Characterisations of Testing Preorders for a Finite Probabilistic pi-Calculus

TL;DR

This paper characterizes testing preorders for a probabilistic pi-calculus using probabilistic simulations and modal logic, highlighting the importance of early simulation and characteristic formulas for process comparison.

Contribution

It introduces two novel characterizations of testing preorders for probabilistic pi-calculus, emphasizing the early simulation relation and the use of characteristic formulas and tests.

Findings

Earliest simulation relation captures testing preorders.

Characteristic formulas enable process characterization.

Extension with mismatch operator is essential for characteristic tests.

Abstract

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.