Go and mathematics

The game of Go is one of the most popular games in the world. As a result of its elegant and simple rules, the game has long been an inspiration for mathematical research. Shen Kuo, a Chinese scholar in 11th century, estimated that the number of possible board positions is around 10172 in The Dream Pool Essays. In more recent years, research of the game by John H. Conway led to the invention of the surreal numbers and contributed to development of combinatorial game theory (with Go Infinitesimals[1] being a specific example of its use in Go).

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Computational complexity

Generalized Go is played on n × n boards, and the computational complexity of determining the winner in a given position of generalized Go depends crucially on the ko rules.

Go is “almost” in PSPACE, since in normal play, moves are not reversible, and it is only through capture that there is the possibility of the repeating patterns necessary for a harder complexity.

Without ko

Without ko, Go is PSPACE-hard.[2] This is proved by reducing True Quantified Boolean Formula, which is known to be PSPACE-complete, to generalized geography, to planar generalized geography, to planar generalized geography with maximum degree 3, finally to Go positions.

Go with superko is not known to be in PSPACE. Though actual games seem never to last longer than moves, in general it is not known if there were a polynomial bound on the length of Go games. If there were, Go would be PSPACE-complete. As it currently stands, it might be PSPACE-complete, EXPTIME-complete, or even EXPSPACE-complete.

Japanese ko rule

Japanese ko rules state that only the basic ko, that is, a move that reverts the board to the situation one move previously, is forbidden. Longer repetitive situations are allowed, thus potentially allowing a game to loop forever, such as the triple ko, where there are three kos at the same time, allowing a cycle of 12 moves.

With Japanese ko rules, Go is EXPTIME-complete.[3]

Superko rule

The superko rule (also called the positional superko rule) states that a repetition of any board position that has previously occurred is forbidden. This is the ko rule used in most Chinese and US rulesets.

It is an open problem what the complexity class of Go is under superko rule. Though Go with Japanese ko rule is EXPTIME-complete, both the lower and the upper bounds of Robson’s EXPTIME-completeness proof[3] break when the superko rule is added.

It is known that it is at least PSPACE-hard, since the proof in[2] of the PSPACE-hardness of Go does not rely on the ko rule, or lack of the ko rule. It is also known that Go is in EXPSPACE.[4]

Robson[4] showed that if the superko rule, that is, “no previous position may ever be recreated”, is added to certain two-player games that are EXPTIME-complete, then the new games would be EXPSPACE-complete. Intuitively, this is because an exponential amount of space is required even to determine the legal moves from a position, because the game history leading up to a position could be exponentially long.

As a result, superko variants (moves that repeat a previous board position are not allowed) of generalized chess and checkers are EXPSPACE-complete, since generalized chess[5] and checkers[6] are EXPTIME-complete. However, this result does not apply to Go.[4]

Complexity of certain Go configurations

A Go endgame begins when the board is divided into areas that are isolated from all other local areas by living stones, such that each local area has a polynomial size canonical game tree. In the language of combinatorial game theory, it happens when a Go game decomposes into a sum of subgames with polynomial size canonical game trees.

With that definition, Go endgames are PSPACE-hard.[7]

This is proven by converting the Quantified Boolean Formula problem, which is PSPACE-complete, into a sum of small (with polynomial size canonical game trees) Go subgames. Note that the paper does not prove that Go endgames are in PSPACE, so they might not be PSPACE-complete.

Determining which side wins a ladder capturing race is PSPACE-complete, whether Japanese ko rule or superko rule is in place.[8] This is proven by simulating QBF, known to be PSPACE-complete, with ladders that bounce around the board like light beams.

Since each location on the board can be either empty, black, or white, there are a total of 3n2 possible board positions on a square board with length n; however not all of them are legal. Tromp and Farnebäck derived a recursive formula for legal positions of a rectangle board with length m and n.[9] The exact number of was obtained in 2016.[10] They also find an asymptotic formula , where , and . It has been estimated that the observable universe contains around 1080 atoms, far fewer than the number of possible legal positions of regular board size (m=n=19). As the board gets larger, the percentage of the positions that are legal decreases.

Board size n×n 3n2 Percent legal (legal positions) (A094777)[11]
1 × 1 3 33.33% 1
2 × 2 81 70.37% 57
3 × 3 19,683 64.40% 12,675
4 × 4 43,046,721 56.49% 24,318,165
5 × 5 847,288,609,443 48.90% 414,295,148,741
9 × 9 4.43426488243 × 1038 23.44% 1.03919148791 × 1038
13 × 13 4.30023359390 × 1080 8.66% 3.72497923077 × 1079
19 × 19 1.74089650659 × 10172 1.20% 2.08168199382 × 10170

Game tree complexity

The computer scientist Victor Allis notes that typical games between experts last about 150 moves, with an average of about 250 choices per move, suggesting a game-tree complexity of 10360.[12] For the number of theoretically possible games, including games impossible to play in practice, Tromp and Farnebäck give lower and upper bounds of 101048 and 1010171 respectively.[9] The lower bound was improved to a Googolplex by Walraet and Tromp.[13] The most commonly quoted number for the number of possible games, 10700[14] is derived from a simple permutation of 361 moves or 361! = 10768. Another common derivation is to assume N intersections and L longest game for NL total games. For example, 400 moves, as seen in some professional games, would be one out of 361400 or 1 × 101023 possible games.

The total number of possible games is a function both of the size of the board and the number of moves played. While most games last less than 400 or even 200 moves, many more are possible.

Game size Board size N (intersections) N! Average game length L NL Maximum game length (# of moves) Lower limit of games Upper limit of games
2 × 2 4 24 3 64 386,356,909,593[15] 386,356,909,593
3 × 3 9 3.6×105 5 5.9×104
4 × 4 16 2.1×1013 9 6.9×1010
5 × 5 25 1.6×1025 15 9.3×1020
9 × 9 81 5.8×10120 45 7.6×1085
13 × 13 169 4.3×10304 90 3.2×10200
19 × 19 361 1.0×10768 200 3×10511 1048 101048 1010171
21 × 21 441 2.5×10976 250 1.3×10661

The total number of possible games can be estimated from the board size in a number of ways, some more rigorous than others. The simplest, a permutation of the board size, (N)L, fails to include illegal captures and positions. Taking N as the board size (19 × 19 = 361) and L as the longest game, NL forms an upper limit. A more accurate limit is presented in the Tromp/Farnebäck paper.

Longest game L (19 × 19) (N)L Lower limit of games Upper limit of games Notes
1 361 361 361 White resigns after first move, 361 ignoring all symmetry including y = x else (distances from corner) 10×10−10=90 90/2=45 +10 (adding back x = y points of symmetry) = 55.
2 722 721 Black resigns after white's first move, 721 ignoring all symmetry including y=x else 19×19−19=342 342/2=171 +19 = 190 − 1 = 189.
50 2.1×10126 7.5×10127
100 1.4×10249 5.6×10255
150 6.4×10367 4.2×10383
200 1.9×10481 3.2×10511
250 8.2×10587 2.4×10639
300 2.8×10684 7.8×10766
350 3.6×10760 1.3×10895
361 1.4×10768 1.8×10923 Longest game using 181 black and 180 white stones
411 n/a 1.3×101051 Longest professional game[16]
500 n/a 5.7×101278
1000 n/a 3.2×102557
47 million n/a 10108 3613 moves
1048 n/a 101048 1010171 Longest game

10700 is thus an overestimate of the number of possible games that can be played in 200 moves and an underestimate of the number of games that can be played in 361 moves. Since there are about 31 million seconds in a year, it would take about ​2 1⁄4 years, playing 16 hours a day at one move per second, to play 47 million moves.

See also

Notes
  1. Go Infinitesimals
  2. Lichtenstein, David; Sipser, Michael (April 1980). "Go Is Polynomial-Space Hard" (PDF). Journal of the ACM. 27 (2): 393–401. doi:10.1145/322186.322201.
  3. Robson, John (1983). "The complexity of Go". Proceedings of the IFIP 9th World Computer Congress on Information Processing: 413–417.
  4. Robson, J (1984). Combinatorial games with exponential space complete decision problems. Proceedings of the Mathematical Foundations of Computer Science. Lecture Notes in Computer Science. 176. pp. 498–506. doi:10.1007/BFb0030333. ISBN 978-3-540-13372-8.
  5. Aviezri Fraenkel and D. Lichtenstein (1981). "Computing a perfect strategy for n×n chess requires time exponential in n". J. Comb. Th. A. 31 (31): 199–214. doi:10.1016/0097-3165(81)90016-9.
  6. J. M. Robson (1984). "N by N checkers is Exptime complete". SIAM Journal on Computing. 13 (2): 252–267. doi:10.1137/0213018.
  7. Wolfe, David (2002). Nowakowski, Richard J. (ed.). "Go endgames are PSPACE-hard" (PDF). More Games of No Chance, Mathematical Sciences Research Institute Publications 42: 125–136.
  8. Crâşmaru, Marcel; Tromp, John (2000). Ladders are PSPACE-Complete. Computers and Games. Lecture Notes in Computer Science. 2063. Springer. pp. 241–249. CiteSeerX 10.1.1.24.4665. doi:10.1007/3-540-45579-5_16. ISBN 978-3-540-43080-3.
  9. Tromp, J; Farnebäck, G (2007), Combinatorics of Go, Springer, Berlin, Heidelberg, doi:10.1007/978-3-540-75538-8_8, ISBN 978-3-540-75537-1
  10. https://tromp.github.io/go/legal.html 208 168 199 381 979 984 699 478 633 344 862 770 286 522 453 884 530 548 425 639 456 820 927 419 612 738 015 378 525 648 451 698 519 643 907 259 916 015 628 128 546 089 888 314 427 129 715 319 317 557 736 620 397 247 064 840 935
  11. https://tromp.github.io/go/gostate.pdf
  12. Allis 1994
  13. Walraet, M; Tromp, J (2016), A Googolplex of Go Games, Springer, Berlin, Heidelberg, doi:10.1007/978-3-319-50935-8_18, ISBN 978-3-319-50934-1
  14. AGA – top ten reason to play Go
  15. Tromp 1999
  16. https://homepages.cwi.nl/~aeb/go/misc/gostat.html

References
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