Binding bigraphs as symmetric monoidal closed theories

TL;DR

This paper reconstructs Milner's bigraphs within symmetric monoidal closed theories, clarifying their structure and scope, and extending their framework to include terms and higher-order contexts.

Contribution

It introduces a categorical reconstruction of bigraphs using SMC theories, clarifying their scoping discipline and expanding their expressive power.

Findings

Bigraphs can be modeled as symmetric monoidal closed theories.

The reconstruction enforces the scoping discipline directly from SMC structure.

The resulting category is larger, including terms and higher-order contexts.

Abstract

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.