偏序集博弈的赢家翻转
Flipping the Winner of a Poset Game
摘要：
偏序集博弈（英语：Partially-ordered set games），又称偏序集博弈，是一类两人组合博弈。竞技场由一组元素组成，其中一些元素大于其他元素。两个玩家轮流移除一个元素和所有大于它的元素，谁拿走最后一个元素谁就赢了。偏序集博弈的例子包括Nim和Chomp。研究了偏序集博弈中博弈方获胜策略的计算复杂性。我们给出了一个归纳过程，修改偏序集游戏，以改变尼姆值，非正式地捕捉游戏中的获胜策略。对于一般偏序集对策G，我们描述了一个有效的方法来构造一个非G的对策，使得第一个局中人有获胜策略当且仅当第二个局中人在G上有获胜策略。这解决了长期存在的问题，即这种建设是否可以有效地进行。这种构造也允许我们将布尔公式类减少到偏序集博弈，建立偏序集博弈复杂性的下界。
英文摘要：
Partially-ordered set games, also called poset games, are a class of two-player combinatorial games. The playing field consists of a set of elements, some of which are greater than other elements. Two players take turns removing an element and all elements greater than it, and whoever takes the last element wins. Examples of poset games include Nim and Chomp. We investigate the complexity of computing which player of a poset game has a winning strategy. We give an inductive procedure that modifies poset games to change the nim- value which informally captures the winning strategies in the game. For a generic poset game G, we describe an efficient method for constructing a game not G such that the first player has a winning strategy if and only if the second player has a winning strategy on G. This solves the long-standing problem of whether this construction can be done efficiently. This construction also allows us to reduce the class of Boolean formulas to poset games, establishing a lower bound on the complexity of poset games.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
