AI 中文总结
本文解决了分配格上Ungar游戏结果判定的计算复杂性问题，证明其在$l(P) \le 3$的二分图Hasse图偏序集上为PSPACE完全，$l(P) \le 2$时为NP完全，$l(P) \le 1$时属于LOGSPACE。
AI 中文摘要
Ungar游戏由Defant等人提出，是一种在有限格上进行的组合游戏，其起源于Ungar为解决Scott斜率问题引入的几何变换。当在分配格$L$上进行该游戏时，根据Birkhoff表示定理，$L$可表示为偏序集$P$的序理想构成的格$J(P)$，此时Ungar游戏简化为反复选择并移除有限偏序集$P$中的非空极大元集合。Defant等人提出了一个开放问题：确定$J(P)$上Ungar游戏的结果是否关于$|P|$是PSPACE完全的。本文肯定地解决了该开放问题，更准确地说，我们证明即使在满足以下条件的偏序集$P$上，该问题仍是PSPACE完全的：最大链长度$l(P) \le 3$，且$P$的Hasse图的基础图是二分图，其中链的长度定义为链中元素数量减一。此外，我们表明当$l(P) \le 2$时该问题是NP完全的，而$l(P) \le 1$的情况已知属于LOGSPACE。
英文摘要
The Ungar game, introduced by Defant et al., is a combinatorial game played on a finite lattice, which originates from a geometric transformation introduced by Ungar to resolve Scott's slope problem. When played on a distributive lattice $L$, the Ungar game reduces to repeatedly choosing and removing a non-empty set of maximal elements from a finite poset $P$, where $L$ is represented as the lattice $J(P)$ of the order ideals of $P$ by Birkhoff's representation theorem. Defant et al. raised the open question of whether determining the outcome of the Ungar game on $J(P)$ is PSPACE-complete with respect to $|P|$. In this paper, we settle this open question in the affirmative. More precisely, we prove that the problem is PSPACE-complete even when restricted to posets $P$ with maximum chain length $l(P) \le 3$ such that the underlying graph of the Hasse diagram of $P$ is bipartite, where the length of a chain is defined as the number of elements in the chain minus one. Furthermore, we show that the problem is NP-complete for $l(P) \le 2$, whereas the case $l(P) \le 1$ is known to belong to LOGSPACE.
