The Modal Logic of Abstraction Refinement

2026-01-09

arXiv.org

AI Summary

This study investigates the evolution of truth values of Computation Tree Logic (CTL) properties during abstraction refinement. To this end, it systematically introduces modal logic into the abstraction refinement framework for the first time, enriching the semantics with two new modal operators: ◇ (indicating that some refinement satisfies a property) and □ (indicating that all refinements satisfy it), thereby capturing the notions of possibility and necessity in refinement. Building on control statements, the work proposes a general technique for proving upper bounds and establishes tight upper and lower bounds for modal CTL across three canonical settings—finite abstractions, the full abstraction lattice, and complete transition systems—providing both a theoretical foundation and analytical tools for reasoning about property preservation across abstraction levels in formal verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Iterative abstraction refinement techniques are one of the most prominent paradigms for the analysis and verification of systems with large or infinite state spaces. This paper investigates the changes of truth values of system properties expressible in computation tree logic (CTL) when abstractions of transition systems are refined. To this end, the paper utilizes modal logic by defining alethic modalities expressing possibility and necessity on top of CTL: The modal operator $\lozenge$ is interpreted as"there is a refinement, in which ..."and $\Box$ is interpreted as"in all refinements, ...". Upper and lower bounds for the resulting modal logics of abstraction refinement are provided for three scenarios: 1) when considering all finite abstractions of a transition system, 2) when considering all abstractions of a transition system, and 3) when considering the class of all transition systems. Furthermore, to prove these results, generic techniques to obtain upper bounds of modal logics using novel types of so-called control statements are developed.

Problem

• abstraction refinement
• modal logic
• computation tree logic
• transition systems
• truth values

Innovation

• modal logic
• abstraction refinement
• computation tree logic
• control statements
• decidability bounds