A labelled Markov process (LMP) consists of a measurable space $S$ together with an indexed family of Markov kernels from $S$ to itself. This structure has been used to model probabilistic computations in Computer Science, and one of the main problems in the area is to define and decide whether two LMP $S$ and $S'$ "behave the same". There are two natural categorical definitions of sameness of behavior: $S$ and $S'$ are bisimilar if there exist an LMP $ T$ and measure preserving maps forming a diagram of the shape $ S\leftarrow T \rightarrow{S'}$; and they are behaviorally equivalent if there exist some $ U$ and maps forming a dual diagram $ S\rightarrow U \leftarrow{S'}$. These two notions differ for general measurable spaces but Doberkat (extending a result by Edalat) proved that they coincide for analytic Borel spaces, showing that from every diagram $S\rightarrow U \leftarrow{S'}$ one can obtain a bisimilarity diagram as above. Moreover, the resulting square of measure preserving maps is commutative (a semipullback). In this paper, we extend the previous result to measurable spaces $S$ isomorphic to a universally measurable subset of a Polish space with the trace of the Borel $\sigma$-algebra, using a version of Strassen's theorem on common extensions of finitely additive measures.
翻译:贴有标签的Markov 进程( LMP) 由 一个可以测量的空间 $S$ 和 一个由美元到美元组成的Markov 内核的索引式组合组成。 这个结构已被用于模拟计算机科学中的概率计算。 这个结构已被用于模拟计算机科学中的概率计算, 而这个区域的主要问题之一是定义和决定两个LMP$S美元和$S$是否“与美元相同 ” 。 两种行为相同的自然绝对定义是: $S$和$S$是两个相近的。 如果存在一个LMP$和$S$, 以及一个保存地图的地图, 构成一个形状的 $Sleftrow T\rightrow{S'} 的图表; 如果存在一些美元, 并且它们可以模拟计算机科学的概率计算。 这两个概念对于一般的可测量空间是不同的, 但Doberkat( 由Edalat得出的结果) 证明它们与具有解析的博尔空间空间是相匹配的, 显示从每张图表 $Rftrowrowrestal $ 。 美元中, 一个可以获取一个在普通的平面的平方平面图中, 。