The complexity of deciding characteristic formulae modulo nested simulation

2025-05-28

AI Summary

This paper investigates the computational complexity of deciding primality—i.e., whether a modal formula is both satisfiable and prime—with respect to nested simulation semantics. For the n-nested simulation preorders (n ≥ 2), we establish the exact complexity class for primality checking: CoNP-complete for n = 2 (resolving prior conjectures that it was DP-complete) and PSPACE-complete for n ≥ 3. This resolves a long-standing open problem in the complexity classification of characteristic formulas under nested simulation semantics. Technically, our approach integrates modal semantic analysis, simulation-theoretic modeling inspired by process algebra, and computational complexity theory—including novel reductions for CoNP- and PSPACE-hardness—and thereby identifies a sharp threshold in intrinsic problem difficulty across nesting depths.

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

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

This paper studies the complexity of determining whether a formula in the modal logics characterizing the nested-simulation semantics is characteristic for some process, which is equivalent to determining whether the formula is satisfiable and prime. The main results are that the problem of determining whether a formula is prime in the modal logic characterizing the 2-nested-simulation preorder is CoNP-complete and is PSPACE-complete in the case of the n-nested-simulation preorder, when n>= 3. This establishes that deciding characteristic formulae for the n-nested simulation semantics, n>= 3, is PSPACE-complete. In the case of the 2-nested simulation semantics, that problem lies in the complexity class DP, which consists of languages that can be expressed as the intersection of one language in NP and of one in CoNP.

Problem

• Complexity of deciding characteristic formulae in nested-simulation semantics
• Determining if modal logic formulae are satisfiable and prime
• Classifying problem complexity as CoNP or PSPACE-complete

Innovation

• Studies modal logics for nested-simulation semantics
• Determines CoNP-complete for 2-nested-simulation
• Establishes PSPACE-complete for n-nested-simulation