Reflections on Termination of Linear Loops

TL;DR

This paper leverages linear dynamical systems techniques to analyze the termination behavior of general loops, providing new methods for modeling and reasoning about loop invariance and termination conditions.

Contribution

It introduces a framework connecting linear dynamical systems with loop termination analysis, including models for affine transition systems and invariant conditions.

Findings

Loops expressible in linear integer arithmetic have a deterministic affine transition system model

For systems with integer eigenvalues, invariant conditions can be exactly characterized by linear integer formulas

Develops a monotone conditional termination analysis method for general loops

Abstract

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.