A Bigraph Relational Model

Milner’s bigraphical reactive systems [7], or BRSs in short, are formulated in terms of category theory. They encompass earlier models, such as CCS [6], the π-calculus [3], and Petri nets [5]. However, as with other categorical models, it is not immediately clear how to implement a logical framework that could check, for example, the well-formedness or correctness of a BRS, or just execute reaction rules. On the other hand there are mature implementations of logical frameworks, e.g. Celf [8] that already provide many of the algorithms that one would need for such an implementation. In particular, Celf provides support for linearity and concurrency using a kind of structural congruence that arises naturally from the definition of equivalence in the type theory CLF [2]. In this paper we show that the two are deeply connected. In particular, we formulate a bigraph relational model for BRSs and demonstrate how to piggy-bag on Celf’s implementation by reusing algorithms, such as unification, type checking, type inference, logic programming, and multi-set rewriting. A BRS consists of a bigraph and a set of reaction rules. The bigraph consists of a place graph, that usually models the hierarchical (physical) structure of the concurrent system to be modeled, and the link graph that establishes the communication structure between the different places. By the virtue of this definition alone, a bigraph does not have any dynamic properties. It is best understood as a snapshot of a concurrent system at a particular point of time. What makes a bigraph reactive is the accompanying set of reaction rules. A reaction rule can be thought of as a rewrite rule, except that the left and the right hand side are graphs rather than terms. Consequently, matching the left hand side of a reaction rule with a subgraph of the current bigraph is conceptually and computationally not as straightforward as for example first-order unification. As an alternative, we relate bigraphical reactive systems to something that we understand well: unification modulo structural congruence in the setting of CLF [9]. CLF is a type theory that conservatively extends the λ -calculus and serves to model truly concurrent systems. CLF follows the standard judgements-as-types encoding paradigm, which means that derivations of the validity of a BRS, traces of the operational semantics, etc. are encoded as CLF objects of the corresponding CLF type. The extensions include, for example, type families that are indexed by objects, a dependent type constructor Πx : A.B(x) as a generalisation of the usual function type constructor, linear type constructors, for example A ( B, that capture the nature of resource consumption, and also a monadic type constructor {A}. This type is