On the Expressive Power of Restriction in CCS with Replication

Busi et al [8] showed that CCS! (CCS with replication instead of recursion) is Turing powerful by providing an encoding of Random Access Machines (RAMs) which preserves and reflects convergence (i.e., the existence of terminating computations). The encoding uses an unbounded number of restrictions arising from having restriction operators under the scope of replication. We study the expressive power of restriction and its interplay with replication. We do this by considering several syntactic variants of CCS! which differ from each other in the use of restriction with respect to replication. We consider three syntactic variations of CCS! which do not allow the use of an unbounded number of restrictions: CCS−!ν ! is the fragment of CCS! not allowing restrictions under the scope of a replication. CCS −ν ! is the restriction-free fragment of CCS!. The third variant is CCS −!ν !+pr which extends CCS−!ν ! with Phillip’s priority guards. We show that the use of unboundedly many restrictions in CCS! is necessary for obtaining Turing expressiveness in the sense of Busi et al. We do this by showing that there is no encoding of RAMs into CCS−!ν ! which preserves and reflects convergence. We also prove that up to failures equivalence, there is no encoding from CCS! into CCS −!ν ! nor from CCS−!ν ! into CCS −ν ! . As lemmata for the above results we prove that convergence is decidable for CCS−!ν ! and that language equivalence is decidable for CCS−ν ! . As corollary it follows that convergence is decidable for restriction-free CCS. Finally, we show the expressive power of priorities by providing an encoding of RAMs in CCS−!ν !+pr. Not only does the encoding preserve and reflect convergence but it also preserves and reflects divergence (the existence of infinite computations). Busi et al showed that there is no encoding of RAMs into CCS! which preserves and reflects divergence.