137 research outputs found
Distribution-based bisimulation for labelled Markov processes
In this paper we propose a (sub)distribution-based bisimulation for labelled
Markov processes and compare it with earlier definitions of state and event
bisimulation, which both only compare states. In contrast to those state-based
bisimulations, our distribution bisimulation is weaker, but corresponds more
closely to linear properties. We construct a logic and a metric to describe our
distribution bisimulation and discuss linearity, continuity and compositional
properties.Comment: Accepted by FORMATS 201
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
Quantitative Approximation of the Probability Distribution of a Markov Process by Formal Abstractions
The goal of this work is to formally abstract a Markov process evolving in
discrete time over a general state space as a finite-state Markov chain, with
the objective of precisely approximating its state probability distribution in
time, which allows for its approximate, faster computation by that of the
Markov chain. The approach is based on formal abstractions and employs an
arbitrary finite partition of the state space of the Markov process, and the
computation of average transition probabilities between partition sets. The
abstraction technique is formal, in that it comes with guarantees on the
introduced approximation that depend on the diameters of the partitions: as
such, they can be tuned at will. Further in the case of Markov processes with
unbounded state spaces, a procedure for precisely truncating the state space
within a compact set is provided, together with an error bound that depends on
the asymptotic properties of the transition kernel of the original process. The
overall abstraction algorithm, which practically hinges on piecewise constant
approximations of the density functions of the Markov process, is extended to
higher-order function approximations: these can lead to improved error bounds
and associated lower computational requirements. The approach is practically
tested to compute probabilistic invariance of the Markov process under study,
and is compared to a known alternative approach from the literature.Comment: 29 pages, Journal of Logical Methods in Computer Scienc
Bisimulations Meet PCTL Equivalences for Probabilistic Automata
Probabilistic automata (PAs) have been successfully applied in formal
verification of concurrent and stochastic systems. Efficient model checking
algorithms have been studied, where the most often used logics for expressing
properties are based on probabilistic computation tree logic (PCTL) and its
extension PCTL^*. Various behavioral equivalences are proposed, as a powerful
tool for abstraction and compositional minimization for PAs. Unfortunately, the
equivalences are well-known to be sound, but not complete with respect to the
logical equivalences induced by PCTL or PCTL*. The desire of a both sound and
complete behavioral equivalence has been pointed out by Segala in 1995, but
remains open throughout the years. In this paper we introduce novel notions of
strong bisimulation relations, which characterize PCTL and PCTL* exactly. We
extend weak bisimulations that characterize PCTL and PCTL* without next
operator, respectively. Further, we also extend the framework to simulation
preorders. Thus, our paper bridges the gap between logical and behavioral
equivalences and preorders in this setting.Comment: Long version of CONCUR'11 with the same title: added extension to
simulations, countable state
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