AI 中文总结
本文提出对抗串行尼姆（PARTIZAN-SERIAL-NIM）规则集，证明其局面结果在m≥2时不依赖于m，给出O(n²)与O(n)时间的结果计算算法，且证明其局面原子权重为整数。
AI 中文摘要
组合博弈是一种无隐藏信息或随机元素的双人游戏，组合博弈理论的核心目标是确定给定组合博弈局面的结果，即哪位玩家拥有获胜策略。尼姆（NIM）是组合博弈理论中知名且基础的规则集。本文提出一种新型的尼姆对抗变体，名为PARTIZAN-SERIAL-NIM，其定义如下：存在n堆编号为1,2,…,n的石子；两位玩家分别拥有(1,2,…,n)的排列σ^L与σ^R；玩家的操作是从其排列中最小值对应的非空堆中移除任意正数量的石子；无法进行操作的玩家落败。该规则集是串行尼姆（SERIAL-NIM）与对抗终局尼姆（PARTIZAN-END-NIM）的推广。我们给出一种算法，在每次算术运算与比较操作均以O(1)时间执行的前提下，可在O(n²)时间内计算PARTIZAN-SERIAL-NIM给定局面的结果。此外，针对所有非空堆均有相同数量m（m≥2）石子的情况，我们证明该结果不依赖于m，并提出一种可在O(n)时间内计算结果的算法。进一步地，我们证明PARTIZAN-SERIAL-NIM的每个局面的原子权重均为整数。
英文摘要
A combinatorial game is a two-player game without hidden information or chance elements. The main object of combinatorial game theory is to determine the outcome (i.e., which player has a winning strategy) of a given position in combinatorial games. Nim is a well-known and fundamental ruleset in combinatorial game theory. This paper proposes a novel partizan variant of Nim called Partizan-Serial-Nim, defined as follows: there are $n$ piles of stones indexed by $1, 2, \ldots, n$; the two players have permutations $\mathbfσ^L$ and $\mathbfσ^R$ of $(1, 2, \ldots, n)$, respectively; a move is to remove any positive number of stones from the non-empty pile with the minimum value in the player's permutation; the player who cannot make a move loses. This ruleset is a generalization of Serial-Nim and Partizan-End-Nim. We give an algorithm to compute the outcome of a given position in Partizan-Serial-Nim in $O(n^2)$ time, provided that each arithmetic and comparison operation is performed in $O(1)$ time. Also, for the case where all non-empty piles have the same number $m$ of stones, we prove that the outcome does not depend on $m$ for $m \geq 2$ and present an algorithm to compute the outcome in $O(n)$ time. Further, we prove that the atomic weight of every position in Partizan-Serial-Nim is an integer.
