Some Finite-Graph Models for Process Algebra

In this paper, we present a number of closely related models of process algebra [2, 3, 4], called finite-graph models. In a finite-graph model of process algebra, each process is a bisimulafion class of a particular kind of process graphs, called recursive process graphs. Just as in the standard graph model [I], each guarded recursive specification has exactly one solution in a finite-graph model, but in contrast to the standard graph model, this solution can be shown to contain a finite recursive process graph as element. The finite-graph models were defined in order to be able to build an editor that can manipulate process graphs. It is well-known that there are finite guarded specifications that have no finite solution in the standard graph model; the specification of a stack is an example [1, page 63]. Figure 1 shows an approximation of a graph of a (terminating) stack process and figure 2 shows a recursive process graph that, in a finite-graph model, bisimulates with it. The intuitive reading of figure 2 is that after a pushi event, there is a choice between a popi or doing process S itself. Details are given below. Figure 2 is a finite graph that can be drawn on a finite screen. The finite-graph models presented in this paper can represent more processes in a finite manner than the standard graph model. Note that recursive specifications can also be finitely represented in a graphical way, by means of parse trees of the terms. However, such parse trees have a structure that is quite different from the structure of recursive process graphs, and are more difficult to understand. In section 2, we define the set ~ ofrecursive process graphs, a bisimulation relation ~ on ~, and prove that ~/*-* is a model of B P A . It is shown that a subalgebra of ~R/____~ is isomorphic with the standard graph model of finitely-branching processes. Section 3 extends this to process algebra with the parallel composition operator. We also show the relation with some forms of true concurrency. Section 4 concludes the paper. All proofs are omitted from the paper, but are given in [9, 10].