Algebra of Parameterised Graphs

One of the difficulties in designing modern hardware systems is the necessity for comprehending and dealing with a very large number of system configurations, operational modes, and behavioural scenarios. It is often infeasible to consider and specify each individual mode explicitly, and one needs methodologies and tools to exploit similarities between the individual modes and work with groups of modes rather than individual ones. The modes and groups of modes have to be managed in a compositional way: the specification of the system should be composed from specifications of its blocks. This includes both structural and behavioural composition. Furthermore, one should be able to transform and optimise the specifications in a formal way. In this article, we propose a new formalism, called parameterised graphs. It extends the existing conditional partial order graphs (CPOGs) formalism in several ways. First, it deals with general graphs rather than just partial orders. Moreover, it is fully compositional. To achieve this, we introduce an algebra of parameterised graphs by specifying the equivalence relation by a set of axioms, which is proved to be sound, minimal, and complete. This allows one to manipulate the specifications as algebraic expressions using the rules of this algebra. We demonstrate the usefulness of the developed formalism on several case studies coming from the area of microelectronics design.