Metrics for Labeled Markov Systems

Abstract

Partial Labeled Markov Chains are simultaneously generalizations of process algebra and of traditional Markov chains. They provide a foundation for interacting discrete probabilistic systems, the interaction being synchronization on labels as in process algebra. Existing notions of process equivalence are too sensitive to the exact probabilities of various transitions. This paper addresses contextual reasoning principles for reasoning about more robust notions of "approximate" equivalence between concurrent interacting probabilistic systems.• We develop a family of metrics between partial labeled Markov chains to formalize the notion of distance between processes.• We show that processes at distance zero are bisimilar.• We describe a decision procedure to compute the distance between two processes.• We show that reasoning about approximate equivalence can be done compositionally by showing that process combinators do not increase distance.• We introduce an asymptotic metric to capture asymptotic properties of Markov chains; and show that parallel composition does not increase asymptotic distance.

Approximating Markov Processes by Averaging

“…However, equality, being a true/false statement, is very brittle and unable to tell when two processes are, in some sense, close. Thus, pseudometrics were developed in order to capture such a notion [DGJP99,DGJP04,vBW01b,vBW01a]. One important property of these pseudometrics is that the kernel precisely corresponds to bisimulation.…”

Quantifying Timing Leaks and Cost Optimisation

“…However, this notion turns out to be still too strict and a number of researchers developed "approximate" versions; among them we just name the approaches by Desharnais et.al. [17,18] and van Breugel [19] and our work [10,20] (an extensive bibliography on this issue can be found in [21]). We based this current paper on the latter approach because it allows for an implementation of the semantics of pWhile via linear operators, i.e.…”

Approximate reasoning for real-time probabilistic processes

“…In an early treatment of metrics for LMPs [DGJP99,DGJP04] the metric was defined through a class of functions closely related to a modal logic. The idea was that, in a probabilistic setting, random variables play a role analogous to modal formulas.…”

“…As has been argued before [JS90,DGJP99,DGJP04], this style of reasoning is fragile in the sense of being too dependent on the exact numerical values of times and probabilities. Previously this had pointed out for probability, but the same remarks apply, mutis mutandis, to real time as well.…”