Section: New Results
Semantics of probabilistic systems
One of the goals of Comète is to investigate the foundations of probabilistic calculi, and in particular the probabilistic asynchronous -calculus described in Section 3.1.2 .
In  we have studied a process calculus which combines both nondeterministic and probabilistic behavior in the style of Segala and Lynch's probabilistic automata. We have considered various strong and weak behavioral equivalences, and we have provided complete axiomatizations for finite-state processes, restricted to guarded definitions in case of the weak equivalences. We conjecture that in the general case of unguarded recursion the ``natural'' weak equivalences are undecidable.
This has been the first work, to our knowledge, to provide a complete axiomatization for weak equivalences in the presence of recursion and both nondeterministic and probabilistic choice.
In systems that model quantitative processes, steps are associated with a given quantity, such as the probability that the step will happen or the resources (e.g. time or cost) needed to perform that step. The standard notion of bisimulation can be adapted to these systems by treating the quantities as labels, but this does not provide a robust relation, since quantities are matched only when they are identical. Processes that differ for a very small probability, for instance, would be considered just as different as processes that perform completely different actions. This is particularly relevant to security systems where specifications can be given as perfect, but impractical processes and other, practical processes are considered safe if they only differ from the specification with a negligible probability.
To find a more flexible way to differentiate processes, we have considered the notion of metric, which is a function that associates a real number (distance) with a pair of elements. In  , we have studied metric semantic for a general framework that we call Action-labeled Quantitative Transition Systems (AQTS). This framework subsumes some other well-known quantitative systems such as probabilistic automata  , reactive and generative models  , and (a simplified version of) weighted automata  ,  .
The metric semantics that we have investigated in  is based on rather sophisticated techniques. In particular, we needed to resort to the notion of Hutchinson distance.
Still in  , we have considered two extended examples which show that our results apply to both probabilistic and weighted automata as special cases of AQTS. In particular, we have shown that the operators of the corresponding process algebras are non-expansive, which is the metric correspondent of the notion of congruence.
Probability and guards
In  we have proposed a probabilistic extension of the -calculus whose main novelty is a probabilistic mixed choice operator, that is, a choice construct with a probability distribution on the branches, and where input and output actions can both occur as guards. We have developed the operational semantics of this calculus, and we have investigated its expressiveness. In particular, we have compared it with the sublanguage with the two separate choices , where input and output guards are not allowed together in the same choice construct. Our main result is that the separate choices can encode the mixed one. Further, we have showed that input-guarded choice can encode output-guarded choice and viceversa.
In  we have developed a model of Parametric Probabilistic Transition Systems, where probabilities associated with transitions may be parameters. We have showed how to find instances of the parameters that satisfy a given property and instances that either maximize or minimize the probability of reaching a certain state. As an application, we have modeled a probabilistic non–repudiation protocol with a Parametric Probabilistic Transition System. The theory we have developed allows us to find instances that maximize the probability that the protocol ends in a fair state (i.e. no participant has an advantage over the others).