Characterisations of Testing Preorders for a Finite Probabilistic pi-Calculus

We consider two characterisations of the may and must testing preorders for a probabilistic extension of the finite pi-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 pi-calculus. We base our notions of simulations on the 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 pi-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 pi-calculus by Boreale and De Nicola, we extend the language of the $\pi$-calculus with a mismatch operator, without which the formulation of a characteristic test will not be possible.

💡 Research Summary

This paper presents two complete characterisations of the may‑ and must‑testing preorders for a finite probabilistic extension of the π‑calculus. The first characterisation is based on a notion of probabilistic weak simulation, while the second relies on a probabilistic fragment of Milner‑Parrow‑Walker modal logic that has been suitably extended.
The authors begin by adapting the weak simulation concepts originally developed for probabilistic CSP to the π‑calculus. Because the π‑calculus features name passing and dynamic name generation, the way in which name quantification is scoped becomes a critical design choice. Several variants of simulation—early, late, open—can be defined, but the paper shows that only the “earliest” simulation (the analogue of early bisimulation in the non‑probabilistic setting) correctly captures the testing preorders. In an earliest simulation, the choice of a fresh name is made before the transition is observed, ensuring that the simulating process never commits to a more specific name binding than the simulated one.
The second characterisation extends the MPW modal logic with probabilistic choice (⊕_p) and expectation operators, while preserving the original possibility (◇) and concurrency constructs. For each process P a “characteristic formula” Φ(P) is constructed inductively; Φ(P) precisely describes all possible actions of P together with their probability distributions. The authors prove that a process Q satisfies Φ(P) if and only if Q is related to P by the may‑testing preorder, and similarly for must‑testing using a suitably strengthened formula.
A crucial technical device is the introduction of a mismatch operator