套索博弈（Lasso Games）
Lasso Games
AI总结：
本文研究有限图上的套索博弈，对比环与套索两种胜者判定语义，分析其计算复杂度与最优博弈内存需求，证明套索语义对博弈性质的显著影响及相关博弈的复杂度下界。
摘要：
我们研究有限图上的博弈，这类博弈的边带有标签或权重，博弈会在首次形成环时立即终止。最终的博弈过程呈现套索形状：一条路径（头部）后接一个环（循环）。我们考虑两种确定胜者的语义：环语义（仅环起作用）和套索语义（考虑整个博弈过程）。研究表明，采用套索语义会对博弈的复杂度和最优博弈所需的内存产生显著影响：套索语义下的均值收益博弈不再具有位置确定性，阈值问题（判定博弈值是否至少为0）是PSPACE完全的；相比之下，套索奇偶博弈等价于弱奇偶博弈，仍具有位置确定性且可在多项式时间内求解。此外，我们证明若一个具有位置确定性的获胜目标在循环置换下封闭，且该目标及其补集在拼接下均封闭，则它等价于奇偶目标。最后，我们考虑模k博弈（k≥2，胜者取决于路径长度模k的结果），证明这类博弈在两种语义下均为PSPACE完全。除计算复杂度外，我们还分析了最优博弈所需的内存：对于所有在环语义下具有位置确定性的博弈，套索语义下大小为2^n的内存足够；同时，套索均值收益博弈和模k博弈存在指数级内存下界。
英文摘要：
We study games on finite graphs with labelled or weighted edges in which plays end as soon as the first loop is formed. The resulting play has a lasso shape: a path (head) followed by a cycle (loop). We consider two semantics for determining the winner: the loop semantics, where only the loop matters, and the lasso semantics, where the entire play is taken into account. We show that adopting the lasso semantics may have a dramatic impact on both the complexity of the game and the memory required to play optimally. Mean-payoff games with lasso semantics are no longer positionally determined, and the threshold problem – deciding whether the value of the game is at least 0 – is PSPACE-complete. In contrast, lasso parity games are equivalent to weak parity games, remain positionally determined and are solvable in polynomial time. Furthermore, we show that if a positionally determined winning objective is closed under cyclic permutation and both the objective and its complement are closed under concatenation, then it is equivalent to the parity objective. Finally, we consider mod-k games, where the winner depends on the length of the path modulo k >= 2. We show that these games, in both semantics, are PSPACE-complete. Alongside computational complexity, we analyze the memory required to play optimally. We establish that for any game positionally determined in loop semantics, memory of size 2^n is sufficient in lasso semantics. We complement this by showing exponential memory lower bounds for lasso mean-payoff and mod-k games.
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