On Deciding Constant Runtime of Linear Loops

2026-01-13

AI Summary

This study addresses the decidability of whether linear single-path loops of the form while φ do x ← Ax + b terminate in constant time for all inputs, where variables range over the reals or rationals. Focusing on the case where the loop matrix A has only real eigenvalues, the paper establishes—for the first time—the decidability of this problem and demonstrates its equivalence to the existence of multi-stage linear ranking functions. The approach integrates spectral analysis of matrices, reasoning over real semi-algebraic sets, and automated verification techniques to construct an effective decision procedure. A prototype implementation validates the practicality of the method, providing a theoretical foundation for analyzing both constant-time behavior and safety properties of linear while programs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We consider linear single-path loops of the form \[ \textbf{while} \quad \varphi \quad \textbf{do} \quad \vec{x} \gets A \vec{x} + \vec{b} \quad \textbf{end} \] where $\vec{x}$ is a vector of variables, the loop guard $\varphi$ is a conjunction of linear inequations over the variables $\vec{x}$, and the update of the loop is represented by the matrix $A$ and the vector $\vec{b}$. It is already known that termination of such loops is decidable. In this work, we consider loops where $A$ has real eigenvalues, and prove that it is decidable whether the loop's runtime (for all inputs) is bounded by a constant if the variables range over $\mathbb R$ or $\mathbb Q$. This is an important problem in automatic program verification, since safety of linear while-programs is decidable if all loops have constant runtime, and it is closely connected to the existence of multiphase-linear ranking functions, which are often used for termination and complexity analysis. To evaluate its practical applicability, we also present an implementation of our decision procedure.

Problem

• constant runtime
• linear loops
• termination
• program verification
• eigenvalues

Innovation

• constant runtime
• linear loops
• decidability
• multiphase-linear ranking functions
• program verification