Verifying First-Order Temporal Properties of Infinite-State Systems via Timers and Rankings

2026-01-19

AI Summary

This work addresses the challenge of verifying first-order temporal properties in infinite-state systems by proposing a unified deductive verification framework that does not rely on fairness assumptions. The approach reduces the verification of arbitrary first-order temporal properties to termination problems, introducing prophecy timer variables to predict the number of steps until the target formula holds. By integrating well-founded orderings with implicit ranking techniques extended to infinite domains, the method enables fully automatic verification of complex systems. Leveraging SMT solvers, the framework generates concise, automatically checkable proofs and successfully verifies a range of challenging temporal properties across multiple benchmark infinite-state systems.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Temporal Planning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present a unified deductive verification framework for first-order temporal properties based on well-founded rankings, where verification conditions are discharged using SMT solvers. To that end, we introduce a novel reduction from verification of arbitrary temporal properties to verification of termination. Our reduction augments the system with prophecy timer variables that predict the number of steps along a trace until the next time certain temporal formulas, including the negated property, hold. In contrast to standard tableaux-based reductions, which reduce the problem to fair termination, our reduction does not introduce fairness assumptions. To verify termination of the augmented system, we follow the traditional approach of assigning each state a rank from a well-founded set and showing that the rank decreases in every transition. We leverage the recently proposed formalism of implicit rankings to express and automatically verify the decrease of rank using SMT solvers, even when the rank is not expressible in first-order logic. We extend implicit rankings from finite to infinite domains, enabling verification of more general systems and making them applicable to the augmented systems generated by our reduction, which allows us to exploit the decrease of timers in termination proofs. We evaluate our technique on a range of temporal verification tasks from previous works, giving simple, intuitive proofs for them within our framework.

Problem

• temporal verification
• infinite-state systems
• first-order temporal properties
• termination
• timers

Innovation

• prophecy timers
• implicit rankings
• temporal verification
• termination reduction
• SMT-based verification