Operational Semantics for PBC with Asynchronous Communication

This paper presents two related algebras which can be used to specify and analyse concurrent systems with synchronous and asynchronous communications. The first algebra is based on a class of P/T-nets, called boxes, and their standard transition firing rule. It is an extension of the Petri Box Calculus (PBC). Essentially, the original model is enriched with the introduction of special ‘buffer’ places where different transitions (processes) may deposit and remove tokens, together with an explicit asynchronous communication operator, denoted by tie, allowing to make them ‘private’. We also introduce an algebra of process expressions corresponding to such a net algebra, by augmenting the existing syntax of PBC expressions, and defining a system of SOS rules providing their operational semantics. The two algebras are related through a mapping which, for any extended box expression, returns a corresponding box with an isomorphic transition system. INTRODUCTION This paper is concerned with the theme of relating process algebras, such as CCS [9] and CSP [6], and Petri nets [11]. In general, the approaches proposed in the literature aim at providing a Petri net semantics to process algebras whose definition has been given independently of any Petri nets semantics (see, e.g., [5]). Another way is to translate elements from Petri nets into process algebras such as ACP [1] (see, e.g., [2]). A specific framework and the starting point of the current paper is a concrete process algebra, called the Petri Box Calculus (PBC [3, 4]), designed with the aim of allowing a compositional Petri net semantics. PBC is composed of an algebra of process expressions (called box expressions) with a fully compositional translation into labelled safe Petri nets (called boxes). In this paper, the original PBC is extended by the introduction of special ‘buffer’ places where different transitions (processes) may deposit and remove tokens, together with an explicit asynchronous communication operator, denoted by tie, allowing to make them ‘private’. In the variant of the original PBC considered here, places are labelled by their status symbols (e for entry, x for exit, and i for internal) while transitions are labelled by CCS-like synchronous communication actions, such as a, â, and τ (similarly as in CCS, τ represents an invisible action). The operators considered are: sequence E; E (the execution of E is followed by that of E); choice E2E (either E or E can be executed); parallel composition E‖E ′ (E and E can be executed concurrently); iteration E~E ′ (E can be executed an arbitrary number of times, and is followed by E ); and scoping E sc a (all handshake synchronisations involving pairs of aand â–labelled transitions are enforced, and after that they may no longer be executed alone). We illustrate some of the PBC constructs in figure 1, where the upper net is the translation of the box expression (d2a)‖((â~c)‖a).