Factoring stochastic relations

Abstract

When a system represented through a stochastic model is observed, the equivalence of behavior is described through the observation that equivalent inputs lead to equivalent outputs. This paper has a look at the systems that arise when the stochastic model is factored through the congruence. Congruences may refine each other, and we show that this refinement is reflected through factoring. We also show that factoring a factor does not give rise to any new constructions, since we are kept in the realm of factors for the original system. Thus we cannot have infinite long chains of factors, so that no new behavior can arise from the original system upon factoring (a system and its factors are bisimilar, after all).

The Hennessy–Milner equivalence for continuous time stochastic logic with mu-operator

“…Morphisms will be used for comparing interpretations; they will be based on morphisms for stochastic relations [9,13], cp. [6].…”

“…All these constructions may be undertaken in the uniform framework of a suitable category [9,13], making for a more concise formulation. We refrain from a systematic algebraic interpretation here, however, and use this notation only occasionally.…”

“…This condition is sometimes more practical than the one given in Definition 5.1, see e.g. [9]. Smooth relations are a helpful tool for the theory of Borel sets [16,25], for the theory of stochastic relations [9,13], and for the theory of labelled Markov transition systems [6,8].…”

Semi-pullbacks for stochastic relations over analytic spaces

“…It becomes easier to investigate the bisimilarity of stochastic relations: for example, it can be shown that two stochastic relations are bisimilar provided they have isomorphic factors; the converse holds under the assumption of compactness as well, so one might speculate whether bisimilarity and having isomorphic factors are equivalent, see (Doberkat 2004b) for a more comprehensive discussion. These investigations require us to study congruences for stochastic relations (Doberkat 2004a).…”

“…Analytic spaces have the nice property that they are closed under factorisation of smooth equivalence relations that occur quite naturally for stochastic relations. The existence of semi-pullbacks helps in investigating these congruences, see Doberkat (2004a) for some first results. Based on the result derived here, and on an Isomorphism Theorem for them, it can be shown that two stochastic relations are bisimilar provided they have isomorphic factors; the converse also holds under the assumption of compactness -see Doberkat (2004b).…”