congruences for stochastic relations

We discuss congruences for stochastic relations, stressing the equivalence of smooth equivalence relations and countably generated ó-algebras. Factor spaces are constructed for congruences and for morphisms. Semi-pullbacks are needed when investigating the interplay between congruences and bisimulations, and it is shown that semi-pullbacks exist for stochastic relations over analytic spaces, generalizing a previous result and answering an open question. Equivalent congruences are investigated, and it is shown that stochastic relations that have equivalent congruences are bisimilar. The well-known equivalence relation coming from a Hennessy-Milner logic for labelled Markov transition systems is shown to be a special case in this development

Stochastic Relations Interpreting Modal Logic

We propose an interpretation of modal logic through stochastic relations, providing a probabilistic complement to the usual nondeterministic interpretations using Kripke models. A simple temporal logic and a logic with a countable number of diamonds illustrate the approach. The main technical result of this paper is a probabilistic analogon to the well-known Hennessy-Milner Theorem characterizing models that have the same theories for their states and bisimilarity as equivalent properties. This requires the study of congruences for stochastic relations that underly the interpretation, for which a general bisimilarity result is also established. The results depend on the existence of semi-pullbacks for stochastic relations over analytic spaces

Unprovability of the Logical Characterization of Bisimulation

We quickly review labelled Markov processes (LMP) and provide a counterexample showing that in general measurable spaces, event bisimilarity and state bisimilarity differ in LMP. This shows that the logic in Desharnais [*] does not characterize state bisimulation in non-analytic measurable spaces. Furthermore we show that, under current foundations of Mathematics, such logical characterization is unprovable for spaces that are projections of a coanalytic set. Underlying this construction there is a proof that stationary Markov processes over general measurable spaces do not have semi-pullbacks. ([*] J. Desharnais, Labelled Markov Processes. School of Computer Science. McGill University, Montr\'eal (1999))Comment: Extended introduction and comments; extra section on semi-pullbacks; 11 pages Some background details added; extra example on the non-locality of state bisimilarity; 14 page

Abstractions of Stochastic Hybrid Systems

In this paper we define a stochastic bisimulation concept for a very general class of stochastic hybrid systems, which subsumes most classes of stochastic hybrid systems. The definition of this bisimulation builds on the concept of zigzag morphism defined for strong Markov processes. The main result is that this stochastic bisimulation is indeed an equivalence relation. The secondary result is that this bisimulation relation for the stochastic hybrid system models used in this paper implies the same kind of bisimulation for their continuous parts and respectively for their jumping structures

Factoring Stochastic Relations

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 re.ne each other, and we show that this re.nement is re.ected 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 in.nite 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)

Characterizing the Eilenberg-Moore Algebras for a Monad of Stochastic Relations

We investigate the category of Eilenberg-Moore algebras for the Giry monad associated with stochastic relations over Polish spaces with continuous maps as morphisms. The algebras are characterized through convex partitions of the space of all probability measures. Examples are investigated, and it is shown that .nite spaces usually do not have algebras at all

Bisimilarity is not Borel

We prove that the relation of bisimilarity between countable labelled transition systems is $\Sigma_1^1$-complete (hence not Borel), by reducing the set of non-wellorders over the natural numbers continuously to it. This has an impact on the theory of probabilistic and nondeterministic processes over uncountable spaces, since logical characterizations of bisimilarity (as, for instance, those based on the unique structure theorem for analytic spaces) require a countable logic whose formulas have measurable semantics. Our reduction shows that such a logic does not exist in the case of image-infinite processes.Comment: 20 pages, 1 figure; proof of Sigma_1^1 completeness added with extended comments. I acknowledge careful reading by the referees. Major changes in Introduction, Conclusion, and motivation for NLMP. Proof for Lemma 22 added, simpler proofs for Lemma 17 and Theorem 30. Added references. Part of this work was presented at Dagstuhl Seminar 12411 on Coalgebraic Logic