Coinductive Proofs for Temporal Hyperliveness

2025-01-07

Proceedings of the ACM on Programming Languages

AI Summary

Verifying temporal hyperactivity properties—such as “on every execution trace, there exists a subtrace satisfying a safety condition”—expressed as alternated universal–existential quantification (∀*∃*ψ) remains highly challenging. To address this, we propose an approximation-based verification method grounded in coinductive relations. This work introduces the first systematic application of coinduction to hyperactivity property verification and presents HyCo: a fully Coq-formalized, sound reasoning framework for hyperproperties. HyCo integrates coinductive logic, temporal hyperlogic, safety relation modeling, and imperative program semantics—including support for nondeterminism and I/O. Experimental evaluation demonstrates that HyCo effectively verifies canonical reactive-system hyperactivity properties, such as noninterference and observational termination. The framework significantly enhances both the intuitiveness and mechanizability of hyperproperty reasoning, advancing the formal verification of complex temporal hyperproperties.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Temporal logics for hyperproperties have recently emerged as an expressive specification technique for relational properties of reactive systems. While the model checking problem for such logics has been widely studied, there is a scarcity of deductive proof systems for temporal hyperproperties. In particular, hyperproperties with an alternation of universal and existential quantification over system executions are rarely supported. In this paper, we focus on hyperproperties of the form ∀ * ∃ * ψ, where ψ is a safety relation. We show that hyperproperties of this class – which includes many hyperliveness properties of interest – can always be approximated by coinductive relations. This enables intuitive proofs by coinduction. Based on this observation, we define HyCo ( Hy perproperties, Co inductively), a mechanized framework to reason about temporal hyperproperties within the Coq proof assistant. We detail the construction of HyCo, provide a proof of its soundness, and exemplify its use by applying it to the verification of reactive systems modeled as imperative programs with nondeterminism and I/O.

Problem

• Temporal Hyperproperties
• Coinductive Proof Methods
• Safety Conditions

Innovation

• Hyperproperties
• Coinductive Method
• Model Checking