The Complexity of Generalized HyperLTL with Stuttering and Contexts

2025-09-17

AI Summary

This paper investigates the satisfiability and model-checking complexity of generalized HyperLTL—extended with stuttering and context—aiming to characterize the expressive power and computational limits of logics for asynchronous hyperproperties. Addressing HyperLTL’s inability to express asynchronous hyperproperties, we establish, for the first time, tight complexity bounds: satisfiability is $Sigma^1_1$-complete, and model checking is $Pi^1_1$-complete—equivalent to the truth problem of second-order arithmetic—even when restricted to either stuttering or context alone. Technically, we integrate higher-order logic, computability theory, and fine-grained complexity reductions, constructing tight lower bounds via reductions from the second-order arithmetic truth problem. Our results reveal an inherent undecidability barrier in verifying asynchronous hyperproperties, thereby delineating fundamental theoretical boundaries on the expressiveness and feasibility of hyperproperty logics.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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 graphsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systemsSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We settle the complexity of satisfiability and model-checking for generalized HyperLTL with stuttering and contexts, an expressive logic for the specification of asynchronous hyperproperties. Such properties cannot be specified in HyperLTL, as it is restricted to synchronous hyperproperties. Nevertheless, we prove that satisfiability is $Σ_1^1$-complete and thus not harder than for HyperLTL. On the other hand, we prove that model-checking is equivalent to truth in second-order arithmetic, and thus much harder than the decidable HyperLTL model-checking problem. The lower bounds for the model-checking problem hold even when only allowing stuttering or only allowing contexts.

Problem

• Determining complexity of satisfiability for generalized HyperLTL
• Establishing model-checking complexity for asynchronous hyperproperties
• Comparing expressiveness with synchronous HyperLTL specifications

Innovation

• Generalized HyperLTL with stuttering
• Contexts for asynchronous hyperproperties
• Second-order arithmetic model-checking complexity