Abstract. We study the behavioural theory of πP, a π-calculus featuring restriction as the only binder. In contrast with calculi such as Fusions and Chi, reduction in πP generates a preorder on names rather than an equivalence relation. We present two characterisations of barbed congruence in πP: the first is based on a compositional LTS, and the second is an axiomatisation. The results in this paper bring out basic properties of πP, mostly related to the interplay between the restriction operator and the preorder on names.
IntroductionThe π-calculus expresses mobility via name passing, and has two binders: the input prefix binds the value to be received, and restriction is used to delimit the scope of a private name. The study of Fusions [13], Chi [6], Explicit Fusions [16] and Solos [10] has shown that using restriction as the only binder is enough to express name passing. In such calculi (which, reusing a terminology from [8], we shall refer to as fusion calculi ), the bound input prefix, a(x).P , is dropped in favour of free input, ab.P . This yields a pleasing symmetry between input and output prefixes; moreover, one can encode bound input in terms of free input as (νx)ax.P .In [8], the analysis of capability types [14] for fusion calculi has shown that an important modification to existing fusion calculi is necessary in order to be able to define meaningful type systems, beyond the simple discipline of sorting. This has led to the introduction of πP, a π-calculus with preorders. Like existing fusion calculi, πP features restriction as the only binder, and free input and output prefixes. The calculus differs however from existing name-passing calculi. The most important difference is that interaction does not have the effect of equating (or fusing) two names, but instead generates an arc process, according to the following reduction relation:The arc a/b expresses the fact that anything that can be done using name b can be done using a as well (but not the opposite): we say that a is above b. Arcs induce a preorder relation on names, which can evolve along reductions.Arcs can modify interaction possibilities: in presence of a/b, a is above b, hence a process emitting on b can also make an output transition along channel a. In general, an output on channel c can interact with an input on d provided c and d are joinable, written c d, which means that there is some name that is above both c and d according to the preorder relation To formalise these observations, the operational semantics exploits conditions, which are either of the form b a (a is above b), or a b (a and b are joinable).[8] shows that capabilities make sense in a fusion calculus if preorders replace equivalence relations ('fusions'). Accordingly, [8] presents a type system with input/output types and subtyping for πP. Name preorders have an impact on how processes express behaviours. It is necessary, beyond types, to understand the consequences of introducing this new construct, whose behaviour does not seem to be reducible to known situations....