We consider the following problem: given $d \times d$ rational matrices $A_1, \ldots, A_k$ and a polyhedral cone $\mathcal{C} \subset \mathbb{R}^d$, decide whether there exists a non-zero vector whose orbit under multiplication by $A_1, \ldots, A_k$ is contained in $\mathcal{C}$. This problem can be interpreted as verifying the termination of multi-path while loops with linear updates and linear guard conditions. We show that this problem is decidable for commuting invertible matrices $A_1, \ldots, A_k$. The key to our decision procedure is to reinterpret this problem in a purely algebraic manner. Namely, we discover its connection with modules over the polynomial ring $\mathbb{R}[X_1, \ldots, X_k]$ as well as the polynomial semiring $\mathbb{R}_{\geq 0}[X_1, \ldots, X_k]$. The loop termination problem is then reduced to deciding whether a submodule of $\left(\mathbb{R}[X_1, \ldots, X_k]\right)^n$ contains a ``positive'' element.

翻译：考虑如下问题：给定 $d \times d$ 有理矩阵 $A_1, \ldots, A_k$ 和一个多面体锥 $\mathcal{C} \subset \mathbb{R}^d$，判断是否存在非零向量，其在 $A_1, \ldots, A_k$ 乘法作用下的轨道包含于 $\mathcal{C}$。该问题可解释为验证具有线性更新和线性守卫条件的多路径 while 循环的终止性。我们证明对于交换的可逆矩阵 $A_1, \ldots, A_k$，该问题是可判定的。判定过程的关键在于从纯代数角度重新诠释该问题：具体而言，我们发现了其与多项式环 $\mathbb{R}[X_1, \ldots, X_k]$ 及多项式半环 $\mathbb{R}_{\geq 0}[X_1, \ldots, X_k]$ 上模的联系。从而将循环终止性问题归约为判断 $\left(\mathbb{R}[X_1, \ldots, X_k]\right)^n$ 的子模是否包含一个“正”元素。