A Bisimulation-Invariance-Based Approach to the Separation of Polynomial Complexity Classes

2026-01-27

AI Summary

This work explores a novel approach to separating the complexity classes P and NP or PSPACE under the constraint of bisimulation invariance. Building on Otto’s theorem, it characterizes bisimulation-invariant queries in P as those definable in the polyadic μ-calculus and reduces this definability question—via power graph transformations—to one in the modal μ-calculus. The paper introduces the notion of “relative regularity” to analyze the associated tree language families and establishes that a bisimulation-invariant query lies in P if and only if its corresponding tree language is not relatively regular. This method circumvents the reliance on linear orderings typical in descriptive complexity, offering a fresh perspective on complexity class separation grounded in model checking and logical definability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We investigate the possibility to separate the bisimulation-invariant fragment of P from that of NP, resp. PSPACE. We build on Otto's Theorem stating that the bisimulation-invariant queries in P are exactly those that are definable in the polyadic mu-calculus, and use a known construction from model checking in order to reduce definability in the polyadic $\mu$-calculus to definability in the ordinary modal mu-calculus within the class of so-called power graphs, giving rise to a notion of relative regularity. We give examples of certain bisimulation-invariant queries in NP, resp. PSPACE, and characterise their membership in P in terms of relative non-regularity of particular families of tree languages. A proof of non-regularity for all members of one such family would separate the corresponding class from P, but the combinatorial complexity involved in it is high. On the plus side, the step into the bisimulation-invariant world alleviates the order-problem that other approaches in descriptive complexity suffer from when studying the relationship between P and classes above.

Problem

• bisimulation-invariance
• complexity classes
• P vs NP
• descriptive complexity
• tree languages

Innovation

• bisimulation-invariance
• polyadic mu-calculus
• relative regularity
• power graphs
• descriptive complexity