Probabilistic Trace and Testing Semantics: The Importance of Being Coherent

For nondeterministic and probabilistic processes, the validity of some desirable properties of probabilistic trace semantics depends both on the class of schedulers used to resolve nondeterminism and on the capability of suitably limiting the power of the considered schedulers. Inclusion of probabilistic bisimilarity, compositionality with respect to typical process operators, and backward compatibility with trace semantics over fully nondeterministic or fully probabilistic processes, can all be achieved by restricting to coherent resolutions of nondeterminism. Here we provide alternative characterizations of probabilistic trace post-equivalence and pre-equivalence in the case of coherent resolutions. The characterization of the former is based on fully coherent trace distributions, whereas the characterization of the latter relies on coherent weighted trace sets.

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.

We provide two interpretations, over nondeterministic and probabilistic processes, of PML, the probabilistic version of Hennessy-Milner logic used by Larsen and Skou to characterize bisimilarity of probabilistic processes without internal nondeterminism. We also exhibit two new bisimulation-based equivalences, which are in full agreement with the two different interpretations of PML. The new equivalences are coarser than the bisimilarity for nondeterministic and probabilistic processes proposed by Segala and Lynch, which instead is in agreement with a version of Hennessy-Milner logic extended with an additional probabilistic operator interpreted over state distributions rather than over individual states. The modal logic characterizations provided for the new equivalences thus offer a uniform framework for reasoning on purely nondeterministic processes, reactive probabilistic processes, and 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.KeywordsProbabilistic FailureSuccess ProbabilityTest EquivalenceProbabilistic ChoiceLabel Transition SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Action hiding and probabilistic choice have independently established their roles in process algebraic modeling and verification of concurrent systems. While action hiding allows abstraction from unimportant details and model reduction, and the induced nondeterminism enables modeling uncertainty in the system behaviour, probabilistic choice allows quantification of the nondeterminism. However, as not all of the nondeterministic behaviour has a random nature, we are faced with the challenge to combine the above two aspects of concurrent systems, such that one can take maximal advantage of both. This thesis addresses two problems regarding concurrent processes that exhibit both hidden and probabilistic behaviour, or probabilistic processes for short. Namely, a proper reduction of a model, by elimination of the hidden actions, requires a semantical equivalence that preserves the process properties of interest and is a congruence for the process operators. For non-probabilistic processes it has been shown that such an equivalence is branching bisimilarity. However, in the presence of probabilistic choice, more concretely in the alternating model of probabilistic processes, the intuitive notion of branching bisimulation is not a congruence for parallel composition. In this thesis a new branching bisimulation for this model is defined, and it is shown that this is the coarsest congruence for parallel composition that is included in the former. To achieve the congruence result, a hidden action preceding directly a non-trivial probabilistic choice cannot be eliminated. The new branching bisimulation preserves the properties expressible in the probabilistic computation tree logic, and is decidable in polynomial time. Similar to the non-probabilistic case, a single axiom characterizes branching bisimilarity for finite probabilistic processes. The previous results imply that branching bisimilarity, although potentially useful for model reduction, may be in fact too strong to serve as an equivalence relation for probabilistic processes. Another view, taken in the may/must testing theory (as well as in the process calculus CSP), is to distinguish two processes only if they can be distinguished when interacting with their environment, i.e. with another process. However, although processes that differ only in the moment an internal (nondeterministic) choice is made are not distinguished by this theory, for probabilistic processes this is no longer valid. The problem stems from an earlier observation that the schedulers that resolve the nondeterminism in concurrent probabilistic processes are too powerful and yield unrealistic overestimations of the probabilities with which a process can pass a test. The power of the schedulers comes from the fact that they allow the same choice to be resolved in different manners in different futures. In order to restrict the schedulers and thus to obtain the right probabilities, this thesis proposes integrating the information, based on which a nondeterministic choice is resolved, in labels on the nondeterministic transitions. In this way, choices using the same information are resolved in the same way, regardless of the considered future. As a result, the new testing preorder relation can be characterized by a probabilistic ready-trace preorder, a relation that is insensitive to the moment an internal choice is made, yet sensitive to deadlock and to action priorities. In other words, it combines useful features of both the bisimulation-style and the trace-style relations. The parallel composition is also generalized here to include both interleaving and action hiding after synchronization, and it is shown that probabilistic ready-trace preorder is a precongruence with respect to it. Finally, the CSP-style axiomatic characterization shows that all the distributivity laws for nondeterministic choice from CSP are preserved and no new laws are added.

We present a spectrum of trace-based, testing, and bisimulation equivalences for nondeterministic and probabilistic processes whose activities are all observable. For every equivalence under study, we examine the discriminating power of three variants stemming from three approaches that differ for the way probabilities of events are compared when nondeterministic choices are resolved via deterministic schedulers. We show that the first approach -- which compares two resolutions relatively to the probability distributions of all considered events -- results in a fragment of the spectrum compatible with the spectrum of behavioral equivalences for fully probabilistic processes. In contrast, the second approach -- which compares the probabilities of the events of a resolution with the probabilities of the same events in possibly different resolutions -- gives rise to another fragment composed of coarser equivalences that exhibits several analogies with the spectrum of behavioral equivalences for fully nondeterministic processes. Finally, the third approach -- which only compares the extremal probabilities of each event stemming from the different resolutions -- yields even coarser equivalences that, however, give rise to a hierarchy similar to that stemming from the second approach.

Algebraic theory of probabilistic and nondeterministic processes

ULTraS at work: Compositionality metaresults for bisimulation and trace semantics

Testing Probabilistic and Nondeterministic Processes

The logic PML is a probabilistic version of Hennessy---Milner logic introduced by Larsen and Skou to characterize bisimilarity over probabilistic processes without internal nondeterminism. In this paper, two alternative interpretations of PML over nondeterministic and probabilistic processes as models are considered, and two new bisimulation-based equivalences that are in full agreement with those interpretations are provided. The new equivalences include as coarsest congruences the two bisimilarities for nondeterministic and probabilistic processes proposed by Segala and Lynch. The latter equivalences are instead known to agree with two versions of Hennessy---Milner logic extended with an additional probabilistic operator interpreted over state distributions in place of individual states. The new interpretations of PML and the corresponding new bisimilarities are thus the first ones to offer a uniform framework for reasoning on processes that are purely nondeterministic or reactive probabilistic or that mix nondeterminism and probability in an alternating/nonalternating way.

A uniform framework for modeling nondeterministic, probabilistic, stochastic, or mixed processes and their behavioral equivalences

Some probabilistic models of simple choice and ranking

Transition systems are a basic semantic model for formal description, specification, and analysis of concurrent and distributed systems. In order to describe and analyze aspects of reliability, such as the likelihood of trace and failure, this model has been extended in various ways to handle probabilistic behavior. To use these models for specification and stepwise development of systems, it is important to develop appropriate refinement preorders. In the paper, we develop refinement preorders based on a framework of testing for a model that represents both nondeterministic and probabilistic choices as independent concepts [YL92]. Our main contribution is a notion of reward testing, and a denotational characterization of a testing preorder, which corresponds to a natural probabilistic extension of the trace model [Hoa85].

A process algebra for probabilistic and nondeterministic processes

Relating strong behavioral equivalences for processes with nondeterminism and probabilities