A Myhill-Nerode Theorem for Higher-Dimensional Automata

We establish a Myhill-Nerode type theorem for higher-dimensional automata (HDAs), stating that a language is regular precisely if it has finite prefix quotient. HDAs extend standard automata with additional structure, making it possible to distinguish between interleavings and concurrency. We also introduce deterministic HDAs and show that not all HDAs are determinizable, that is, there exist regular languages that cannot be recognised by a deterministic HDA. Using our theorem, we develop an internal characterisation of deterministic languages

Sculptures in Concurrency

We give a formalization of Pratt's intuitive sculpting process for higher-dimensional automata (HDA). Based on this, we show that sculptures, Pratt's Chu spaces, and Johansen's ST-structures are in close correspondence. We also develop an algorithm to decide whether a HDA can be sculpted and use this to show that some natural acyclic HDA are not sculptures. We believe that our result shed new light on the intuitions behind sculpting as a method of modeling concurrent behavior, showing the precise reaches of its expressiveness. We also show that there are sculptures whose unfoldings cannot besculpted, and that sculptures are the same as Euclidean cubical complexes. This exposes a close connection between geometric and combinatorial models for concurrency which may be of use for both areas

Sculptures in Concurrency

We give a formalization of Pratt's intuitive sculpting process for higher-dimensional automata (HDA). Intuitively, an HDA is a sculpture if it can be embedded in (i.e., sculpted from) a single higher dimensional cell (hypercube). A first important result of this paper is that not all HDA can be sculpted, exemplified through several natural acyclic HDA, one being the famous "broken box" example of van Glabbeek. Moreover, we show that even the natural operation of unfolding is completely unrelated to sculpting, e.g., there are sculptures whose unfoldings cannot be sculpted. We investigate the expressiveness of sculptures, as a proper subclass of HDA, by showing them to be equivalent to regular ST-structures (an event-based counterpart of HDA) and to (regular) Chu spaces over 3 (in their concurrent interpretation given by Pratt). We believe that our results shed new light on the intuitions behind sculpting as a method of modeling concurrent behavior, showing the precise reaches of its expressiveness. Besides expressiveness, we also develop an algorithm to decide whether an HDA can be sculpted. More importantly, we show that sculptures are equivalent to Euclidean cubical complexes (being the geometrical counterpart of our combinatorial definition), which include the popular PV models used for deadlock detection. This exposes a close connection between geometric and combinatorial models for concurrency which may be of use for both areas

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science