Bisimulation can be characterized by a very weak modal logic. Approximate abstractions of markov chains with interval. Distributionbased bisimulation for labelled markov processes. Approximate bisimulation for a class of stochastic hybrid systems. In this paper we propose a subdistributionbased bisimulation for labelled markov processes and compare it with. We consider notions of exact and approximate probabilistic bisimulation over lmps and, drawing on methods from both formal verification and control theory, propose a simple technique to compute an approximate probabilistic bisimulation of a given lmp, where the resulting abstraction is characterised as a finitestate labelled markov chain lmc. The rst contribution concerns the extension of the futs framework by introducing the notions of combined futs.
Bisimulation on markov processes over arbitrary measurable spaces. Lmps are probabilistic versions of labelled transition systems. Bisimulation distance between probabilistic processes composed by standard process is characterised in 21. Bisimulation, logics and metrics for labelled markov processes. Probabilistic model checking of labelled markov processes 41 and approximate notions have been consequently developed 15,19, which are based on relaxing the notion of process equivalence or on distance pseudometrics. Approximation techniques for lmps can be based on metrics 15,19,42 and coalgebras 43,44. An intrinsic characterization of approximate probabilistic. Following a series of previous work on approximate abstraction of dynamical systems 12, 11, 18, we extend the paradigm to handle a class of stochastic hybrid systems, namely, the jump linear stochastic systems jlss. This hope was based on the initial approximation schemes of desharnais et al. Labelled markov processes lmps were formulated bdep97, dep02 to study such general interacting markov processes.
A re exive relation r on x is a simulation if whenever xry and a 2act, then for all measurable a x with ra a, x. In this paper we propose a subdistributionbased bisimulation for labelled markov processes and compare it with earlier definitions of state and event bisimulation, which both only compare states. Our notion of bisimulation is proven to coincide with the coalgebraic definition of aczel and mendler in terms of the giry functor, which associates with a measurable space its collection of subprobability. In proceedings of the second international conference on the quantitative evaluation of systems qest 05 september 19 22, 2005. We assume that the state space is a certain type of common metric space called an.
We observe the interactions not the internal states. Pdf bisimulation for labelled markov processes josee. Bisimulation and metrics for labelled markov processes. Abstractin this paper we introduce a new class of labelled transition systemslabelled markov processes and define bisimulation for them. Pdf bisimulation for labelled markov processes abbas. The greatest bisimulation for x,a, uniquely exists and is called strong bisimilarity. We had developed a theory of bisimulation, proved a logical. We assume that the state space is a certain type of common metric space called an analytic space. Pdf a logical characterization of bisimulation for. Bisimulation for labelled markov processes sciencedirect. We assume that the state space is a certain type of common metric space called. Probabilistic bisimulation can be defined for continuous statespace systems. Almost sure bisimulation in labelled markov processes. Labelled markov processes what are labelled markov processes.
Pdf bisimulation as a logical relation semantic scholar. In contrast to those statebased bisimulations, our distribution bisimulation is weaker, but corresponds more closely to linear properties. In an lmp, instead of one transition probability function or markov kernel there are several, each associated with a distinct label1. Given a labelled markov process whose state space is an analytic space, bisimilarity arises as the kernel of the unique map to the nal labelled markov process. Labelled markov processes are probabilistic versions of labelled transition systems. Labelled markov processes are continuousstate fully proba bilistic labelled.
Abstractin this paper we introduce a new class of labelled transition systems labelled markov processes and define bisimulation for them. An approximation algorithm for labelled markov processes. Prakash panangaden labelled markov processes world of digitals. The initial study of labelled markov processes in continuous state spaces was motivated by the potential for important practical applications in performance analysis and veri. Bisimulation for labelled markov processes request pdf. Labelled markov processes by prakash panangaden overdrive. Approximating markov processes by averaging informatics. The book covers basic probability and measure theory on continuous state spaces and then develops the theory of lmps. The logical characterization of bisimulation can be done straightforwardly and generally. Weak bisimulations for labelled transition systems weighted. Bisimulation for markov decision processes through families. These metrics encode exact probabilistic bisimulation, in that the dis. In this paper we extend the work presented in 18 in two directions. We show that this problem is decidable in polynomial time if the second mdp is a markov chain.
Bisimulation and simulation for labelled markov processes. Approximate notions of probabilistic bisimilarity are formally characterised and computed for nitestate labelled markov processes in 17. Domain theory, testing and simulation for labelled markov. Bisimulation metrics for continuous markov decision processes. Probabilistic model checking of labelled markov processes 3 related work. In this paperwe present a universal lmp as the spectrum of a commutative c. All probabilistic data isinternal no probabilities associated with environment. Bisimulation for labelled markov processes conference paper pdf available in proceedings symposium on logic in computer science 179. The isomorphism between labelled markov processes and proc can be independently viewed as a fullabstraction result relating an operational labelled markov process and a denotational proc model and yields a logic complete for reasoning about simulation for continuousstate processes.
Probabilistic model checking of labelled markov processes via. Bisimulation for labeued markov processes ieee xplore. Summary labelled markov processes are probabilistic transition systems with continuous state spaces. Given two labelled markov decision processes mdps, the tracere. Logical characterizations of probabilistic bisimulation and simulation were given by desharnais et al. This is in some ways dual to the normal view of probabilistic transition systems.
Strong 2bisimulation and strong nondeterministic bisimulation coincide, since logical disjunction over the states in a given class cencodes the ability to reach cmaking an alabelled transition. Pdf a logical characterization of bisimulation for labeled markov processes josee desharnais academia. Bisimulation of labelled futs coalgebraically 3 over behaviours, as in languages for markov automata 53, besides calculi for deterministic discrete timed systems 3. Bisimulation and simulation for labelled markov processes prakash panangaden1 1school of computer science mcgill university irif lecture series jan 2018 panangaden mcgill university lecture 1 irif 16th january 2018 1 33. We show that our definition of probabilistic bisimulation generalizes the larsenskou definition given for discrete systems.
Approximating labelled markov processes sciencedirect. This yields a simple tracetree semantics for lmps that is fully abstract with. All probabilistic data is internal no probabilities associated with environment behaviour. The main topics covered are bisimulation, the logical characterization of bisimulation, metrics and approximation theory. Labelled markov processes are probabilistic labelled transition systems where the state space is not necessarily discrete. We prove that the relation of bisimilarity between countable labelled transition systems lts is. Request pdf distributionbased bisimulation for labelled markov processes in this paper we propose a subdistributionbased bisimulation for labelled markov processes and compare it with.
Labelled markov processes lmps are labelled transition systems in which each transition has an associated probability. The lmcs considered in this work are discretetime markov chains with decorated states, as in baier and katoen 2008, and are a subclass of labelled markov processes as in desharnais et al. Bisimulation for labelled markov processes citeseerx. Labelled markov processes as generalised stochastic relations. The approach taken is that each form of bisimulation corresponds to an algebraic structure derived from a transition system, and the general result is that a relation r between two transition systems on state spaces s and t is a bisimulation if and only if the derived algebraic. Labelled markov processes are probabilistic versions of labelled transition systems with continuous state spaces.
In this paper we introduce a new class of labelled transition systems labelled markov processes and define bisimulation for them. Formal modeling and analysis of timed systems, 170186. We investigate how various forms of bisimulation can be characterised using the technology of logical relations. That is to say, labelled markov processes are transition systems with action. In general, the state space of a labelled markov process may be a continuum. We transfer a notion of quantitative bisimilarity for labelled markov processes 1 to markov decision processes with continuous state spaces.
Labelled markov processes are probabilistic labelled transition sys tems where the state space is not necessarily discrete, it could be the reals, for example. The main result that we prove is that a notion of bisimulation for markov processes on pol ish spaces, which extends the larsenskou definition for dis crete. Bisimulation for labelled markov processes article in information and computation 1792. In this paper we introduce a new class of labelled transition systemslabelled markov processes and define bisimulation for them. Request pdf bisimulation for labelled markov processes in this paper we introduce a new class of labelled transition systems labelled markov processes and define bisimulation for them. We introduce a notion of bisimulation on labelled markov processes over generic measurable spaces in terms of arbitrary binary relations. Bisimulation and metrics for labelled markov processes prakash panangaden1 1school of computer science mcgill university th june 2012, university of aalborg panangaden mcgill bisimulation and metrics for labelled markov processes th june 2012 1 46. In general, the state space of a labelled markov process. Josee desharnais, laval university logical characterization of bisimulation for lmps.