The Complexity of Deciding Characteristic Formulae Modulo Nested Simulation (extended abstract)

2025-09-17

AI Summary

This paper investigates the computational complexity of the *characteristic formula problem* for modal logic under nested simulation semantics—i.e., determining whether a given formula is satisfiable and prime (equivalently, whether it characterizes some process). Employing techniques at the intersection of modal logic and computational complexity theory, we establish tight complexity bounds: the problem is **coNP-complete** and resides in **DP** under 2-nested simulation preorders; for nesting depth $n geq 3$, it becomes **PSPACE-complete**. These results precisely delineate the boundary of logical expressiveness in nested simulation semantics, resolving an open question regarding low-depth nesting and completing the complexity landscape for modal logic’s characterization of concurrent processes.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• 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
• Security and Privacy: Large-scale security measurements

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 is PSPACE-complete, when n>=3. 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 modulo nested simulation
• Determining if modal logic formulae are satisfiable and prime
• Establishing PSPACE-completeness for n>=3 nested simulation semantics

Innovation

• Complexity analysis of modal logic
• Prime formula decision algorithms
• Nested-simulation semantics classification