A multiset semantics for the pi-calculus with replication [chapter]

Joost Engelfriet
1993 Lecture Notes in Computer Science  
A multiset (or Petri net) semantics is de ned for thecalculus with replication. The semantic mapping is a strong bisimulation, and structurally congruent processes have the same semantics. This paper is readable without knowledge of Petri nets. ? The research of the author was supported by the Esprit Basic Working Group No.6067 CALIBAN. 1 a chemical soup of molecules; but we will use Petri nets rather than the recent CHemical Abstract Machines (CHAM's) of 4], which have some unnecessary
more » ... and have been less well studied. The semantics of the \small -calculus" is presented in 13] (and in 14]) in a novel way, inspired by the CHAM. First a so-called structural congruence is de ned on the process terms that is meant to capture the fact that two processes are structurally, i.e., statically, the same. In other words, the processes have the same ow graph (see 12, 16]), which roughly means that they can be decomposed into the same concurrent subprocesses. Then, an interleaving transition system is given in which structurally congruent processes are given the same behaviour, by de nition. This separation of \physical" structure and behaviour is intuitively clear, and simpli es the transition system to a large extent. In particular, the commutativity and associativity of parallel composition are handled on the structural level, and replication is even handled completely on the structural level (reducing it to parallel composition). In this paper, and its sequel 9], we wish to put forward the general idea that the multiset (or Petri net) semantics of a process algebra should also be used to express the structure of the processes: we would like two processes to have the same structure if and only if they have the same multiset semantics. Intuitively, the syntax of process terms that is needed to describe a multiset of concurrent subprocesses, should not be present in that multiset; the syntactic laws needed to describe multisets should in fact be sound, and preferably even complete. We de ne one \large" multiset transition system (or Petri net), called M , and we de ne a (compositional) semantic mapping that associates a state of M with each process of the small -calculus. Thus, the meaning of a process is a multiset (or marking of the net M ); intuitively, it is the multiset of all its concurrent subprocesses. The Petri net M has one type of transition only, which corresponds to the basic action in the small -calculus: a communication between two subprocesses. In this way M is similar to the \object-oriented" interleaving transition system of the small -calculus. Our main results on this semantics are: (A) the semantic mapping is a strong bisimulation between the interleaving transition system of the small -calculus and the multiset transition system M , and (B) if two processes of the small -calculus are structurally congruent, then they have the same semantics in M . Result (A) ensures that a process and its corresponding multiset in M have the same (interleaving) behaviour. Result (B) means that two processes that have the same structure also have the same multiset semantics. The converse of (B) does not hold and thus the laws of structural congruence of the small -calculus are sound, but not complete relative to the multiset semantics. We suggest that the structural congruence should be extended in such a way that (B) does hold in both directions. In fact, we present four natural laws for structural congruence that are not valid in the small -calculus. After extending structural congruence with these new laws, we show that results (A) and (B) still hold, and we show in
doi:10.1007/3-540-57208-2_2 fatcat:dhz36kar5jbqdnd3nysc7j7dcm