The Complexity of Generalized HyperLTL with Stuttering and Contexts

2025-04-11

AI Summary

This paper investigates the satisfiability and model checking complexity of generalized HyperLTL—extended with stuttering and context mechanisms—for specifying asynchronous hyperproperties. Addressing HyperLTL’s inability to express such properties, we establish: (1) satisfiability of generalized HyperLTL is Σ₁¹-complete; (2) model checking is equivalent to deciding truth in second-order arithmetic, hence undecidable; and (3) adding either stuttering or context alone suffices to render model checking undecidable. Technically, our approach integrates higher-order logical semantics, formalizations of hyperproperties, and the arithmetical hierarchy. The core contribution lies in precisely characterizing the computational essence of asynchronous hyperproperty verification: while satisfiability retains its inherent complexity, model checking transcends the decidable boundary—revealing its fundamental intractability. This work thus provides the first tight complexity-theoretic classification for asynchronous hyperproperty reasoning within a temporal logic framework.

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

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 $Sigma_1^1$-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

• Determine complexity of satisfiability for generalized HyperLTL.
• Compare model-checking complexity with second-order arithmetic.
• Analyze asynchronous hyperproperties with stuttering and contexts.

Innovation

• Generalized HyperLTL with stuttering
• Contexts for asynchronous hyperproperties
• Σ₁¹-complete satisfiability complexity