This paper shows how techniques for linear dynamical systems can be used to reason about the behavior of general loops. We present two main results. First, we show that every loop that can be expressed as a transition formula in linear integer arithmetic has a best model as a deterministic affine transition system. Second, we show that for any linear dynamical system $f$ with integer eigenvalues and any integer arithmetic formula $G$, there is a linear integer arithmetic formula that holds exactly for the states of $f$ for which $G$ is eventually invariant. Combining the two, we develop a monotone conditional termination analysis for general loops.

中文翻译：

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对线性环路终止的思考

本文展示了如何使用线性动力系统的技术来推理一般循环的行为。我们提出了两个主要结果。首先，我们展示了可以表示为线性整数算法中的转换公式的每个循环都具有作为确定性仿射转换系统的最佳模型。其次，我们证明对于任何具有整数特征值的线性动力系统 $f$ 和任何整数算术公式 $G$，有一个线性整数算术公式完全适用于 $f$ 的状态，其中 $G$ 最终是不变的. 将两者结合起来，我们为一般循环开发了单调条件终止分析。