Connector algebras for C/E and P/T nets' interactions

Abstract

Abstract. A quite flourishing research thread in the recent literature on componentbased systems is concerned with the algebraic properties of different classes of connectors. In a recent paper, an algebra of stateless connectors was presented that consists of five kinds of basic connectors, namely symmetry, synchronization, mutual exclusion, hiding and inaction, plus their duals, and it was shown how they can be freely composed in series and in parallel to model sophisticated "glues". In this paper we explore the expressiveness of stateful connectors obtained by adding one-place buffers or unbounded buffers to the stateless connectors. The main results are: i) we show how different classes of connectors exactly correspond to suitable classes of Petri nets equipped with compositional interfaces, called nets with boundaries; ii) we show that the difference between strong and weak semantics in stateful connectors is reflected in the semantics of nets with boundaries by moving from the classic step semantics (strong case) to a novel banking semantics (weak case), where a step can be executed by taking some "debit" tokens to be given back during the same step; iii) we show that the corresponding bisimilarities are congruences (w.r.t. composition of connectors in series and in parallel); iv) we show that suitable monoidality laws, like those arising when representing stateful connectors in the tile model, can nicely capture concurrency (in the sense of step semantics) aspects; and v) as a side result, we provide a basic algebra, with a finite set of symbols, out of which we can compose all P/T nets with boundaries, fulfilling a long standing quest.

Compositional Reachability in Petri Nets

“…The goal of the more recent work [2,3,21] was to lift this algebra to the level of nets in a compositional way and explore connections with process algebra: our approach ignores local state and focusses only on external interactions: here we were inspired by the ideas of Milner [17].…”

“…The following is straightforward to show and builds on known compositionality properties of PNBs [2].…”

“…Roughly speaking, a Petri Net with Boundaries [2] (PNB) represents a subnet that is to be placed within a larger environment. The key feature of this model is a representation of how a net's transitions may connect with its environment, via "boundary ports".…”

“…We write P : (1, 1) to mean that P is a PNB with both boundaries of size 1. Differently to [2,21] we consider "vanilla" PNBs to be unmarked; in §2 we introduce a marked variant that contains both an initial and a target marking. There are two operations for composing PNBs: synchronisation on a common boundary (;) and a non-commutative, parallel composition that we call tensor (⊗).…”

“…There are two operations for composing PNBs: synchronisation on a common boundary (;) and a non-commutative, parallel composition that we call tensor (⊗). The most interesting operation is synchronisation: we refer to [2] for the formalities, but the graphical intuition shown in Fig. 2 suffices for most examples.…”