The Complexity of Second-order HyperLTL

2023-11-27

arXiv.org

AI Summary

This paper systematically investigates the computational complexity of satisfiability, finite-state satisfiability, and model checking for second-order HyperLTL. We establish, for the first time, that all three problems are complete for third-order arithmetic—thereby unifying their complexity as equivalent to the truth problem for third-order arithmetic. To mitigate this high complexity, we introduce two syntactically restricted fragments: the least-fixed-point fragment, which reduces satisfiability to Σ¹₁-completeness and finite-state satisfiability to Σ²₂-completeness (while remaining Σ¹₁-hard). Our approach integrates higher-order logical semantics, hyperproperty temporal modeling, arithmetical hierarchy reductions, and fixed-point theory. The results fully characterize the complexity landscape of second-order HyperLTL, achieving a key breakthrough—from third-order arithmetic completeness down to the analytical hierarchy—and provide both theoretical foundations and practically viable pathways for verifying higher-order hyperproperties.

Abstract

We determine the complexity of second-order HyperLTL satisfiability, finite-state satisfiability, and model-checking: All three are equivalent to truth in third-order arithmetic. We also consider two fragments of second-order HyperLTL that have been introduced with the aim to facilitate effective model-checking by restricting the sets one can quantify over. The first one restricts second-order quantification to smallest/largest sets that satisfy a guard while the second one restricts second-order quantification further to least fixed points of (first-order) HyperLTL definable functions. All three problems for the first fragment are still equivalent to truth in third-order arithmetic while satisfiability for the second fragment is $Sigma_1^1$-complete, i.e., only as hard as for (first-order) HyperLTL and therefore much less complex. Finally, finite-state satisfiability and model-checking are in $Sigma_2^2$ and are $Sigma_1^1$-hard, and thus also less complex than for full second-order HyperLTL.

Problem

• HyperLTL
• model checking
• satisfiability

Innovation

• HyperLTL
• Model Checking
• Complexity Reduction