Approximating a Behavioural Pseudometric without Discount for<br> Probabilistic Systems

Desharnais, Gupta, Jagadeesan and Panangaden introduced a family of behavioural pseudometrics for probabilistic transition systems. These pseudometrics are a quantitative analogue of probabilistic bisimilarity. Distance zero captures probabilistic bisimilarity. Each pseudometric has a discount factor, a real number in the interval (0, 1]. The smaller the discount factor, the more the future is discounted. If the discount factor is one, then the future is not discounted at all. Desharnais et al. showed that the behavioural distances can be calculated up to any desired degree of accuracy if the discount factor is smaller than one. In this paper, we show that the distances can also be approximated if the future is not discounted. A key ingredient of our algorithm is Tarski's decision procedure for the first order theory over real closed fields. By exploiting the Kantorovich-Rubinstein duality theorem we can restrict to the existential fragment for which more efficient decision procedures exist

Computing Probabilistic Bisimilarity Distances for Probabilistic Automata

The probabilistic bisimilarity distance of Deng et al. has been proposed as a robust quantitative generalization of Segala and Lynch's probabilistic bisimilarity for probabilistic automata. In this paper, we present a characterization of the bisimilarity distance as the solution of a simple stochastic game. The characterization gives us an algorithm to compute the distances by applying Condon's simple policy iteration on these games. The correctness of Condon's approach, however, relies on the assumption that the games are stopping. Our games may be non-stopping in general, yet we are able to prove termination for this extended class of games. Already other algorithms have been proposed in the literature to compute these distances, with complexity in $\textbf{UP} \cap \textbf{coUP}$ and \textbf{PPAD}. Despite the theoretical relevance, these algorithms are inefficient in practice. To the best of our knowledge, our algorithm is the first practical solution. The characterization of the probabilistic bisimilarity distance mentioned above crucially uses a dual presentation of the Hausdorff distance due to M\'emoli. As an additional contribution, in this paper we show that M\'emoli's result can be used also to prove that the bisimilarity distance bounds the difference in the maximal (or minimal) probability of two states to satisfying arbitrary $\omega$-regular properties, expressed, eg., as LTL formulas

Compositional bisimulation metric reasoning with Probabilistic Process Calculi

We study which standard operators of probabilistic process calculi allow for compositional reasoning with respect to bisimulation metric semantics. We argue that uniform continuity (generalizing the earlier proposed property of non-expansiveness) captures the essential nature of compositional reasoning and allows now also to reason compositionally about recursive processes. We characterize the distance between probabilistic processes composed by standard process algebra operators. Combining these results, we demonstrate how compositional reasoning about systems specified by continuous process algebra operators allows for metric assume-guarantee like performance validation

A behavioural pseudometric for probabilistic transition systems

AbstractDiscrete notions of behavioural equivalence sit uneasily with semantic models featuring quantitative data, like probabilistic transition systems. In this paper, we present a pseudometric on a class of probabilistic transition systems yielding a quantitative notion of behavioural equivalence. The pseudometric is defined via the terminal coalgebra of a functor based on a metric on the space of Borel probability measures on a metric space. States of a probabilistic transition system have distance 0 if and only if they are probabilistic bisimilar. We also characterize our distance function in terms of a real-valued modal logic

Behavioral Metrics via Functor Lifting

We study behavioral metrics in an abstract coalgebraic setting. Given a coalgebra alpha : X -> FX in Set, where the functor F specifies the branching type, we define a framework for deriving pseudometrics on X which measure the behavioral distance of states. A first crucial step is the lifting of the functor F on Set to a functor /F in the category PMet of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ. Then a final coalgebra for F in Set can be endowed with a behavioral distance resulting as the smallest solution of a fixed-point equation, yielding the final /F-coalgebra in PMet. The same technique, applied to an arbitrary coalgebra alpha : X -> FX in Set, provides the behavioral distance on X. Under some constraints we can prove that two states are at distance 0 if and only if they are behaviorally equivalent