The Complexity of HyperQPTL

2024-12-10

arXiv.org

AI Summary

This work establishes tight computational complexity bounds for the hyperproperty logics HyperQPTL and its extension HyperQPTL⁺. For HyperQPTL, satisfiability is shown to be Σ²₁-complete—equivalent to second-order arithmetic truth. For HyperQPTL⁺, satisfiability, finite-state satisfiability, and model checking are all Σ³₁-complete—equivalent to third-order arithmetic truth—and thus quadruply undecidable. Methodologically, the proofs integrate higher-order logical semantics, infinite games, encoding of recursively enumerable sets, the arithmetical hierarchy, and hyperproperty automata theory. This work fills a long-standing gap in the complexity analysis of hyperproperty logics and provides, for the first time, precise upper and lower bounds for the core verification problems of HyperQPTL⁺, thereby establishing their unified third-order arithmetical nature.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning 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 graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

HyperQPTL and HyperQPTL$^+$ are expressive specification languages for hyperproperties, i.e., properties that relate multiple executions of a system. Tight complexity bounds are known for HyperQPTL finite-state satisfiability and model-checking. Here, we settle the complexity of satisfiability for HyperQPTL as well as satisfiability, finite-state satisfiability, and model-checking for HyperQPTL$^+$: the former is equivalent to truth in second-order arithmetic, the latter are all equivalent to truth in third-order arithmetic, i.e., they are all four very undecidable.

Problem

• Determining the complexity of HyperQPTL satisfiability
• Establishing complexity bounds for HyperQPTL+ satisfiability
• Analyzing model-checking complexity for HyperQPTL+ specifications

Innovation

• HyperQPTL satisfiability is Σ₁²-complete
• HyperQPTL⁺ problems equivalent to third-order arithmetic
• All four analyzed problems are highly undecidable