Josée Desharnais

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. Labelled Markov processes are probabilistic labelled transition systems where the state space is not necessarily discrete, it could be the reals, for example. We assume that it is a Polish space (the underlying topological space for a complete separable metric space). The mathematical theory of such systems is completely new from the point of view of the extant literature on probabilistic process algebra; of course, it uses classical ideas from measure theory and Markov process theory. The notion of bisimulation builds on the ideas of Larsen and Skou and of Joyal, Nielsen and Winskel. The main result that we prove is that a notion of bisimulation for Markov processes on Polish spaces, which extends the Larsen-Skou definition for discrete systems, is indeed an equivalence relation. This turns our to be a rather hard mathematical result which, as far as we know, embodies a new result in pure probability theory. This work heavily uses continuous mathematics which is becoming an important part of work on hybrid systems.

Metrics for labelled Markov processes

The notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions. Thus, a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent. This instability is due to the quantitative nature of probabilistic processes. In a situation where the process behavior has a quantitative aspect there should be a more robust approach to process equivalence. This paper studies a metric between labelled Markov processes. This metric has the property that processes are at zero distance if and only if they are bisimilar. The metric is inspired by earlier work on logics for characterizing bisimulation and is related, in spirit, to the Hutchinson metric.

The metric analogue of weak bisimulation for probabilistic processes

We observe that equivalence is not a robust concept in the presence of numerical information - such as probabilities-in the model. We develop a metric analogue of weak bisimulation in the spirit of our earlier work on metric analogues for strong bisimulation. We give a fixed point characterization of the metric. This makes available conductive reasoning principles and allows us to prove metric analogues of the usual algebraic laws for process combinators. We also show that quantitative properties of interest are continuous with respect to the metric, which says that if two processes are close in the metric then observable quantitative properties of interest are indeed close. As an important example of this we show that nearby processes have nearby channel capacities - a quantitative measure of their propensity to leak information.

A logical characterization of bisimulation for labeled Markov processes

This paper gives a logical characterization of probabilistic bisimulation for Markov processes. Bisimulation can be characterized by a very weak modal logic. The most striking feature is that one has no negation or any kind of negative proposition. Bisimulation can be characterized by several inequivalent logics; we report five in this paper and there are surely many more. We do not need any finite branching assumption yet there is no need of infinitely conjunction. We give an algorithm for deciding bisimilarity of finite state systems which constructs a formula that witnesses the failure of bisimulation.

Approximating labelled Markov processes

Labelled Markov processes are probabilistic versions of labelled transition systems. In general, the state space of a labelled Markov process may be a continuum. In this paper, we study approximation techniques for continuousstate labelled Markov processes.We show that the collection of labelled Markov processes carries a Polish-space structure with a countable basis given by finite-state Markov chains with rational probabilities: thus permitting the approximation of quantitative observations (e.g., an integral of a continuous function) of a continuous-state labelled Markov process by the observations on finite-state Markov chains. The primary technical tools that we develop to reach these results are • A variant of a finite-model theorem for the modal logic used to characterize bisimulation, and • an isomorphism between the poset of Markov processes (ordered by simulation) with the ω-continuous dcpo Proc (defined as the solution of the recursive domain equation Proc = ΠL PPr(Proc)). The isomorphism between labelled Markov processes and Proc can be independently viewed as a full-abstraction result relating an operational (labelled Markov process) and a denotational (Proc) model and yields a logic complete for reasoning about simulation for continuous-state processes.

Continuous stochastic logic characterizes bisimulation of continuous-time Markov processes

In a recent paper Baier et al. [Lecture Notes in Computer Science, Springer-Verlag, 2000, p. 358] analyzed a new way of model-checking formulas of a logic for continuous-time processes—called continuous stochastic logic (henceforth CSL)—against continuous-time Markov chains—henceforth CTMCs. One of the important results of that paper was the proof that if two CTMCs were bisimilar then they would satisfy exactly the same formulas of CSL. This raises the converse question—does satisfaction of the same collection of CSL formulas imply bisimilarity? In other words, given two CTMCs which are known to satisfy exactly the same formulas of CSL does it have to be the case that they are bisimilar? We prove that the answer to the question just raised is “yes”. In fact we prove a significant extension, namely that a subset of CSL suffices even for systems where the state space may be a continuum. Along the way we prove a result to the effect that the set of Zeno paths has measure zero provided that the transition rates are bounded.

Approximating labeled Markov processes

We study approximate reasoning about continuous-state labeled Markov processes. We show how to approximate a labeled Markov process by a family of finite-state labeled Markov chains. We show that the collection of labeled Markov processes carries a Polish space structure with a countable basis given by finite state Markov chains with rational probabilities. The primary technical tools that we develop to reach these results are: a finite-model theorem for the modal logic used to characterize bisimulation; and a categorical equivalence between the category of Markov processes (with simulation morphisms) with the /spl omega/-continuous dcpo Proc, defined as the solution of the recursive domain equation Proc=/spl Pi//sub Labels/ P/sub Prob/(Proc). The correspondence between labeled Markov processes and Proc yields a logic complete for reasoning about simulation for continuous-state processes.

Weak bisimulation is sound and complete for pCTL

Abstract We investigate weak bisimulation of probabilistic systems in the presence of nondeterminism, i.e. labelled concurrent Markov chains (LCMC) with silent transitions. We develop an approach based on allowing convex combinations of computations, similar to Segala and Lynch’s use of randomized schedulers. The definition of weak bisimulation destroys the additivity property of the probability distributions, yielding instead capacities. The mathematics behind capacities naturally captures the intuition that when we deal with nondeterminism we must work with bounds on the possible probabilities rather than with their exact values. Our analysis leads to three new developments: • We identify a characterization of “image finiteness” for countable-state systems and present a new definition of weak bisimulation for these LCMCs. We prove that our definition coincides with that of Philippou, Lee and Sokolsky for finite state systems. • We show that bisimilar states have matching computations. The notion of matching involves convex combinations of transitions. • We study a minor variant of the probabilistic logic pCTL ∗ – the variation arises from an extra path formula to address action labels. We show that bisimulation is sound and complete for this variant of pCTL ∗ . This is an extended complete version of a paper that was presented at CONCUR 2002.

Approximate Analysis of Probabilistic Processes: Logic, Simulation and Games

We tackle the problem of non robustness of simulation and bisimulation when dealing with probabilistic processes. It is important to ignore tiny deviations in probabilities because these often come from experiments or estimations. A few approaches have been proposed to treat this issue, for example metrics to quantify the non bisimilarity (or closeness) of processes. Relaxing the definition of simulation and bisimulation is another avenue which we follow. We define a new semantics to a known simple logic for probabilistic processes and show that it characterises a notion of epsi-simulation. We also define two-players games that correspond to these notions: the existence of a winning strategy for one of the players determines epsi-(bi)simulation. Of course, for all the notions defined, letting epsi = 0 gives back the usual notions of logical equivalence, simulation and bisimulation. However, in contrast to what happens in fully probabilistic systems when epsi = 0, two-way e-simulation for epsi > 0 is not equal to epsi-bisimulation. Next we give a polynomial time algorithm to compute a naturally derived metric: distance between states s and t is defined as the smallest epsi such that s and t are epsi-equivalent. This is the first polynomial algorithm for a non-discounted metric. Finally we show that most of these notions can be extended to deal with probabilistic systems that allow non-determinism as well.

Computing Distances between Probabilistic Automata

We present relaxed notions of simulation and bisimulation on Probabilistic Automata (PA), that allow some error epsilon. When epsilon is zero we retrieve the usual notions of bisimulation and simulation on PAs. We give logical characterisations of these notions by choosing suitable logics which differ from the elementary ones, L with negation and L without negation, by the modal operator. Using flow networks, we show how to compute the relations in PTIME. This allows the definition of an efficiently computable non-discounted distance between the states of a PA. A natural modification of this distance is introduced, to obtain a discounted distance, which weakens the influence of long term transitions. We compare our notions of distance to others previously defined and illustrate our approach on various examples. We also show that our distance is not expansive with respect to process algebra operators. Although L without negation is a suitable logic to characterise epsilon-(bi)simulation on deterministic PAs, it is not for general PAs; interestingly, we prove that it does characterise weaker notions, called a priori epsilon-(bi)simulation, which we prove to be NP-difficult to decide.