On the expressive power of recursion, replication and iteration in process calculi

Abstract

In this paper we investigate the expressive power of three alternative approaches to the definition of infinite behaviours in process calculi, namely, recursive definitions, replication and iteration. We prove several results discriminating between the calculi obtained from a core CCS by adding the three mechanisms mentioned above. These results are derived by considering the decidability of four basic properties: termination (that is, all computations are finite); convergence (that is, the existence of a finite computation); barb (that is, the ability to perform an action on a given channel) and weak bisimulation. Our results, which are summarised in Table 1, show that the three calculi form a strict expressiveness hierarchy in that: all the properties mentioned are undecidable in CCS with recursion; only termination and barb are decidable in CCS with replication; all the properties are decidable in CCS with iteration. As a corollary, we also obtain a strict expressiveness hierarchy with respect to weak bisimulation, since there exist weak bisimulation preserving encodings of iteration in replication and of replication in recursion, whereas there are no weak bisimulation preserving encodings in the other directions.

On the Expressiveness of Forwarding in Higher-Order Communication

“…In HOCORE such an upper bound does not exist. This was essential for obtaining the decidability result; for this, we appealed to the approach developed in [9], which relies on the theory of well-structured transition systems [8]. As far as we are aware, this approach for expressiveness has not previously been used in the higher-order setting.…”

“…In sharp contrast with HOCORE, termination in HO −f is decidable. This result is obtained by appealing to the theory of well-structured transition systems [8], following the approach used in [9].…”

“…We use P to denote that there is no P such that P −→ P . Following [9] we now define process convergence and process termination. Observe that process termination implies process convergence while the opposite does not hold.…”

Reactive Turing machines

“…An absolute expressiveness result establishes that a process calculus can or cannot express a particular process or a particular type of processes. It is then, e.g., shown that a process calculus is, or is not, Turing-powerful [11]. More refined forms of absolute expressiveness may also be exploited to compare the relative expressiveness of two process calculi, by showing that a certain type of process can be expressed in the one calculus but not in the other (see, e.g., [5,7,4]).…”

Turing universality of the Biochemical Ground Form

“…No other direct implications among these criteria exist. For instance, in (Busi et al 2009) it is proved that CCS (Milner 1989) with replication instead of recursion is Turing complete according to Criterion 2 but not according to Criteria 1 or 3.…”