The complexity of bisimilarity on pointmass processes

2026-04-07

AI Summary

This study investigates the descriptive complexity of bisimilarity for nondeterministic labelled Markov processes with finite-support measure assignments over uncountable standard Borel spaces. Employing tools from descriptive set theory—specifically the theories of Borel and analytic sets together with reduction techniques—it establishes, for the first time, that bisimilarity for well-founded processes is Borel definable. Moreover, by reducing the tail equivalence relation $E_0$ on binary sequences to this bisimulation problem, the work provides a sharp lower bound on its complexity. The results demonstrate that bisimilarity in the well-founded case resides precisely within the Borel hierarchy and further show that the countable fragment of basic modal logic cannot fully characterize bisimilarity even for low-rank processes, thereby strictly limiting the expressive power of modal logic for capturing such behavioral equivalences.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

We assess the descriptive complexity of *bisimilarity* or "equality of behavior" on a family of Markov decision processes over uncountable standard Borel spaces, namely *nondeterministic labelled Markov processes* (NLMP). We show that bisimilarity is analytic for processes with a uniform assignment of finitely-supported measures to each state and label. More finely, we obtain that bisimilarity on the space of countable Kripke frames (or labelled transition systems) is classifiable by countable structures. We show that bisimilarity of well-founded ("terminating") processes is Borel. We also provide a lower complexity bound by reducing the relation of eventual equality of binary sequences $E_0$ to the former. As a consequence, there is no countable fragment of basic modal logic with denumerable conjunctions that characterizes bisimilarity for processes of small rank. We finally apply the previous Borel definability to study the well-founded part of discrete uniform processes over uncountable spaces.

Problem

• bisimilarity
• Markov decision processes
• descriptive complexity
• nondeterministic labelled Markov processes
• Borel spaces

Innovation

• bisimilarity
• descriptive complexity
• nondeterministic labelled Markov processes
• Borel definability
• modal logic