In a series of articles, Prakash Panangaden and his collaborators, observed that standard equivalences are not robust notions in quantitative settings, because of the sensitivity they impose on the numerical values. In order to relax the notion of equivalence, they developed several metrics, which are distances between processes, for the probabilistic settings. Following the structure of their article, I will first present a negation-free logic characterizing probabilistic bisimulation and an alternative characterization of probabilistic bisimulation in terms of functionals. Then I will present a continuous satisfaction degree of these functionals and how a family of metrics can be built using these satisfaction degree. I will also present an algorithm to compute the distance between two processes up to any given error and some compositional reasoning technics.
In the second part, I will present a connection I have established between the chemical master equation and the backward stochastic bisimulation. I propose a notion of equivalence motivated only from the point of view of the chemical master equation. Interestingly, it happens that this chemical equivalence corresponds to an equivalence already known in computer: the backward stochastic bisimulation.
I will finish with a presentation of my ongoing work: what are the interests and the difficulties in order to build a metric upon this equivalence connecting chemistry and computer science?