Generalized game
Encyclopedia
In computational complexity theory
, a generalized game is a game that has been generalized so that it can be played on a board of any size. For example, generalized chess is the game of chess
played on an n-by-n board, with 2n pieces on each side.
Complexity theory studies the asymptotic difficulty of problems, so generalizations of games are needed, as games on a fixed size of board are finite problems.
For many generalized games which last for a number of moves polynomial in the size of the board, the problem of determining if there is a win for the first player in a given position is PSPACE-complete
. Generalized hex
and reversi
are PSPACE-complete.
For many generalized games which may last for a number of moves exponential in the size of the board, the problem of determining if there is a win for the first player in a given position is EXPTIME-complete. Generalized chess
, go
and checkers are EXPTIME-complete.
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
, a generalized game is a game that has been generalized so that it can be played on a board of any size. For example, generalized chess is the game of chess
Chess
Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...
played on an n-by-n board, with 2n pieces on each side.
Complexity theory studies the asymptotic difficulty of problems, so generalizations of games are needed, as games on a fixed size of board are finite problems.
For many generalized games which last for a number of moves polynomial in the size of the board, the problem of determining if there is a win for the first player in a given position is PSPACE-complete
PSPACE-complete
In complexity theory, a decision problem is PSPACE-complete if it is in the complexity class PSPACE, and every problem in PSPACE can be reduced to it in polynomial time...
. Generalized hex
Hex (board game)
Hex is a board game played on a hexagonal grid, theoretically of any size and several possible shapes, but traditionally as an 11x11 rhombus. Other popular dimensions are 13x13 and 19x19 as a result of the game's relationship to the older game of Go...
and reversi
Reversi
Reversi is a board game involving abstract strategy and played by two players on a board with 8 rows and 8 columns and a set of distinct pieces for each side. Pieces typically are disks with a light and a dark face, each face belonging to one player...
are PSPACE-complete.
For many generalized games which may last for a number of moves exponential in the size of the board, the problem of determining if there is a win for the first player in a given position is EXPTIME-complete. Generalized chess
Chess
Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...
, go
Go (board game)
Go , is an ancient board game for two players that originated in China more than 2,000 years ago...
and checkers are EXPTIME-complete.
External links
- David EppsteinDavid EppsteinDavid Arthur Eppstein is an American computer scientist and mathematician. He is professor of computer science at University of California, Irvine. He is known for his work in computational geometry, graph algorithms, and recreational mathematics.-Biography:Born in England of New Zealander...
's page on Computational Complexity of Games and Puzzles