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There exist two notions of equivalence of behavior between states of aLabelled Markov Process (LMP): state bisimilarity and event bisimilarity. Thefirst one can be considered as an appropriate generalization to continuousspaces of Larsen and Skou's probabilistic bisimilarity, while the second one ischaracterized by a natural logic. C. Zhou expressed state bisimilarity as thegreatest fixed point of an operator $\mathcal{O}$, and thus introduced anordinal measure of the discrepancy between it and event bisimilarity. We callthis ordinal the "Zhou ordinal" of $\mathbb{S}$, $\mathfrak{Z}(\mathbb{S})$.When $\mathfrak{Z}(\mathbb{S})=0$, $\mathbb{S}$ satisfies the Hennessy-Milnerproperty. The second author proved the existence of an LMP $\mathbb{S}$ with$\mathfrak{Z}(\mathbb{S}) \geq 1$ and Zhou showed that there are LMPs having aninfinite Zhou ordinal. In this paper we show that there are LMPs $\mathbb{S}$over separable metrizable spaces having arbitrary large countable$\mathfrak{Z}(\mathbb{S})$ and that it is consistent with the axioms of$\mathit{ZFC}$ that there is such a process with an uncountable Zhou ordinal.

The Zhou Ordinal of Labelled Markov Processes over Separable Spaces