Binding bigraphs as symmetric monoidal closed theories

Milner's bigraphs are a general framework for reasoning about distributed and concurrent programming languages. Notably, it has been designed to encompass both the pi-calculus and the Ambient calculus. This paper is only concerned with bigraphical syntax: given what we here call a bigraphical signature K, Milner constructs a (pre-) category of bigraphs BBig(K), whose main features are (1) the presence of relative pushouts (RPOs), which makes them well-behaved w.r.t. bisimulations, and that (2) the so-called structural equations become equalities. Examples of the latter include, e.g., in pi and Ambient, renaming of bound variables, associativity and commutativity of parallel composition, or scope extrusion for restricted names. Also, bigraphs follow a scoping discipline ensuring that, roughly, bound variables never escape their scope. Here, we reconstruct bigraphs using a standard categorical tool: symmetric monoidal closed (SMC) theories. Our theory enforces the same scoping discipline as bigraphs, as a direct property of SMC structure. Furthermore, it elucidates the slightly mysterious status of so-called links in bigraphs. Finally, our category is also considerably larger than the category of bigraphs, notably encompassing in the same framework terms and a flexible form of higher-order contexts.Comment: 17 pages, uses Paul Taylor's diagram

Variable binding, symmetric monoidal closed theories, and bigraphs

This paper investigates the use of symmetric monoidal closed (SMC) structure for representing syntax with variable binding, in particular for languages with linear aspects. In our setting, one first specifies an SMC theory T, which may express binding operations, in a way reminiscent from higher-order abstract syntax. This theory generates an SMC category S(T) whose morphisms are, in a sense, terms in the desired syntax. We apply our approach to Jensen and Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This leads to an alternative category of bigraphs, which we compare to the original.Comment: An introduction to two more technical previous preprints. Accepted at Concur '0

On Hierarchical Graphs: Reconciling Bigraphs, Gs-monoidal Theories and Gs-graphs

Abstract. Compositional graph models for global computing systems must account for two relevant dimensions, namely nesting and linking. In Milner’s bigraphs the two dimensions are made explicit and represented as loosely coupled structures: the place graph and the link graph. Here, bigraphs are compared with an earlier model, gs-graphs, based on gs-monoidal theories and originally conceived for modelling the syntactical structure of agents with α-convertible declarations. We show that gs-graphs are quite convenient also for the new purpose, since the two dimensions can be recovered by introducing two types of nodes. With respect to bigraphs, gs-graphs can be proved essentially equivalent, with minor differences at the interface level. We argue that gs-graphs have a simpler and more standard algebraic structure for representing both states and transitions, and can be equipped with a simple type system (in the style of relational separation logic) to check the well-formedness of bounded gs-graphs. Another advantage concerns a textual form in terms of sets of assignments, which can make implementation easier in rewriting frameworks like Maude. Vice versa, the reactive system approach developed for bigraphs needs yet to be addressed in gs-graphs.

Open Graphs and Monoidal Theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits "enough adhesivity" from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are "type-safe" in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discreet, finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parametrised by graphical signatures, similar to the monoidal signatures of Joyal and Street, which define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs, and more general monoidal theories. Thus open-graphs give us a handle for mechanised reasoning in monoidal categories.Comment: 31 pages, currently technical report, submitted to MSCS, waiting review

Completeness of Nominal PROPs

We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams.Comment: arXiv admin note: text overlap with arXiv:1904.0753

Completeness of Nominal PROPs

We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams

Bigraphs and Their Algebra

AbstractBigraphs are a framework in which both existing process calculi and new models of behaviour can be formulated, yielding theory that is shared among these models. A short survey of the main features of bigraphs is presented, showing how they can be developed from standard graph theory using elementary category theory. The algebraic manipulation of bigraphs is outlined with the help of illustrations. The treatment of dynamics is then summarised. Finally, origins and some related work are discussed. The paper provides a motivating introduction to bigraphs

Bigraphs with sharing and applications in wireless networks

Bigraphs are a fully graphical process algebraic formalism, capable of representing both the position in space of agents and their inter-connections. However, they assume a topology based on sets of trees and thus cannot represent spatial locations that are shared among several entities in a simple or intuitive way. This is a problem, because shared locations are often a requirement, for example, when modelling scenarios in the physical world or in modern complex computer systems such as wireless networks and spatial-aware applications in ubiquitous computing. We propose bigraphs with sharing, a generalisation of the original definition of bigraphs, to allow for overlapping topologies. The new locality model is based on directed acyclic graphs. We demonstrate the new formalism can be defined in the general framework of bigraphical theories and wide reactive systems, as originally devised by Robin Milner. We do so by defining a categorical interpretation of bigraphs with sharing, an axiomatisation derived from the equations of a bialgebra over finite ordinals, and a normal form to express bigraphical terms. We illustrate how sharing is essential for modelling overlapping localities by presenting two example case studies in the field of wireless networking. We show that bigraphs with sharing can be used realistically in a production environment by describing the implementation of an efficient matching algorithm and a software tool for the definition, simulation, visualisation and analysis of bigraphical reactive systems

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems