Generalized game
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In computational complexity theory, a generalized game is a game or puzzle that has been generalized so that it can be played on a board or grid of any size. For example, generalized chess is the game of chess played on an [math]\displaystyle{ n\times n }[/math] board, with [math]\displaystyle{ 2n }[/math] pieces on each side. Generalized Sudoku includes Sudokus constructed on an [math]\displaystyle{ n\times n }[/math] grid.
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.[1][2]
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 (with Japanese ko rules), Quixo,[3] and checkers are EXPTIME-complete.[4][5][6]
See also
References
- ↑ Reisch, Stefan (1981), "Hex ist PSPACE-vollständig", Acta Informatica 15 (2): 167–191, doi:10.1007/bf00288964
- ↑ Iwata, Shigeki; Kasai, Takumi (January 1994), "The Othello game on an [math]\displaystyle{ n\times n }[/math] board is PSPACE-complete", Theoretical Computer Science 123 (2): 329–340, doi:10.1016/0304-3975(94)90131-7
- ↑ Mishiba, Shohei; Takenaga, Yasuhiko (2020-07-02). "QUIXO is EXPTIME-complete" (in en). Information Processing Letters 162: 105995. doi:10.1016/j.ipl.2020.105995. ISSN 0020-0190.
- ↑ Fraenkel, Aviezri S.; Lichtenstein, David (September 1981), "Computing a perfect strategy for [math]\displaystyle{ n\times n }[/math] chess requires time exponential in [math]\displaystyle{ n }[/math]", Journal of Combinatorial Theory, Series A 31 (2): 199–214, doi:10.1016/0097-3165(81)90016-9
- ↑ Robson, J. M. (1983), "The complexity of Go", Proceedings of the IFIP 9th World Computer Congress on Information Processing: 413–417
- ↑ Robson, J. M. (May 1984), "[math]\displaystyle{ N }[/math] by [math]\displaystyle{ N }[/math] checkers is Exptime complete", SIAM Journal on Computing (Society for Industrial & Applied Mathematics ({SIAM})) 13 (2): 252–267, doi:10.1137/0213018
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