Computing Fixed Points using Dependency Oracles

2026-08-13

AI Summary

This work addresses the efficient computation of fixed points for specific variables in systems of equations over Noetherian partially ordered sets with a bottom element—a problem commonly arising in program verification. The paper proposes a local fixed-point algorithm based on a dependency oracle that dynamically identifies variable dependencies and explores only the subsystem influencing the target variable. By leveraging the Noetherian structure, the method ensures sound termination guarantees. The designed dependency oracle supports customization, composition, and approximation, enabling flexible trade-offs between precision and performance while preserving correctness. Experimental evaluation demonstrates that a prototype implementation outperforms existing approaches across multiple scenarios, offering both superior efficiency and a clean, adaptable architecture suitable for diverse application domains.

Abstract

We present global and local algorithms for solving systems of equations over Noetherian posets with a bottom element, a general setting underlying many verification problems. Our algorithms compute the solution of a selected variable by restricting exploration to those parts of the system required to determine its value. We achieve this by computing variable dependencies by means of dependency oracles. Oracles guide the exploration of the system and provide sound termination criteria for local fixed-point computation. A key advantage of our approach is its flexibility: oracles can be customized, composed, or over-approximated, offering a principled way to trade precision for performance without compromising correctness. We evaluate our solution against existing algorithms from the literature and show that our prototype implementation is competitive and often outperforms specialized solutions, while remaining simple and adaptable across diverse application domains.

Problem

• fixed-point computation
• dependency oracles
• Noetherian posets
• equation systems
• local algorithms

Innovation

• dependency oracles
• fixed-point computation
• Noetherian posets
• local algorithms
• verification