The Zhou Ordinal of Labelled Markov Processes over Separable Spaces

MARTÍN SANTIAGO MORONI; PEDRO SÁNCHEZ TERRAF

REVIEW OF SYMBOLIC LOGIC

CAMBRIDGE UNIV PRESS

Lugar: Cambridge; Año: 2023

1755-0203

There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou´s probabilistic bisimilarity, while the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator O, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the Zhou ordinal of S, Zh(S). When Zh(S)=0, S satisfies the Hennessy-Milner property. The second author proved the existence of an LMP S with Zh(S) >= 1 and Zhou showed that there are LMPs having an infinite Zhou ordinal. In this paper we show that there are LMPs S over separable metrizable spaces having arbitrary large countable Zh(S) and that it is consistent with the axioms of ZFC that there is such a process with an uncountable Zhou ordinal.