Comparing the expressive power of the synchronous and asynchronous $pi$-calculi

The Asynchronous $\pi$-calculus, proposed in Honda and Tokoro (1991) and, independently, in Boudol (1992), is a subset of the $\pi$-calculus (Milner et al. 1992), which contains no explicit operators for choice and output prefixing. The communication mechanism of this calculus, however, is powerful enough to simulate output prefixing, as shown in Honda and Tokoro (1991) and Boudol (1992), and input-guarded choice, as shown in Nestmann and Pierce (2000). A natural question arises, then, as to whether or not it is as expressive as the full $\pi$-calculus. We show that this is not the case. More precisely, we show that there does not exist any uniform, fully distributed translation from the $\pi$-calculus into the asynchronous $\pi$-calculus, up to any ‘reasonable’ notion of equivalence. This result is based on the incapability of the asynchronous $\pi$-calculus to break certain symmetries that may be present in the initial communication graph. By similar arguments, we prove a separation result between the $\pi$-calculus and CCS, and between the $\pi$-calculus and the $\pi$-calculus with internal mobility, a subset of the $\pi$-calculus proposed by Sangiorgi where the output actions can only transmit private names.