Higher Dimensional Automata Research Articles

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 if and only 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. Lastly, we develop analogues of the Myhill-Nerode construction and of determinacy for HDAs with interfaces.

Connectivity of spaces of directed paths in geometric models for concurrent computation

Higher Dimensional Automata (HDA) are higher dimensional relatives to transition systems in concurrency theory taking into account to which degree various actions commute. Mathematically, they take the form of labelled cubical complexes. It is important to know, and challenging from a geometric/topological perspective, whether the space of directed paths (executions in the model) between two vertices (states) is connected; more generally, to estimate higher connectivity of these path spaces.This paper presents an approach for such an estimation for particularly simple HDA arising from PV programs and modelling the access of a number of processors to a number of resources with given limited capacity each. It defines the spare capacity of a concurrent program with prescribed periods of access of the processors to the resources using only the syntax of individual programs and the capacities of shared resources. It shows that the connectivity of spaces of directed paths can be estimated (from above) by spare capacities. Moreover, spare capacities can also be used to detect deadlocks and critical states in such a simple HDA.The key theoretical ingredient is a transition from the calculation of local connectivity bounds (of the upper links of vertices of an HDA) to global ones by applying a version of the nerve lemma due to Anders Björner.

On symmetric higher-dimensional automata and bisimilarity

It is shown that there exists a hereditary history-preserving bisimulation between a higher-dimensional automaton (HDA) and the symmetric HDA freely generated by it. Consequently, up to hereditary history-preserving bisimilarity, ordinary HDAs and symmetric HDAs are models of concurrency with the same expressive power.

Weak equivalence of higher-dimensional automata

This paper introduces a notion of equivalence for higher-dimensional automata, called weak equivalence. Weak equivalence focuses mainly on a traditional trace language and a new homology language, which captures the overall independence structure of an HDA. It is shown that weak equivalence is compatible with both the tensor product and the coproduct of HDAs and that, under certain conditions, HDAs may be reduced to weakly equivalent smaller ones by merging and collapsing cubes.

Languages of higher-dimensional automata

Abstract We introduce languages of higher-dimensional automata (HDAs) and develop some of their properties. To this end, we define a new category of precubical sets, uniquely naturally isomorphic to the standard one, and introduce a notion of event consistency. HDAs are then finite, labeled, event-consistent precubical sets with distinguished subsets of initial and accepting cells. Their languages are sets of interval orders closed under subsumption; as a major technical step, we expose a bijection between interval orders and a subclass of HDAs. We show that any finite subsumption-closed set of interval orders is the language of an HDA, that languages of HDAs are closed under binary unions and parallel composition, and that bisimilarity implies language equivalence.

Strictifying and taming directed paths in Higher Dimensional Automata

AbstractDirected paths have been used by several authors to describe concurrent executions of a program. Spaces of directed paths in an appropriate state space contain executions with all possible legal schedulings. It is interesting to investigate whether one obtains different topological properties of such a space of executions if one restricts attention to schedulings with “nice” properties, e.g. involving synchronisations. This note shows that this is not the case, i.e. that one may operate with nice schedulings without inflicting any harm. Several of the results in this note had previously been obtained by Ziemiański in Ziemiański (2017. Applicable Algebra in Engineering, Communication and Computing28 497–525; 2020a. Journal of Applied and Computational Topology4 (1) 45–78). We attempt to make them accessible for a wider audience by giving an easier proof for these findings by an application of quite elementary results from algebraic topology; notably the nerve lemma.

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.

Labeled homology of higher-dimensional automata

We construct labeling homomorphisms on the cubical homology of higher-dimensional automata and show that they are natural with respect to cubical dimaps and compatible with the tensor product of HDAs. We also indicate two possible applications of labeled homology in concurrency theory.

Higher-Dimensional Timed Automata

We introduce a new formalism of higher-dimensional timed automata, based on van Glabbeek’s higher-dimensional automata and Alur’s timed automata. We prove that their reachability is PSPACE-complete and can be decided using zone-based algorithms. We also show how to use tensor products to combat state-space explosion and how to extend the setting to higher-dimensional hybrid automata.

Spaces of directed paths on pre-cubical sets

The spaces of directed paths on the geometric realizations of pre-cubical sets, called also square -sets, can be interpreted as the spaces of possible executions of Higher Dimensional Automata, which are models for concurrent computations. In this paper we construct, for a sufficiently good pre-cubical set K, a CW-complex W(K)_v^w that is homotopy equivalent to the space of directed paths between given vertices v, w of K. This construction is functorial with respect to K, and minimal among all functorial constructions. Furthermore, explicit formulas for incidence numbers of the cells of W(K)_v^w are provided.

Topological abstraction of higher-dimensional automata

Higher-dimensional automata constitute a very expressive model for concurrent systems. In this paper, we discuss “topological abstraction” of higher-dimensional automata, i.e., the replacement of HDAs by smaller ones that can be considered equivalent from the point of view of both computer science and topology. By definition, topological abstraction preserves the homotopy type, the trace category, and the homology graph of an HDA. We establish conditions under which cube collapses yield topological abstractions of HDAs.

Weak morphisms of higher dimensional automata

We introduce weak morphisms of higher dimensional automata and use them to define preorder relations for HDAs, among which are homeomorphic abstraction and trace equivalent abstraction. It is shown that homeomorphic abstraction is essentially always stronger than trace equivalent abstraction. We also define the trace language of an HDA and show that, for a large class of HDAs, it is invariant under trace equivalent abstraction.

Cubical Sets and Trace Monoid Actions

This paper is devoted to connections between trace monoids and cubical sets. We prove that the category of trace monoids is isomorphic to the category of generalized tori and it is a reflective subcategory of the category of cubical sets. Adjoint functors between the categories of cubical sets and trace monoid actions are constructed. These functors carry independence preserving morphisms in the independence preserving morphisms. This allows us to build adjoint functors between the category of weak asynchronous systems and the category of higher dimensional automata.

Simplicial models for trace spaces II: General higher dimensional automata

Higher Dimensional Automata (HDA) are topological models for the study of concurrency phenomena. The state space for an HDA is given as a pre-cubical complex in which a set of directed paths (d-paths) is singled out. The aim of this paper is to describe a general method that determines the space of directed paths with given end points in a pre-cubical complex as the nerve of a particular category.

Execution spaces for simple higher dimensional automata

Higher dimensional automata (HDA) are highly expressive models for concurrency in Computer Science, cf van Glabbeek (Theor Comput Sci 368(1–2):168–194, 2006). For a topologist, they are attractive since they can be modeled as cubical complexes—with an inbuilt restriction for directions of allowable (d-)paths. In Raussen (Algebr Geom Topol 10:1683–1714, 2010), we developed a new method describing, for a certain subclass of HDA, the homotopy type of the space of execution paths (d-paths) as a finite simplicial complex. Several restrictions that were made to ease the presentation in that latter paper will be removed in the present article in order to make the results applicable in greater generality. Furthermore, we take a close look at semaphore models with semaphores all of arity one. It turns out that execution spaces for these are always homotopy discrete with components representing sets of “compatible” permutations. Finally, we describe a model for the complement of the execution space seen as a subspace of a product of spheres—with the aim to make the calculation of topological invariants easier and faster.

Unifying Equivalences for Higher Dimensional Automata

The intention of the paper is to show how several categorical (open morphisms, path morphisms and coalgebraic morphisms based) approaches to an abstract characterization of bisimulation relate to each other and to behavioral bisimulations, in the setting of higher dimensional automata. Such a relating makes it possible to develop a metatheory designed for unified definition and study of equivalences in true concurrency semantics.

Simplicial models of trace spaces

In this paper, we develop a systematic approach describing spaces of directed paths ‐ up to homotopy equivalence ‐ as finite prodsimplicial complexes, ie with products of simplices as building blocks. This method makes use of a certain poset category of binary matrices related to a given model space. It applies to a class of directed spaces that arise from a certain class of models of computation ‐ still restricted but with a fair amount of generality. In the final section, we outline a generalization to model spaces known as Higher Dimensional Automata. In particular, we describe algorithms that allow us to determine not only the fundamental category of such a model space, but all homological invariants of spaces of directed paths within it. The prodsimplical complexes and their associated chain complexes are finite, but they will, in general, have a huge number of cells and generators.

Path categories and resolutions

Path categories are defined, and their basic properties are described, for simplicial and cubical sets. A calculational method for describing the path category P(K) of a finite oriented simplicial complex K is introduced, which involves a finite 2-category which can be specified by generators and relations. This method specializes to higher dimensional automata via the triangulation functor from cubical to simplicial sets, and leads to calculations of their associated execution paths.

Combinatorics of labelling in higher-dimensional automata

The main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled n -cube, in exactly one way. The main ingredient is the non-functorial construction called the labelled directed coskeleton. It is defined as a subobject of the labelled coskeleton, the latter coinciding in the unlabelled case with the right adjoint to the truncation functor. This non-functorial construction is necessary since the labelled coskeleton functor of the category of labelled precubical sets does not fulfil the above requirement. We prove in this paper that it is possible to force the labelled coskeleton functor to be well behaved by working with labelled transverse symmetric precubical sets. Moreover, we prove that this solution is the only one. A transverse symmetric precubical set is a precubical set equipped with symmetry maps and with a new kind of degeneracy map called transverse degeneracy. Finally, we also prove that the two settings are equivalent from a directed algebraic topological viewpoint. To illustrate, a new semantics of the calculus of communicating systems (CCS), equivalent to the old one, is given.

Cubical local partial orders on cubically subdivided spaces—Existence and construction

The geometric models of higher dimensional automata (HDA) and Dijkstra's PV-model are cubically subdivided topological spaces with a local partial order. If a cubicalization of a topological space is free of immersed cubic Möbius bands, then there are consistent choices of direction in all cubes, such that any n-cube in the cubic subdivision is dihomeomorphic to [ 0 , 1 ] n with the induced partial order from R n . After subdivision once, any cubicalized space has a cubical local partial order. In particular, all triangularized spaces have a cubical local partial order. This implies in particular that the underlying geometry of an HDA may be quite complicated.