论文标题
可分离空间上标记的马尔可夫过程的周的序数
The Zhou Ordinal of Labelled Markov Processes over Separable Spaces
论文作者
论文摘要
标有马尔可夫过程（LMP）的状态之间存在两个概念：状态双相和事件双相似性。第一个可以被视为对Larsen和Skou概率双相似性连续空间的适当概括，而第二个则具有自然逻辑的特征。 C. Zhou表示状态双性恋是运算符$ \ Mathcal {O} $的最大固定点，因此引入了对IT和事件双相似性之间差异的有序度量。我们称此序列为$ \ Mathbb {s} $，$ \ Mathfrak {z}（\ Mathbb {s}）$的“周”。当$ \ mathfrak {z}（\ mathbb {s}）= 0 $，$ \ mathbb {s} $满足Hennessy-Milner属性。第二作者证明了LMP $ \ Mathbb {s} $的存在，其中$ \ Mathfrak {Z}（\ Mathbb {s}）\ geq 1 $，Zhou表明有LMP具有Infinite Zhou Ordinal的LMP。在本文中，我们表明，在可分离的Metrizable空间上有LMP $ \ MATHBB {S} $，具有任意的大型可计数$ \ Mathfrak {z}（\ Mathbb {s}）$，并且它与$ \ Mathit {Zfc} $的axioms是一致的。
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.
