Core Concepts
For every connected polyhedron, there exists a shared-variable concurrent system whose higher-dimensional automaton model has the same homotopy type as the polyhedron.
Abstract
The paper presents a result showing that the topology of higher-dimensional automata (HDAs), which are a combinatorial-topological model for concurrent systems, can be arbitrarily complex. Specifically, the authors prove that for every connected polyhedron, there exists a shared-variable concurrent system whose HDA model has the same homotopy type as the polyhedron.
The key steps are:
The authors construct the cubical barycentric subdivision of a simplicial complex as a precubical set and show that its geometric realization is homeomorphic to the original polyhedron.
They then turn this precubical set into an HDA, which is shown to be the HDA model of its 1-skeleton (transition system) with respect to a strict total order on the labels.
To make this HDA accessible (i.e., all states are reachable), the authors modify it by adding new edges and cubes, while preserving the homotopy type.
Finally, they show that this accessible HDA is isomorphic to the HDA model of a shared-variable concurrent system.
The result demonstrates the expressive power of HDAs and their connection to topology, which can be leveraged to analyze the structure and properties of concurrent systems.