On the Discriminating Power of Testing Equivalences for Reactive Probabilistic Systems: Results and Open Problems

International audienceTesting equivalences have been deeply investigated on fully nondeterministic processes, as well as on processes featuring probabilities and internal nondeterminism. This is not the case with reactive probabilistic processes, for which it is only known that the discriminating power of probabilistic bisimilarity is achieved when admitting a copying capability within tests. In this paper, we introduce for reactive probabilistic processes three testing equivalences without copying, which are respectively based on reactive probabilistic tests, fully nondeterministic tests, and nondeterministic and probabilistic tests. We show that the three testing equivalences are strictly finer than probabilistic failure-trace equivalence, and that the one based on nondeterministic and probabilistic tests is strictly finer than the other two, which are incomparable with each other. Moreover, we provide a number of facts that lead us to conjecture that (i) may testing and must testing coincide on reactive probabilistic processes and (ii) nondeterministic and probabilistic tests reach the same discriminating power as probabilistic bisimilarity

The Theory of Traces for Systems with Nondeterminism, Probability, and Termination

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories.Comment: This paper is an extended version of a LICS 2019 paper "The Theory of Traces for Systems with Nondeterminism and Probability". It contains all the proofs, additional explanations, material, and example

The Theory of Traces for Systems with Nondeterminism and Probability

International audienceThis paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories

Revisiting Trace and Testing Equivalences for Nondeterministic and Probabilistic Processes

One of the most studied extensions of testing theory to nondeterministic and probabilistic processes yields unrealistic probabilities estimations that give rise to two anomalies. First, probabilistic testing equivalence does not imply probabilistic trace equivalence. Second, probabilistic testing equivalence differentiates processes that perform the same sequence of actions with the same probability but make internal choices in different moments and thus, when applied to processes without probabilities, does not coincide with classical testing equivalence. In this paper, new versions of probabilistic trace and testing equivalences are presented for nondeterministic and probabilistic processes that resolve the two anomalies. Instead of focussing only on suprema and infima of the set of success probabilities of resolutions of interaction systems, our testing equivalence matches all the resolutions on the basis of the success probabilities of their identically labeled computations. A simple spectrum is provided to relate the new relations with existing ones. It is also shown that, with our approach, the standard probabilistic testing equivalences for generative and reactive probabilistic processes can be retrieved

Revisiting Trace and Testing Equivalences for Nondeterministic and Probabilistic Processes

Two of the most studied extensions of trace and testing equivalences to nondeterministic and probabilistic processes induce distinctions that have been questioned and lack properties that are desirable. Probabilistic trace-distribution equivalence differentiates systems that can perform the same set of traces with the same probabilities, and is not a congruence for parallel composition. Probabilistic testing equivalence, which relies only on extremal success probabilities, is backward compatible with testing equivalences for restricted classes of processes, such as fully nondeterministic processes or generative/reactive probabilistic processes, only if specific sets of tests are admitted. In this paper, new versions of probabilistic trace and testing equivalences are presented for the general class of nondeterministic and probabilistic processes. The new trace equivalence is coarser because it compares execution probabilities of single traces instead of entire trace distributions, and turns out to be compositional. The new testing equivalence requires matching all resolutions of nondeterminism on the basis of their success probabilities, rather than comparing only extremal success probabilities, and considers success probabilities in a trace-by-trace fashion, rather than cumulatively on entire resolutions. It is fully backward compatible with testing equivalences for restricted classes of processes; as a consequence, the trace-by-trace approach uniformly captures the standard probabilistic testing equivalences for generative and reactive probabilistic processes. The paper discusses in full details the new equivalences and provides a simple spectrum that relates them with existing ones in the setting of nondeterministic and probabilistic processes

Probabilistic Semantics: Metric and Logical Character¨ations for Nondeterministic Probabilistic Processes

In this thesis we focus on processes with nondeterminism and probability in the PTS model, and we propose novel techniques to study their semantics, in terms of both classic behavioral relations and the more recent behavioral metrics. Firstly, we propose a method for decomposing modal formulae in a probabilistic extension of the Hennessy-Milner logic. This decomposition method allows us to derive the compositional properties of probabilistic (bi)simulations. Then, we propose original notions of metrics measuring the disparities in the behavior of processes with respect to (decorated) trace and testing semantics. To capture the differences in the expressive power of the metrics we order them by the relation `makes processes further than'. Thus, we obtain the first spectrum of behavioral metrics on the PTS model. From this spectrum we derive an analogous one for the kernels of the metrics, ordered by the relation `makes strictly less identification than'. Finally, we introduce a novel technique for the logical characterization of both behavioral metrics and their kernels, based on the notions of mimicking formula and distance on formulae. This kind of characterization allows us to obtain the first example of a spectrum of distances on processes obtained directly from logics. Moreover, we show that the kernels of the metrics can be characterized by simply comparing the mimicking formulae of processes