AI 中文总结
本文证明四向移动方格网格上的 Cardinal Grid Slime Trail 是 PSPACE 完全问题，通过适配 QBF 归约设计网格兼容小工具，还将构造扩展至八向变体，解决了相关开放问题。
AI 中文摘要
Slime Trail 是一款双人组合游戏，玩家交替移动共享棋子至相邻顶点，每次离开的顶点会被永久移除，目标是到达终点节点。Ferland 和 Burke（2017）证明了任意平面图上的 Slime Trail 是 PSPACE 完全问题，并提出实际游戏中使用的网格版本是否也如此的疑问。我们解决这一开放问题，证明 Cardinal Grid Slime Trail（即四向移动的方格网格上的 Slime Trail）是 PSPACE 完全问题。我们将他们的 QBF 归约适配至网格场景，设计了符合四度约束和整数格奇偶性要求的网格兼容小工具。进一步证明该构造经 45 度旋转后可扩展至八向变体。
英文摘要
Slime Trail is a two-player combinatorial game in which the players alternately move a shared token to an adjacent vertex, permanently removing each vertex the token leaves, while attempting to reach a goal node. Ferland and Burke (2017) proved that Slime Trail is PSPACE-complete on arbitrary planar graphs and asked whether the same holds for the grid version actually used in play. We resolve this open problem by proving that Cardinal Grid Slime Trail, that is, Slime Trail on a square grid with four-directional movement, is PSPACE-complete. We adapt their QBF reduction to the grid setting, designing grid-compatible gadgets that respect the degree-4 bound and the parity constraints of the integer lattice. We further show the construction extends, under a 45-degree rotation, to the eight-directional variant.
