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Does Kami No Itte Exist
Keywords: Theory
Moved here from Philosophy Of Go. Let's start with the assumption that a set of kami no itte does in fact exist (I'm undecided, but that's irrelevant for the time being). If there is one, let's consider the methods of discovering it since the dawn of Go. In the game's childhood, players probably found patterns emerging only after many many games (how long would I have to play before I discovered the notion of atari on my own?). Eventually, masters of the game emerged, and they worked long and hard to perfect their art. Their knowledge could be compounded by study, meditation, what have you -- but ultimately could survive only through books and students, each of which could only be developed so much, so fast. And each of which can only develop / sort through / combine so much material so quickly. Add to this the relatively (to today) small numbers of players, and progress toward the kami no itte could be seen as crawling. As the popularity of Go expanded, so too did the level of study, and the recombinant degree of complexity. As time went on, masters were able to study each others' work/art, and critique it and comment on it. All of this was watched intently by the studentia, who then became the new masters, and so on. Enter the computer. Vastly more qualified than the human at simple number-crunching, HAL and his cousins began to race through the relative merits of shape elementals and game threads. But even more significantly, they gave us the tools needed to again raise the level of game analysis exponentially. We can save SGF files, review and comment on them, store them, and ask HAL to crunch them. The race toward the kami no itte speeds up incredibly. As microchips become more complex and AI explodes through its confines, we're probably closer to the kami no itte than ever before. Suppose we then graph the speed of this progress. Wouldn't it look something like this?
...where the horizontal axis is time and the vertical axis is proximity to the kami no itte... If so, then the question is: Will we ever reach it? I'm inclined to say "no," for the reasons outlined above, and because I don't really believe it exists.. -- Scartol Hi! I'm a mathematics student, and while this philosophical discourse is intriguing, I must object in the strongest possible terms to the assertion that kami no itte does not exist. Go is a two-player perfect information game, which always terminates in a finite number of moves. Therefore there is no question whether perfect play exists. It must. Before I give the argument for this, let me do some clean-up about details concerning Go. As you may know, the major Go federations use rules which permit situations like triple ko, in which the game could theoretically continue forever, but then there is also a rule which declares a draw upon the third repetition of the same whole board position. Second, I can give no concrete upper bound on the length of a Go game (practically, no pro game has lasted beyond 450, if what I've read is right). But this is irrelevant. All that matters is whether any single run of play terminates, not whether there is a finite bound on the lengths of all possible runs of play (it is not so hard to devise a combinatorial game which can last 1 or 2 or 45 or 5432459854 moves, or any larger number you care to name, but can not go on forever[1]). Now as promised, I will show why any perfect information game must have a winning strategy for one of the two players. To make argument shorter, we assume there are no draws, though it's trivial to extend the reasoning to cover these cases. We will call the players Black and White, with the convention that Black goes first (it is not necessary to assume that any particular one of the players goes first, or that the possible moves at every point are in any sense symmetric. However, then it becomes not a matter of proving a winning strategy exists for Black or for White, but rather for Black, White, the first player (whoever that may be) or the second player (whoever that may be)). So here we go: consider the set of all possible positions in the game. The most primitive relation between position X and position Y is: Y is an option of X; that is, one of the moves available to the player whose turn it is to move in X, turns X into Y. We assume the concept of position includes information about whose turn it is to move. Consider the initial position P1. Suppose by way of contradiction there is no winning strategy for Black or for White. This means there must be some option P2 of P1, in which there is also no winning strategy for either player. To see why, suppose for the sake of finding a contradiction, that each option of P1 was determined (there was a winning strategy for Black of for White). If one such option was a won position for Black, then Black could simply use his first move to go to that position. If all the options were winners for White, then Black would have no choice but to pick the means of his execution, and White's would just have to wait patiently, and then apply his surefire strategy at whatever option Black chose. So we have shown that the supposedly undetermined initial position P1, begets another undetermined position P2 as one of its options. Likewise, since there is no winning strategy for either side at P2, there must be an option P3 of P2, in which there is no winning strategy for either player. Continuing this way, we generate an infinite run of play P1, P2, P3, P4, ... all of whose positions share the quality of lacking a winning strategy. Consider this carefully!!! I just said we have an infinite run of play such that ... well, it doesn't matter. An infinite run of play is a contradiction in terms, because we know the game must terminate in a finite number of moves (notice how we need not have any idea what this finite number is) To review, we assumed that there is no winning strategy for our two player finite perfect information game, and then we showed that the game must not really be finite after all. How would a thinking galaxy find the kami no itte? It would first consider all possible terminal positions in Go (one is which both players have passed, or otherwise agreed the game is over, or in which one player is forced to concede he has no legal move left). For each such position, the thinking galaxy would note who won, and put it in the huge DONE pile in its memory banks (former red supergiant stars, which have been cleverly re-engineered so that each electron's spin records a bit of information). The DONE pile at any point, consists of all the positions for which the galaxy (or Enlightened Zen master, if you prefer) has already determined who is the winner and what his strategy is. Then the galaxy would work its way backwards, placing in its TO DO pile every possible position whose options are all already in the DONE pile. Suppose it were Black's turn to move in some such position X. Then if one of the options in the DONE pile was a won position for Black, then X would be won for Black as well. But if all the options were won positions for White, then X would be won for White too (extend this reasoning to cover draws and drawing strategies if you like). Either way, the galaxy could move X into the DONE pile. Now there would be a new updated DONE pile, so the TO DO pile would no longer be current, and the galaxy would have to form it again. This process would go on and on and on, with an ever growing DONE pile, perhaps making a new DONE pile every day. Now we make use of the fact that Go is one of those games which is finite in the strong sense that we can pick a specific finite number N ahead of time such that every game of Go must terminate in N moves. Certainly the a game can last no longer than twice the number of possible arrangements of black stones, white stones, and liberties on the board, which in turn is no larger than 3^361. For concreteness, let's just say that the longest game possible lasts a zillion moves. Then, it must be true that on the zillionth day, the galaxy finally finds the initial position itself, and following its usual procedure, evaluates it, and places it in the DONE pile. At this moment the galaxy is enlightened. I might point out again, that when we merely wanted to show that perfect play exists (without identifying a method to find it), we did not have to know there is any particular finite upper bound to how long a game of Go could last, so long as any one game could only go on finitely long. The sequence P1, P2, P3, ... of undetermined positions was like a ladder in Go, with each step in the reasoning like an atari by Black and the response to it by White. If the board were infinite, Black could keep running forever, but so long as the edge is out there somewhere, White's fate is sealed from the beginning (here Black is someone who insists there is no perfect play and White is someone who knows better). [1]If all games terminate and the players only have a finite number of possible moves on each term, as in Go, then there is a time before which all games terminate. For otherwise one of the finitely many choices of the first player must permit infinitely many different lengths of continuation, as must one of the second player's responses to that choice, and so on: in this line of play the game doesn't end. "Another mathematician"
The mathematician again ... In the above, I assumed the concept of kami no itte refers to perfect play, that is, play that assures victory. But if the literal sense of "God's move" is right then maybe the argument is wrong. God might be a moyo player, like Takemiya Masaki, and perhaps His opponent could find ways to refute His strategy, forcing God to either resign Himself to defeat, or to taking real territory on the sides. Either way God, or Takemiya, might not be happy with the result. ___ Since the subject of Zen Enlightenment has been brought up, I was hoping my efforts at teaching some basic combinatorial game theory might be reciprocated with an explanation of this concept. In the West, there was of course a period known as the Enlightenment which consisted largely in the resurgence of rationality as an ideal, and a skepticism about established religious and secular authority. But that is a different way of using the word. So, what is the nature of Enlightenment in the Zen sense? In science, we have somewhat vague but workable criteria of success: expanding the explanatory power and predictive scope of our theories, improving man's control over nature through the development of technology. In Go, a good player is one who often wins or gives a strong challenge to players of high rank, or at least one who creates good positions for himself (certainly there would be something to be said for someone who opens like a shodan, but frequently makes beginner mistakes in middle game and end game, though I gather this isn't common). In art, one can speak of a work being beautiful or expressive. In Zen, there is Enlightenment? But what does this word refer to?
I'd also like to ask some questions about the thought processes other players undergo when playing a game. I wasn't sure where else to put this, so go to MentalProcess using this link if you'd like to respond to my questions. Scartol: I feel like Jeff Goldblum in Jurassic Park. I understand that -- mathematically -- the kami no itte must (does) exist. And yet, in examining the balance between chaos and order (in science, in math, in literature, in societies, in ecology), I grow more skeptical by the day that we humans will ever be able to impose the pure order that we seek, on the goban as everywhere else. To put it another way: A computer may be able to someday determine the kami no itte and use it to defeat all human players. On that day, Go will cease to be a game, and will be seen only historically as a puzzle that had to be solved by machines (see Manuel de Landa's War in the Age of Intelligent Machines for a synopsis of this line of discussion). 'Players' of the future will simply memorize this pattern and play accordingly. 'Skill' will consist merely of knowing the perfect responses to play that deviates from the pattern of white's that comes closest to challenging the kami no itte. Meanwhile, what does that mean for humans? It means that -- until Deep Blue comes up with the ultimate DONE pile -- we have to find the balance between chaos and order. Thus, no kami no itte. Now, about Zen enlightenment. In Zen, enlightenment does not refer to pure rationality, as it was used in Europe. Rather, Zen enlightment is about understanding the way -- the way of the Buddha, the way of the universe, the path, whatever you want to call it. It is neither pure rationality or pure non-rationality. It's about balance and self-awareness. I will let the ancient masters speak for themselves. Here is a story from Paul Reps' compilation Zen Flesh, Zen Bones that does a better job than I ever could: Everyday Life is the Path Joshu asked Nansen: "What is the path?" Nansen said: "Everyday life is the path." Joshu asked: "Can it be studied?" Nansen said: "If you try to study, you will be far away from it." Joshu asked: "If I do not study, how can I know it is the path?" Nansen said: "The path does not belong to the perception world, neither does it belong to the nonperception world. Cognition is a delusion and noncognition is senseless. If you want to reach the true path beyond doubt, place yourself in the same freedom as sky. You name it neither good nor not-good." At these words Joshu was enlightened. I will have to take a look at this "Zen Flesh, Zen Bones" sometime. In the meanwhile, I hope people do appreciate the proof I gave above. It could of course be made much more succinct, if one used mathematical notations and way of expression. At any rate, I think it is a really beautiful proof of something which is not entirely obvious (when I first learned the idea of a combinatorial game and heard this theorem, I felt like I would have tried to refute it if it were phrased as a conjecture. As it was, I nodded my head and proceeded to construct the proof by myself. Now it seems painfully obvious but that's because I've already thought about it so much). And yes, the brute-force methods described above, will probably never be amenable to implementation. That's why I used the metaphor of a thinking galaxy with memory banks made of supergiant stars. I can't really claim to know whether with the number of partciles in the universe, it's thermodynamically possible to construct a perfect player for Go (even if one gave it the age of the Universe to compute). I suspect the answer is no, but for now, I don't really have the calculations to back it up. Certainly, there is a disparity of many orders of magnitude between the number of possible Go games and the number of particles in the world. Then again, it is possible to add really huge numbers on an abacus with nowhere near that many discrete moving parts, so maybe there is some way around this. The idea of order and chaos in Go appeals to me. Chess seems like a more orderly game than Go, in that it's not so hard to evaluate a position once you've mapped out the move tree as far as you can. Just look at material, king safety (possibly backed up by a little bit more move-by-move analysis directed specifically at this one parameter), pawn structure, potential for pawn promotion, strength on open files, and a few other parameters. Yet even there, I believe Deep Blue's programmers had trouble developing anything like a mid-game library to complement the database of opening and endgame patterns. I believe one said that chess is just "too chaotic" with small differences in a position disabling any meaningful analogy making (to a known position in a library) in the mid-game. Or at least analogy making that could be implemented using know algorithms. By the way, see Douglas Hofstadter's "Fluid Concepts and Creative Analogies" for a discussion of an interesting architecture for computer analogy making. The idea in that book is to go beyond 1. implementations inspired by mathematical notions like homomorphism and isomorphism, where the underlying conceptual structure (a group structure, a metric, a topology, etc.) is already given, to 2. analogy making where the conceptual structure is constructed from primitive ingredients together with the mapping between the two analogous sitatuations. But Hofstadter's goal is to take the first baby steps towards an understanding of how the human mind works, not to create an expert system that can outperform a human at any particular task. At any rate, Go is even more frustrating than chess for a programmer because the move tree branches much faster, and positional evaluation is much more difficult. Hundreds of points may hinge on whether a group is alive or dead, and in real play, this may be depend both on its own independent potential for life, its potential for connecting to other groups (themselves of doubtful status), similar questions regarding surrounding enemy stones as well as faraway shapes, and the whole complex interplay of threats and counter-threats based on these considerations. Attempts are being made to remedy the unsatisfactory state of life and death algorithms with goal-directed reasoning. This strikes me as a good idea, true to the spirit of Go. I am reminded of how Davies begins his work "[Tesuji|Tesuji the book" The first principle in reading is to start with a definite purpose. There is no better way to waste time than to say to yourself "I wonder what happens if I play here" and start tracing out sequences aimlessly. Tactics must serve strategy. Start by asking yourself what you would like to accomplish in the position in question, then start hunting around for the sequence that accomplishes it. Once you have your goal clearly in mind the right move, it if exists, will be much easier to find. I think Go is very interesting for the fact that it involves concepts on multiple levels generated from a very simple base, and allowing both formal, numberical and informal, intuitive versions. At the lowest level, we have black, white, liberty, and adjacency. Then we get concepts like capture and connection, as well as life and ko. Then we have sente and gote, strength, good and bad shape, aji, influence, and so on. The excellent player perceives both incisive tactical sequences and grand strategic issues. He is able to leap from one level of abstraction and precision to another. This movement between different levels of thinking and the ability to see their mutual relevance seem to be essential to much of rational thought, but not something which has been implemented very well by machines yet. Kami no itte does exist, as proven above. However, it may be interesting to note that it has not been proven to be unique. That is, there may be more than one way to play perfectly. (Even not taking symmetries into account.) As far as I know, there could be millions of equally 'perfect' games. Does this change anything? Maybe not. Still, it's a nice thought. Maybe tengen is just as good as hoshi. Perhaps those perfect games would even be widely different in style, more so than current professional games? In any case, I feel that perfect game should be jigo. If komi is needed to achieve that, it must be an integer. It's sort of mystery to me why fractional komi is used so often. We know it can't be mathematically cerrect, after all. ;) And preventing ties feels philosophically wrong to me. -Miz SAS: Professional tournaments are often knockouts or other formats where drawn games would be a problem, so fractional komi is pretty much inevitable there. Amateurs have a tendency to follow professional practice (on the grounds that professionals must know what they're doing), so we end up with fractional komi in McMahon tournaments, where it clearly isn't useful. (Part of the discussion here has been moved to Mathematical Bounds of Komi.) Scartol: As a rational individual, I must admit that I can't find a reason to doubt that we will eventually "find" the Kami No Itte. And yet, let us not forget that at one point we believed that we could eventually find a way to predict weather with 100% accuracy -- an idea that chaos theory has more or less voided. Irving: Here's a partial resolution to this argument. Just consider n by n boards. Instead of thinking in terms of perfect moves, think in terms of the perfect general strategy. Then even though the perfect moves clearly exist, it may be provably impossible to find or understand them efficiently. Of course, it could be even more interesting than that. The existence of an efficient solution to Go could be unprovably either way given the current axioms of mathematics, in which case we're back to philosophy... Andre Engels: In theory it's possible. From a position play out all possible games until either they end or they get into a cycle. This will be an enormous but finite number. Because the number is finite, one can calculate the minimax value?s, and ready. It's all very simple in theory, the only problem is that in practice it takes more than the age of the universe. Migeru: Assume that we knew, for each position, which moves result in the largest margin of victory or the smallest margin of defeat. One cannot assume that there is a unique such move for each position, so even "kami no itte" will be a branching tree, and not necessarily a single "perfect" game. However, for each line there must be a point where the tree stops branching or where all branching amounts just to trasposition of yose sequences. There should be such a thing as perfect yose from a sufficiently advanced position. Also, Irving's point is very good, and seems to have escaped most contributors to this page: just knowing the optimal moves doesn't mean that you understand them. This is related to joseki, where "knowing joseki" means not only knowing the optimal sequences, but also how to punish deviations from joseki. Just like people can lose to opponents just because they didn't play joseki, it might be possible for someone who knows the entire "optimal subtree" to be stomped by a player who deviates from kami no itte early. TheDude? So what Irving is suggesting is that the problem of finding the optimal strategy for Go for arbirtrary board sizes may lie outside of P. Even if it lies in P it might practically impossible anyway. For instance the recent solution of Primes in polynomial time was mitigated by the fact that the polynomial time algorithm runs slower than known exponential time algorithms until extremely large numbers are involved . . . making the algorithm unlikely to be useful. JohnAspinall Thought experiment. You wake up 1000 years in the future, revived by alien technology, but you are told that you only have one day to live. What do you want to do with your 24 hours? You want to play a teaching game of go against the best player this new era has to offer. Your hosts put a touch screen in front of you, it displays a go board, and they tell you that you can ask all the questions you want. You don't know whether there's a human, an alien, or a computer at the other end of the network connection. But as far as you can tell, you are playing the nearest equivalent to a player who really understands the kami no itte. What happens? 1. The game is familiar to you. Some joseki have gotten some new wrinkles, and there are a few moves in the fuseki that seem a little strange, but basically, it's a game like you expected. 2. The game is bizarre. The opening is incomprehensible, lines of play are abandoned just when you think you've got an urgent move, and in the middle of (what you think is) the midgame, your teacher suggests you stop now and count, since the rest is obvious. The kami no itte exists. (The proof is above.) But can humans understand it? Is it understandable in terms of all the hand-waving concepts we use today? (Thickness, miai, profit, aji,... ). That's a much more interesting question. This is a copy of the living page "Does Kami No Itte Exist" at Sensei's Library. ![]() |