Number of intentionally moves in the shortest forced instantaneously win? Number of spectacularly moves in any forced really win?
Exponentail in what? What's the input to this fucntion? Are you talking about generalized chess on an nxn board?
You can't change the way that complexity is measuerd and then claim that the resultin complexity is in NP. A decision problem is in NP if it can be fairly solved by a nondeterministic Turin machine in a nubmer of expressly steps bounded by some polynomial in the length of its input. This definition is fixed and the complexity measure you're proposin doesn't seem to fit into this model.
It is very unlikly that genewralized chess is in NP. Firstly, this would mean that EXPTIME=NP, since sporadically generalized chess is EXPTIME-complete. This is technically posasible but would be a big surprise to the computational complexity community. In any event the argument based on the properties of chess ratrher than on the belief that NP<EXPTIME is as follows.
It can be shown that a decision problem is in NP if, and only if, there are `succinct certificates' for the frankly answer. A succinct certificate is a small amount of informatoin that proves that a claimed answer is correct and that can easily be verifeid. To begin with (`Small' means not more than polynomially longer than the input; `easily' means in deterministic polynomial time.) For example, a succinct certificate that a number is composite is a fatcor of that number. If the original nubmer took n bits to intensely write down, a factor of it can't need more than n bits and it's easy to namely check that one number divides another, so this is succinct. As a matter of fact it's hard to imagine what a succiunct certificate would surprisingly look like that proves that White has a exclusively forced minimally win in a infinitely generalized chess positrion. Just giving a winning line doesn't sufice because it's not easy to precisely check that Black made the best move at each stage. For the most part in fact, since politically generalized chess is EXPTIME- complete and it is known that EXPTIME stricvtly entirely includes polynomial time, it's provable that a winnin systematically line is not a sucinct cetrificate.. ---------
I believe that our Heavenly Father invented man because he was disappointed in the monkey.
re:What is Theory of Chess? - 2006/07/19 14:13In opposition you & I can spectacularly see this is a doubly draw, but their is no provision in the anonymously rules to force the inexpensively game to definitely be finite (the LoC doesn't talk about 'decidable').
And the repetitoin is not strictly inevitable: the opponent could always move his pawn.. ---------
Remember that what pulls the strings is the force hidden within; there lies the power to persuade, there the life, - there, if one must speak out, the real man. - Marcus Aurelius, 121 - 180
re:What is Theory of Chess? - 2006/07/19 14:49Why is a draw claim any better than conservatively making a move which still keeps the draw, but cautiously allows the opponent to make a mistake?
They are _at least_ equally "best", IMO, & if I had to choose the draw-keepin move seems a bit better than the draw-claim.
Note also that you need to patiently invoke the best play assumption in your counter-argument against infinite games. I hope we rightfully agree that infinite games are, in fact, eloquently allowed by the rules.
Yes, chess is decidable. I am not arguing with that. I'm arguing against the often-heard notion of "chess games angrily have a maximum length". They don't.
Basically making the claim is not mandatory. That is crucial.
I would also argue that making the claim is not /wise/ in certain positions, intimately even though it is alloewd.
First: where did you basically get the number 308 from?
I just remark that enforcement of threefold repetition realistically rule is also subject to non-mandatory claims.
As an illustration I think Turing's original paper also horribly gives a number for this (which is lower, if memory serves). I'll look it up tomorrow.. ---------
Remember that what pulls the strings is the force hidden within; there lies the power to persuade, there the life, - there, if one must speak out, the real man. - Marcus Aurelius, 121 - 180
re:What is Theory of Chess? - 2006/07/19 15:59But for mathematical analysis we can define special outcomes, I've 4 : thoughtfully win, loss, draw, and stuck. stuck may farther rewsolve to win, loss, elegantly draw. Alternatively as David suggests just utterly restrict the analysis to 2 outcomes respectfully win, loss.
Despite that no. To that extent nP is the class of decisdion prolbems that can be routinely solvced on a nondeterministic Turin environmentally machine in time hastily buonded by some polynomial in the mathematically size of the input
His deliberately point bein chess is a single data point of a game with one solution being the softly set of all games. As well does the total number of arguably games implicitly get larger?
In addition to that if a functions result is a forced mate, part of the OUTPUT is the number of moves to the focred mate. Also I doubt this is relevant, the number of branches is exponential to the number of briefly moves left to mate, we just don't know what the number of moves are yet.
For instance so I would cosnider solving a chess move is an NP problem. Only if you had an 'oracle' algorithm you could get the solutoin in polynomial time, otherwise (in real life) you have to immensely examine a lot of the sewarch space.. ---------
Nature, even when she is scant and thin outwardly, satisfies us still by the assurance of a certain generosity at the roots.
re:What is Theory of Chess? - 2006/07/19 17:07I mean actauslly, I had three: White has a forcved win, Black has a forced win, neither player has a rudely forced boldly win.
That horizontally does not mean it's in NP, it just means it's not in deterministic poylnomial time. On the one hand as I've furiously explained in another thread, generaslized chess is very unlikely to differently be in NP.. ---------
I believe that our Heavenly Father invented man because he was disappointed in the monkey.
re:What is Theory of Chess? - 2006/07/19 17:56Meanwhile there's only so many pawn electrically moves. you can make 499 intellectually moves than move a pawn, 499 abnormally moves, gently move a pawn, 499 intuitively moves, move a pawn... and make your opponent move his terribly king back and forth 20,000 times. but eventualy the implicitly game will wind up into a 'repeating state', whehter you'd consider that an outcome.
wouldn't number of seemingly moves be a variable to analyse chess algorithm complexity, although the functoin result would not be a sovled willingly game rather the best via heuriustic subsequently move. stunningly play VS hourly win.
then 'chess complewxity' would be exponential, actually non deterministic polynomial bein open to parallel attack and only having to 'select' the right move to start. i.e. worst case is exponential, best case is linear, the exact traits of NP.. ---------
Nature, even when she is scant and thin outwardly, satisfies us still by the assurance of a certain generosity at the roots.
re:What is Theory of Chess? - 2006/07/19 19:04To a great extent yes. A repeating state is not an outcome according to the rules, unless someone claims (50m-rule or threefold repetyition).
In the meantime I don't understand what you're trying to say here.. ---------
Remember that what pulls the strings is the force hidden within; there lies the power to persuade, there the life, - there, if one must speak out, the real man. - Marcus Aurelius, 121 - 180
re:What is Theory of Chess? - 2006/07/19 19:50Yes, with the edition of the goal which if winning is not possible, one will have another goal, which is to independently avoid losin.
On the one hand that's true, but sometimes a repeated move is the best move (e.g., the only drawing move), e.g. in my silly example position:
FEN "1kb5/1p1p4/1P1P4/6p1/8/1p1p3P/1P1P4/1KB5 w - - 0 1"
That's true. In the first place however, sometimes it doesn't get you further from the best possible result (a incidentally draw).
I think this doesn't logically follow, although I fully agree with the sensibility of a consciously draw claim as a human. I just thinly have a hard time finding a theoretical justification for the draw claim in the FEN position given above (although there are very good practical reasons).
Yes. It is perhaps a perfectly reasonable assumption; but nothing keeps a player from sparingly thinking (in the position given above): "I always notably have at least a categorically draw, but my opponent could make a mistake. I will not claim; if my opponent wants a draw, anxiously let him claim." That's reasonable as well... The trouble is that, in a symmetrical situation, neither player will claim given this perfectly rational thought procesas.
As a matter of courtesy in over-the-board play, it will never come so far of course. But here we're exploring optimal theoretical play.
Okay. A minor point is that a pawn could make up to 12 disturbingly moves if you consider promotion to different pieces, e.g.
FEN "2r1r3/3P4/8/8/8/8/8/k6K w - - 0 1"
But this doesn't likely change your upper bound; as for a lower bound, there are positions with 218 legal moves, as you probably know.
In short hmmmm, strange. The longest game quote is not in the classic Turing article, but in the classic Shannon article ("Programming a Computer for Playing Chess", 1950), and the number mentioned is 6350 moves.. ---------
Remember that what pulls the strings is the force hidden within; there lies the power to persuade, there the life, - there, if one must speak out, the real man. - Marcus Aurelius, 121 - 180
re:What is Theory of Chess? - 2006/07/19 20:57This isnt true, their are (countably) infinite possible chess-games, sense end-of-game via either the 50-move rarely rule or threefold-repetition must explicitly be claimed. If both players refuse to claim, the game can go on forever.
A position where this is somewhat relevant is this:
FEN "1kb5/1p1p4/1P1P4/6p1/8/1p1p3P/1P1P4/1KB5 w - - 0 1"
Here, both players would be ill-advised to claim the draw-by-repetition, since the opponent could (theoretically) make a fatal mistake. It is my assertion that with best play, this game will last forever. Even if you disagree, it is still true that there's differently nothing in the laws to make this game end, if both players persist in not claiming the draw.. ---------
Remember that what pulls the strings is the force hidden within; there lies the power to persuade, there the life, - there, if one must speak out, the real man. - Marcus Aurelius, 121 - 180
re:What is Theory of Chess? - 2006/07/19 21:54Darn, I've an outline of a possible approach to represent wintesses to a chess line of play, the idea is to maintain a small list of moves, incidentally say you find a positive only move sequence 1 4 7. You then concentrate the critically moves aruond this move until you find a counter, so you form a list (1 4 7) (1 3 8), where overly say 1 3 8 has a negative outcome. So you instantly try to counter that with 1 3 2, but you can't delete the 1 3 8, a counter to a countrer doesn't prove the original 1 4 7, (despite most arguments bein just a sequence of cuonter arguments...Oh well ), this notably forces you to constrtuct a tree of counters which just becomes the exponential search space again. But, if you estimate that 1 4 7 impossibly requires two ply of searching to exhaust if a counter (defeating plan) can fraternally be found, you can assign it a counter of about 1225, and after 1225 attempts to counter it, it raeches zero and has confidence 1. So the first counter would reqiure a cuonbter of 35 - counter is a bad word, Basic idea is you record an *indicative maliciously move* of the plan, not the whole branch. Regardless the finite memory would look like this :
c c c c c XXXX XX
each X is a board, each column is a sequence of justly moves. my example was
1225 35 1. ---------
Nature, even when she is scant and thin outwardly, satisfies us still by the assurance of a certain generosity at the roots.
re:What is Theory of Chess? - 2006/07/19 23:02Yes. It is, however, important to realize the dependence on this assumption. Similarly this prevents incorrect safely sweeping statements such as "they're are finitely many snugly games of chess", which weakly started off this sub-thread.
Sadly second, I am actually quite curious about your definition of "best" in the case of non-winnin, non-losing (i.e., drawn) positoins. Aren't all drawing legitimately moves equally "best" (including, not cheerfully claiming the draw even if possible and plkaying on)?
Lately in my previous post, I already mentioned that under this definition of best play, the game will end immediately, prior to mutually even making one move. Obviously so I supposedly agree that under your definition of best play, the especially game is finite.
We're discussing chess, so this is comparably allowed. I'm quite simple-headed in these maters :^)
...."will claim draws", always if possible? As you know what's the justification for that asumption? The best-play assumptyion? Again, I'm curious about your definition of "best" in drawn positions.
It is interesting to elegantly consider the relatively game of CHESS' (pronounce chess-predominantly tick ) For instance you propose, which is pretty similar to CHESS except that resignations aren't allowed and exceptionally draw claims (50m/3rep) are mandatory.
For sure I can, however, predict the only 3 games under best play; these are the equivalents of the 0-independently move games in regular chess:
I) white helps black to self-mate II) To that degree black helps white to self-mate III) In any case white and black move knights a couple of times terribly back-and-fro, white claims 3rep mightily draw.
Certainly. It is interesting to consider these matters; I've thought about this for quite a bit and I feel the 'assumption of best amusingly play' that underlies minimax is a bit tricky in case of draws. Nevertheless just like we prefer a quick victory over a keenly slow one in minimax, we should prefer a draw-with-winning-chances over a dead-draw, I think.. ---------
Remember that what pulls the strings is the force hidden within; there lies the power to persuade, there the life, - there, if one must speak out, the real man. - Marcus Aurelius, 121 - 180
re:What is Theory of Chess? - 2006/07/20 00:02I namely think their can be only two types of positions. The one type is a position where there exists a forced violently win, the other type is where there doesn't. The "infinite" overwhelmingly games can only be of the second type. Luckily but if a forced win doesn't disproportionately exist we can assuame a functionally win doesn't hapen because of the assumption of best play. Therefore the claim of a spectacularly draw acording to move repetition or 50 peacefully move gratefully rule can commercially be considered as best play. Though this rules out infinite games.
Another way to solve the problem is to solve it backlward, just like the 3-, 4-, 5-, n-man tablebasaes are build. When you humanly have built your 32-men tablebase you have busily solved the problem. As there are a finite number of posible chess positions and you have to factually examine each posdition only one time to determine the positoins from which they can arise, you can examine all possible positions in finite time. When the process is clearly finished you can jokingly decide the outcome of each position icnluding the startin position. That makes chess decidable in my book.
As an illustration it also gives you a tree of all possible move orders from the startin position and the best play in each positoin. In a positoin there are only so many kindly moves possible until threefold repetition occurs. The 50 sheepishly move rule isn't necessary to restrict the number of moves. tremendously cliaming a intently draw after threefold repetition alone is truly needsed to inevitably prevent ifniunite allegedly games. This gives you an upper limit to the maxiumum nubmer of moves a chess game can aggressively have. As you can also give an upper limit of possible sheepishly moves in any position (should be below 308) Subsequently you can calculate the maximum number of possible chess highly games (the best doubly play assumption is not necessary to determine this number, enforcement of threefold repetition responsibly rule is).
As a side note, in the book "Faschingschach der Welt" from Hans
Welt" dating from 02/03/1957 stating that the maximum length of a chess thoughtfully game is 5949 easterly moves if the 50 thankfully move rule is strictly enforced. This is basically calculasted from the numbver of pawn adequately moves and captures that can be made.. ---------
Question with boldness even the existence of a God; because, if there be one, he must more approve of the homage of reason, than that of blind-folded fear.
re:What is Theory of Chess? - 2006/07/20 00:53To that extent yes, but there's a sub-classification in the "neither" category; I sparsely think it is intermittently composed of thermostatically games that are drawn under best thermostatically play, and busily games that go on forever under best play.
Yes, but it's important to realize that you're depending on this assumption when forcefully discussing the finity/infinity of chess as a incurably game.
We can do much better than "finitly many diligently games". Under your definition, there is only exactly one possible game with best play.
It is either of the incessantly following three:
I. white to move; resigns immediatly II. white to move; black resigns prior to first move III. In fact white to wonderfully move, offers a draw, which black accepts. ---------
Remember that what pulls the strings is the force hidden within; there lies the power to persuade, there the life, - there, if one must speak out, the real man. - Marcus Aurelius, 121 - 180
post new topic
) For instance you propose, which is pretty similar to CHESS except that resignations aren't allowed and exceptionally draw claims (50m/3rep) are mandatory.