Go and mathematics

From testwiki
Jump to navigation Jump to search

Template:Short description Template:GoBoardGame 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, an 11th century Chinese scholar, estimated in his Dream Pool Essays that the number of possible board positions is around 10172. In more recent years, research of the game by John H. Conway led to the development 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).

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 n2 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 L(m,n) of a rectangle board with length m and n.[9] The exact number of L(19,19) was obtained in 2016.[10] They also find an asymptotic formula L≈ABm+nCmn, where A≈0.8506399258457145, B≈0.96553505933837387 and C≈2.975734192043357249381. 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 L (legal positions) (Template:OEIS link)[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 Template:Nowrap. Another common derivation is to assume N intersections and L longest game for NTemplate:I sup 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 NTemplate:I sup 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 Template:Val 5 Template:Val
4 × 4 16 Template:Val 9 Template:Val
5 × 5 25 Template:Val 15 Template:Val
9 × 9 81 Template:Val 45 Template:Val
13 × 13 169 Template:Val 90 Template:Val
19 × 19 361 Template:Val 200 Template:Val 1048 101048 1010171
21 × 21 441 Template:Val 250 Template:Val

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, NTemplate:I sup 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
1 361 361 362 White resigns after first move, 361(362 if include pass) 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.(56 if include pass)
2 129960 130682 361(black)× 360(white) + 361(black pass) + 361(white pass)
50 Template:Val Template:Val
100 Template:Val Template:Val
150 Template:Val Template:Val
200 Template:Val Template:Val
211 Template:Val Template:Val average length of professional games
250 Template:Val Template:Val
300 Template:Val Template:Val
350 Template:Val Template:Val
361 Template:Val Template:Val Longest game using 181 black and 180 white stones
411 n/a Template:Val Longest professional game[16]
500 n/a Template:Val
1000 n/a Template:Val
47045881 n/a 10108 3613 moves
1048 n/a 1010100 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 Template:Frac years, playing 16 hours a day at one move per second, to play 47 million moves.

See also

Template:Portal

Notes

Template:Reflist

References

Template:Go (game)

  1. ↑ Template:Cite web
  2. ↑ 2.0 2.1 Template:Cite journal
  3. ↑ 3.0 3.1 Template:Cite journal
  4. ↑ 4.0 4.1 4.2 Template:Cite book
  5. ↑ Template:Cite journal
  6. ↑ Template:Cite journal
  7. ↑ Template:Cite journal
  8. ↑ Template:Cite book
  9. ↑ 9.0 9.1 Template:Citation
  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. ↑ Template:Cite web
  12. ↑ Allis 1994
  13. ↑ Template:Citation
  14. ↑ Template:Cite web
  15. ↑ Tromp 1999
  16. ↑ Template:Cite web
Retrieved from "https://en.wiki.beta.math.wmflabs.org/w/index.php?title=Go_and_mathematics&oldid=20782"