A bigraphical reactive system (BRS) involves bigraphs, in which the nesting of nodes represents locality, independently of the edges connecting them; it also allows bigraphs to reconfigure themselves. BRSs aim to provide a uniform way to model spatially distributed systems that both compute and communicate. In this memorandum we develop their static and dynamic theory. In part I, we illustrate...

A bigraphical reactive system (BRS) involves bigraphs, in which the nesting of nodes represents locality, independently of the edges connecting them; it also allows bigraphs to reconfigure themselves. BRSs aim to provide a uniform way to model spatially distributed systems that both compute and communicate. In this memorandum we develop their static and dynamic theory. In Part I we illustrate...

Motivated by recent work on the derivation of labelled transitions and bisimulation congruences from unlabelled reaction rules, we show how to solve this problem in the DPO (double-pushout) approach to graph rewriting. Unlike in previous approaches, we consider graphs as objects, instead of arrows, of the category under consideration. This allows us to present a very simple way of deriving labelled transitions (called rewriting steps with borrowed context) which smoothly integrates with the DPO approach, has a very constructive nature and requires only a minimum of category theory. The core part of this paper is the proof sketch that the bisimilarity based on rewriting with borrowed contexts is a congruence relation.

Bigraphs are graphs whose nodes may be nested, representing locality, independently of the edges connecting them. They may be equipped with reaction rules, forming a bigraphical reactive system (Brs) in which bigraphs can reconfigure themselves. Following an earlier paper describing link graphs, a constituent of bigraphs, this paper is a devoted to pure bigraphs, which in turn underlie various more refined forms. Elsewhere it is shown that behavioural analysis for Petri nets, π-calculus and mobile ambients can all be recovered in the uniform framework of bigraphs. The paper first develops the dynamic theory of an abstract structure, a wide reactive system (Wrs), of which a Brs is an instance. In this context, labelled transitions are defined in such a way that the induced bisimilarity is a congruence. This work is then specialised to Brss, whose graphical structure allows many refinements of the theory. The latter part of the paper emphasizes bigraphical theory that is relevant to the treatment of dynamics via labelled transitions. As a running example, the theory is applied to finite pure CCS, whose resulting transition system and bisimilarity are analysed in detail. The paper also mentions briefly the use of bigraphs to model pervasive computing and

This paper axiomatises the structure of bigraphs, and proves that the resulting theory is complete. Bigraphs are graphs with double structure, representing locality and connectivity. They have been shown to represent dynamic theories for the #-calculus, mobile ambients and Petri nets, in a way that is faithful to each of those models of discrete behaviour. While the main purpose of bigraphs is to understand mobile systems, a prerequisite for this understanding is a well-behaved theory of the structure of states in such systems. The algebra of bigraph structure is surprisingly simple, as the paper demonstrates; this is because bigraphs treat locality and connectivity orthogonally

The theory of reactive systems, introduced by Leifer and Milner and previously extended by the authors, allows the derivation of well-behaved labelled transition systems (LTS) for semantic models with an underlying reduction semantics. The derivation procedure requires the presence of certain colimits (or, more usually and generally, bicolimits) which need to be constructed separately within each model. In this paper, we o#er a general construction of such bicolimits in a class of bicategories of cospans. The construction sheds light on as well as extends Ehrig and Konig's rewriting via borrowed contexts and opens the way to a unified treatment of several applications.

A notion of bigraph is proposed as the basis for a model of mobile interaction. A bigraph consists of two independent structures: a topograph representing locality and a monograph representing connectivity. Bigraphs are equipped with reaction rules to form bigraphical reactive systems (BRSs), which include versions of the -calculus and the ambient calculus. Bigraphs are shown to be a special case of a more abstract notion, wide reactive systems (WRSs), not assuming any particular graphical or other structure but equipped with a notion of width, which expresses that agents, contexts and reactions may all be widely distributed entities. A behavioural theory is established for WRSs using the categorical notion of relative pushout; it allows labelled transition systems to be derived uniformly, in such a way that familiar behavioural preorders and equivalences, in particular bisimilarity, are congruential under certain conditions. Then the theory of bigraphs is developed, and they are shown to meet these conditions. It is shown that, using certain functors, other WRSs which meet the conditions may also be derived; these may, for example, be forms of BRS with additional structure. Simple examples of bigraphical systems are discussed; the theory is developed in a number of ways in preparation for deeper application studies.

A framework is defined within which reactive systems can be studied formally. The framework is based upon s-categories, a new variety of categories, within which reactive systems can be set up in such a way that labelled transition systems can be uniformly extracted. These lead in turn to behavioural preorders and equivalences, such as the failures preorder (treated elsewhere) and bisimilarity, which are guaranteed to be congruential. The theory rests upon the notion of relative pushout previously introduced by the authors. The framework

G-relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner’s approach to deriving labelled bisimulation congruences from reduction systems. This paper develops the theory of GRPOs further, arguing that they provide a simple and powerful basis towards a comprehensive solution. As an example, we construct GRPOs in a category of ‘bunches and wirings. ’ We then examine the approach based on Milner’s precategories and Leifer’s functorial reactive systems, and show that it can be recast in a much simpler way into the 2-categorical theory of GRPOs.

We introduce a way of viewing Petri nets as open systems. This is done by considering a bicategory of cospans over a category of p/t nets and embeddings. We derive a labelled transition system (LTS) semantics for such nets using GIPOs and characterise the resulting congruence. Technically, our results are similar to the recent work by Milner on applying the theory of bigraphs to Petri Nets. The two main differences are that we treat p/t nets instead of c/e nets and we deal directly with a category of nets instead of encoding them into bigraphs.