Davide Grohmann

Reactive systems over directed bigraphs

We study the construction of labelled transition systems from reactive systems defined over directed bigraphs, a computational meta-model which subsumes other variants of bigraphs. First we consider wide transition systems whose labels are all those generated by the IPO construction; the corresponding bisimulation is always a congruence. Then, we show that these LTSs can be simplified further by restricting to a subclass of labels, which can be characterized syntactically. We apply this theory to the Fusion calculus: we give an encoding of Fusion in directed bigraphs, and describe its simplified wide transition system and corresponding bisimulation.

DBtk: a toolkit for directed bigraphs

We present DBtk, a toolkit for Directed Bigraphs. DBtk supports a textual language for directed bigraphs, the graphical visualization of bigraphs, the calculation of IPO labels, and the calculation of redex matchings. Therefore, this toolkit provides the main functions needed to implement simulators and verification tools.

An Algebra for Directed Bigraphs

We study the algebraic structure of directed bigraphs, a bigraphical model of computations with locations, connections and resources previously introduced as a unifying generalization of other variants of bigraphs. We give a sound and complete axiomatization of the (pre)category of directed bigraphs. Using this axiomatization, we give an adequate encoding of the Fusion calculus, showing the utility of the added directness.

Bigraphical models for protein and membrane interactions

We present a bigraphical framework suited for modeling biological systems both at protein level and at membrane level. We characterize formally bigraphs corresponding to biologically meaningful systems, and bigraphic rewriting rules representing biologically admissible interactions. At the protein level, these bigraphic reactive systems correspond exactly to systems of k-calculus. Membrane-level interactions are represented by just two general rules, whose application can be triggered by proteinlevel interactions in a well-defined and precise way. This framework can be used to compare and merge models at different abstraction levels; in particular, higher-level (e.g. mobility) activities can be given a formal biological justification in terms of low-level (i.e., protein) interactions. As examples, we formalize in our framework the vesiculation and the phagocytosis processes. Cardelli in [8] has convincingly argued that the various biochemical toolkits identified by biologists can be described as a hierarchy of abstract machines, each of which can be modelled using methods and techniques from concurrency theory. These machines are highly interdependent: “to understand the functioning of a cell, one must understand also how the various machines interact” [8]. Like other complex situations, it seems unlikely to find a single notation covering all aspects of a whole organism. In fact, we are in presence of a tower of models [19], each focusing on specific aspects of the biological system, at different levels of abstractions. Higher-level models must be represented, orrealised, at a lower level, and where possible this representation must be proved sound; in addition, we need to combine different models at the same level. To this end, we need a general metamodel, that is, a framework, where these models (possibly at different abstraction levels) can be encoded, and their interactions can be formally described. In this paper, we substantiate Milner’s idea that bigraphs can be successfully used as a framework for systems biology. More precisely, we define a class of biological bigraphs, and biological bigraphical reactive systems (BioRS), for dealing with both protein-level and membrane-level interactions. An important design choice is that this framework has to be biologically sound, i.e., it must admit only systems and reactions which are biologically meaningful, especially at lower level machines (i.e. protein). In this way, encoding a given model, for any abstract machine, as a BioRS provides automatically a formal, biologically sound justification for the model (or “implementation”) in terms of protein reactions and explains how its membrane-level interactions are realised by protein machinery. In order to formalize this “biological soundness”, we need a formal protein model to compare to our framework. We choose Danos and Laneve’s k-calculus, one of the most accepted formal model of protein systems. By suitable sorting conditions, we define a bigraphical framework which allows all and only protein configurations and interactions of the k-calculus. It is important to notice, however, that our methodology is general, and can be applied to other formal protein models. On the other hand, membrane nesting reconfiguration can be performed by just only two general rules, corresponding to the natural phenomena of “pinch” and “fuse” [10]. For encoding a given membrane model one has just to refine this general schema by specifying when these reactions are triggered,

Graph Algebras for Bigraphs

Binding bigraphs are a graphical formalism intended to be a meta-model for mobile, concurrent and communicating systems. In this paper we present an algebra of typed graph terms which correspond precisely to binding bigraphs over a given signature. As particular cases, pure bigraphs and local bigraphs are described by two sublanguages which can be given a simple syntactic characterization. Moreover, we give a formal connection between these languages and Synchronized Hyperedge Replacement algebras and the hierarchical graphs used in Architectural Design Rewriting. This allows to transfer results and constructions among for- malisms which have been developed independently, e.g., the systematic definition of congruent bisimulations for SHR graphs via the IPO construction.

Security, Cryptography and Directed Bigraphs

Bigraphical reactive systemsare an emerging graphical framework proposed by Milner and others [4,5,2,3] as a unifying theory of process models for distributed, concurrent and ubiquitous computing. A bigraphical reactive system consists of a category of bigraphs(usually generated over a given signature of controls) and a set of reaction rules. Bigraphs can be seen as representations of the possible configurations of the system, and the reaction rules specify how these configuration can evolve. The advantage of using bigraphical reactive systems is that they provide general results for deriving a labelled transition system automaticallyfrom the reaction rules, via the so-called IPO construction. Notably, the bisimulation on this transition system is always a congruence.

Deriving Barbed Bisimulations for Bigraphical Reactive Systems

We study the definition of a general abstract notion of barbed bisimilarity for reactive systems on bigraphs. More precisely, given a bigraphical reactive sys- tem, we define the corresponding barbs from the contextual labels given by the IPO construction, in a general and systematic way. These barbs correspond to observe which names on the interface are actually involved in reactions (and how). As examples, we apply this construction to the (bigraphical representation of the) p-calculus and of Mobile Ambients, and compare the resulting barbed equivalences with those previously known for these calculi.

Controlling resource access in Directed Bigraphs

We study directed bigraph with negative ports, a bigraphical framework for representing models for distributed, concurrent and ubiquitous computing. With respect to previous versions, we add the possibility that components may govern the access to resources, like (web) servers control requests from clients. This frame- work encompasses many common computational aspects, such as name or channel creation, references, client/server connections, localities, etc, still allowing to derive systematically labelled transition systems whose bisimilarities are congruences. As application examples, we analyse the encodings of client/server communications through firewalls, of (compositional) Petri nets and of chemical reactions.