Daniel Gebler

Generalized bisimulation metrics

The bisimilarity pseudometric based on the Kantorovich lifting is one of the most popular metrics for probabilistic processes proposed in the literature. However, its application in verification is limited to linear properties. We propose a generalization of this metric which allows to deal with a wider class of properties, such as those used in security and privacy. More precisely, we propose a family of metrics, parametrized on a notion of distance which depends on the property we want to verify. Furthermore, we show that the members of this family still characterize bisimilarity in terms of their kernel, and provide a bound on the corresponding metrics on traces. Finally, we study the case of a metric corresponding to differential privacy. We show that in this case it is possible to have a dual form, easier to compute, and we prove that the typical constructs of process algebra are non-expansive with respect to this metrics, thus paving the way to a modular approach to verification.

Tree rules in probabilistic transition system specifications with negative and quantitative premises

Probabilistic transition system specifications (PTSSs) in the nt f /nt x format provide structural operational semantics for Segala-type systems that exhibit both probabilistic and nondeterministic behavior and guarantee that bisimilarity is a congruence. Similar to the nondeterministic case of the rule format tyft/tyxt, we show that the well-foundedness requirement is unnecessary in the probabilistic setting. To achieve this, we first define a generalized version of the nt f /nt x format in which quantitative premises and conclusions include nested convex combinations of distributions. Also this format guarantees that bisimilarity is a congruence. Then, for a given (possibly non-well-founded) PTSS in the new format, we construct an equivalent well-founded PTSS consisting of only rules of the simpler (well-founded) probabilistic ntree format. Furthermore, we develop a proof-theoretic notion for these PTSSs that coincides with the existing stratification-based meaning in case the PTSS is stratifiable. This continues the line of research lifting structural operational semantic results from the nondeterministic setting to systems with both probabilistic and nondeterministic behavior.

Compositionality of Approximate Bisimulation for Probabilistic Systems

Probabilistic transition system specifications using the rule format ntmuft-ntmuxt provide structural operational semantics for Segala-type systems and guarantee that probabilistic bisimilarity is a congruence. Probabilistic bisimilarity is for many applications too sensitive to the exact probabilities of transitions. Approximate bisimulation provides a robust semantics that is stable with respect to implementation and measurement errors of probabilistic behavior. We provide a general method to quantify how much a process combinator expands the approximate bisimulation distance. As a direct application we derive an appropriate rule format that guarantees compositionality with respect to approximate bisimilarity. Moreover, we describe how specification formats for non-standard compositionality requirements may be derived.

Compositional 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.

Axiomatizing Bisimulation Equivalences and Metrics from Probabilistic SOS Rules

Probabilistic transition system specifications (PTSS) provide structural operational semantics for reactive probabilistic labeled transition systems. Bisimulation equivalences and bisimulation metrics are fundamental notions to describe behavioral relations and distances of states, respectively. We provide a method to generate from a PTSS a sound and ground-complete equational axiomatization for strong and convex bisimilarity. The construction is based on the method of Aceto, Bloom and Vaandrager developed for non-deterministic transition system specifications. The novelty in our approach is to employ many-sorted algebras to axiomatize separately non-deterministic choice, probabilistic choice and their interaction. Furthermore, we generalize this method to axiomatize the strong and convex metric bisimulation distance of PTSS.

Compositionality of probabilistic Hennessy-Milner logic through structural operational semantics

We present a method to decompose HML formulae for reactive probabilistic processes. This gives rise to a compositional modal proof system for the satisfaction relation of probabilistic process algebras. The satisfaction problem of a probabilistic HML formula for a process term is reduced to the question of whether its subterms satisfy a derived formula obtained via the operational semantics.

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.

Modal Decomposition on Nondeterministic Probabilistic Processes

We propose a SOS-based method for decomposing modal formulae for nondeterministic probabilistic processes. The purpose is to reduce the satisfaction problem of a formula for a process to verifying whether its subprocesses satisfy certain formulae obtained from its decomposition. By our decomposition, we obtain (pre)congruence formats for probabilistic bisimilarity, ready similarity and similarity.

Fixed-point Characterization of Compositionality Properties of Probabilistic Processes Combinators

Bisimulation metric is a robust behavioural semantics for probabilistic processes. Given any SOS specification of probabilistic processes, we provide a meth od to compute for each operator of the language its respective metric compositionality property. The compositionality property of an operator is defined as its modulus of continuity which gives the re lative increase of the distance between processes when they are combined by that operator. The compositionality property of an operator is computed by recursively counting how many times the combined processes are copied along their evolution. The compositionality properties allow to derive an upper bound on the distance between processes by purely inspecting the operators used to specify those processes.

Behavioural Pseudometrics for Nondeterministic Probabilistic Systems

For the model of probabilistic labelled transition systems that allow for the co-existence of nondeterminism and probabilities, we present two notions of bisimulation metrics: one is state-based and the other is distribution-based. We provide a sound and complete modal characterisation for each of them, using real-valued modal logics based on Hennessy-Milner logic. The logic for characterising the state-based metric is much simpler than an earlier logic by Desharnais et al. as it uses only two non-expansive operators rather than the general class of non-expansive operators. For the kernels of the two metrics, which correspond to two notions of bisimilarity, we give a comprehensive comparison with some typical distribution-based bisimilarities in the literature.