# Semipullbacks of labelled Markov processes

## TL;DR

This paper extends the concept of semipullbacks for labelled Markov processes to a broader class of measurable spaces, enabling better understanding of behavioral equivalence in probabilistic models.

## Contribution

It generalizes existing results by proving the existence of semipullbacks for a wider class of measurable spaces using Strassen's theorem.

## Key findings

- Semipullbacks exist for spaces isomorphic to universally measurable subsets of Polish spaces.
- Behavioral equivalence can be characterized via bisimilarity in these spaces.
- The results unify categorical and measure-theoretic approaches to probabilistic process equivalence.

## Abstract

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.