Хаванах: Стратегиялық ойын және жасанды интеллект сын-қабатысы

the board game

Havannah is a two player abstract strategy board game invented by Christian Freeling. It belongs to the family of games commonly called connection games; its relatives include Hex and TwixT. Havannah has "a sophisticated and varied strategy" and is best played on a base 10 hexagonal board, 10 hex cells to a side. The game was published for a period in Germany by Ravensburger, with a smaller, base 8 board suitable for beginners. It is nowadays only produced by Hexboards.

In Hex, when the board is completely filled, exactly one player will have a winning connection; in Havannah a completely filled board will have usually more than one winning structures (but the game ends with first winning structure). Unlike in Hex, in Havannah draws are technically possible, in practice they are extremely rare. There has been one known draw between human players. Tactics are much easier to master than strategy, and differences in playing level are considerable.

In 2002 Freeling offered a prize of 1000 euros, available through 2012, for any computer program that could beat him in even one game of a ten game match. For many years, computer programs lagged far behind human players. However, since 2010 several Havannah playing programs have applied Monte Carlo tree search techniques resulting in some notable improvement in playing strength. The "Havannah Challenge 2012" was held October 15–19, 2012 during which Freeling played ten games against three of the strongest Havannah playing programs available, playing (at least) one game as black and one as white against each opponent. Freeling lost the challenge when he had to resign a game with white against the Lajkonik program. Until 2019, the best humans were still by far stronger than computers. However, MetaTotoro, based on Polygames (an open source project, initially developed by Facebook Artificial Intelligence Research and several universities), won four times in a row on the board of size 8 against the human player with the best ELO rank on LittleGolem, who was also the winner of various tournaments. This result was achieved by the same program as the one used for beating best humans at Hex. It is a zero learning based algorithm, as in AlphaZero, but with novelties: boardsize invariance thanks to fully convolutional neural networks (as in U Net) and global pooling. This allows growing architectures, meaning the program can learn on a small board, and then extrapolate on a large board.

Solving Havannah is PSPACE complete with respect to the size of the input graph. The proof is by a reduction from generalized geography and is based on using ring threats to represent the geography graph. In detail, since Lichtenstein and Sipser have proved that generalized geography remained PSPACE hard even if the graph is only bipartite and of degree at most 3, it only remains to construct an equivalent Havannah position from such a graph, which is accomplished by constructing various gadgets in Havannah.