可分空间上标号马氏过程的Zhou序
The Zhou Ordinal of Labelled Markov Processes over Separable Spaces
摘要：
标签马尔可夫过程（LMP）的状态之间存在两种行为等价的概念：状态双相似性和事件双相似性。第一个可以被认为是一个适当的推广到连续空间的Larsen和Skou的概率双相似性，而第二个是由一个自然的逻辑。C. Zhou将状态双相似性表示为算子$\mathcal{O}$的最大不动点，从而引入了一个序测度来度量状态双相似性与事件双相似性之间的差异.我们称这个序数为$\mathbb{S}$的“周序数”，$\mathfrak{Z}(\mathbb{S})$。当$\mathfrak{Z}(\mathbb{S})=0$，$\mathbb{S}$满足Hennessy-Milner性质。第二作者证明了具有$\mathfrak{Z}(\mathbb{S}) \geq 1$的LMP $\mathbb{S}$的存在性，并且Zhou证明了存在具有无限Zhou序数的LMP。本文证明了在具有任意大可数$\mathfrak{Z}(\mathbb{S})$的可分度量化空间上存在LMP $\mathbb{S}$，并且证明了存在具有不可数Zhou序数的LMP过程与$\mathit{ZFC}$的公理是一致的.
英文摘要：
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 $\mathcal{O}$, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the "Zhou ordinal" of $\mathbb{S}$, $\mathfrak{Z}(\mathbb{S})$. When $\mathfrak{Z}(\mathbb{S})=0$, $\mathbb{S}$ satisfies the Hennessy-Milner property. 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 an infinite 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.
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