39 research outputs found
congruences for stochastic relations
We discuss congruences for stochastic relations, stressing the equivalence of smooth equivalence relations and countably generated Ăł-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
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
Probabilistic Bisimulation: Naturally on Distributions
In contrast to the usual understanding of probabilistic systems as stochastic
processes, recently these systems have also been regarded as transformers of
probabilities. In this paper, we give a natural definition of strong
bisimulation for probabilistic systems corresponding to this view that treats
probability distributions as first-class citizens. Our definition applies in
the same way to discrete systems as well as to systems with uncountable state
and action spaces. Several examples demonstrate that our definition refines the
understanding of behavioural equivalences of probabilistic systems. In
particular, it solves a long-standing open problem concerning the
representation of memoryless continuous time by memory-full continuous time.
Finally, we give algorithms for computing this bisimulation not only for finite
but also for classes of uncountably infinite systems
Generalized labelled Markov processes, coalgebraically
Coalgebras of measurable spaces are of interest in probability theory as a formalization of Labelled Markov Processes (LMPs). We discuss some general facts related to the notions of bisimulation and cocongruence on these systems, providing a faithful characterization of bisimulation on LMPs on generic measurable
spaces. This has been used to prove that bisimilarity on single LMPs is an equivalence, without assuming the state space to be analytic. As the second main contribution, we introduce the first specification rule format to define well-behaved composition operators for LMPs. This allows one to define process description languages on LMPs which are always guaranteed to have a fully-abstract semantics
An ordered framework for partial multivalued functors
The category Rel of sets and relations intimately ties the notions of
function, partial multivalued function, and direct image under a function
through the description of Rel as the Kleisli category of the covariant power
set functor on Set. We present a suitable framework to obtain a similar
relationship between the concepts of functor, partial multivalued functor, and
the direct image under a functor.Comment: Accepted for presentation at the Asia-Pacific World Congress on
Computer Science and Engineering 2015, Fij
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