On decidability properties of two fragments of the asynchronous pi-calculus

In (Cacciagrano, et al., 2008) the authors studied the expressiveness of persistence in the asynchronous pi-calculus, henceforth A pi. They considered A pi and three sub-languages of it, each capturing one source of persistence: the persistent-input calculus (PIA pi), the persistent-output calculus (POA pi), and the persistent calculus (PA pi). They prove that, under some general conditions, there cannot be an encoding from A pi into a (semi)-persistent calculus preserving the must-testing semantics, a semantics sensitive to divergence. In this paper we support and strengthen the separation results of (Cacciagrano, et al., 2008) by showing that convergence and divergence are two decidable properties in a fragment of POA pi and PA pi, in contrast to what happen in A pi. Thus, it is shown that there cannot be a (computable) encoding from A pi into PA pi and in such a fragment of POA pi, preserving divergence or convergence. These impossibility results don't presuppose any condition on the encodings and involve directly convergence for first time in the study of the expressiveness of persistence of (A pi).