The complexity of bisimilarity on pointmass processes

TL;DR

This paper investigates the descriptive complexity of bisimilarity on various classes of Markov decision processes and labeled transition systems, revealing its analytic and Borel properties and implications for modal logic characterization.

Contribution

It establishes the complexity classification of bisimilarity on uncountable state spaces and connects it to logical definability and process termination properties.

Findings

Bisimilarity is analytic for processes with finitely-supported measures.

Bisimilarity on countable Kripke frames is classifiable by countable structures.

Bisimilarity of well-founded processes is Borel.

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.