Characterisations of testing preorders for a finite probabilistic φ-calculus

We consider two characterisations of the may and must testing preorders for a probabilistic extension of the finite η-calculus: one based on notions of probabilistic weak simulations, and the other on a probabilistic extension of a fragment of Milner-Parrow-Walker modal logic for the η-calculus. We base our notions of simulations on similar concepts used in previous work for probabilistic CSP. However, unlike the case with CSP (or other non-value-passing calculi), there are several possible definitions of simulation for the probabilistic η-calculus, which arise from different ways of scoping the name quantification.We show that in order to capture the testing preorders, one needs to use the "earliest" simulation relation (in analogy to the notion of early (bi)simulation in the non-probabilistic case). The key ideas in both characterisations are the notion of a "characteristic formula" of a probabilistic process, and the notion of a "characteristic test" for a formula. As in an earlier work on testing equivalence for the η-calculus by Boreale and De Nicola, we extend the language of the η-calculus with a mismatch operator, without which the formulation of a characteristic test will not be possible.