Martin Raussen

On the classification of dipaths in geometric models for concurrency

This paper focusses on the determination of the dihomotopy classes of dipaths in cubical complexes, representing the essentially different computations in a given concurrent program due to different schedules. Several new notions have to be developed, for example, the domains of dependence (Definition 2.4), which are borrowed from relativity theory (Penrose 1972). However, it turns out that the algorithm determining deadlocks and unsafe regions described in Fajstrup et al. (1998a; 1998b) can be modified and applied to do the essential calculational work.

Invariants of Directed Spaces

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.

Directed Algebraic Topology and Concurrency

This monograph presents an application of concepts and methods from algebraic topology to models of concurrent processes in computer science and their analysis. Taking well-known discrete models for concurrent processes in resource management as a point of departure, the book goes on to refine combinatorial and topological models. In the process, it develops tools and invariants for the new discipline directed algebraic topology, which is driven by fundamental research interests as well as by applications, primarily in the static analysis of concurrent programs. The state space of a concurrent program is described as a higher-dimensional space, the topology of which encodes the essential properties of the system. In order to analyse all possible executions in the state space, more than just the topological properties have to be considered: Execution paths need to respect a partial order given by the time flow. As a result, tools and concepts from topology have to be extended to take privileged directions into account. The target audience for this book consists of graduate students, researchers and practitioners in the field, mathematicians and computer scientists alike.

Dihomotopy as a Tool in State Space Analysis Tutorial

Recent geometric methods have been used in concurrency theory for quickly finding deadlocks and unreachable states, see [14] for instance. The reason why these methods are fast is that they contain in germ ingredients for tackling the state-space explosion problem. In this paper we show how this can be made formal. We also give some hints about the underlying algorithmics. Finally, we compare with other well-known methods for coping with the state-space explosion problem.

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.

Geometric analysis of nondeterminacy in dynamical systems: Towards a geometric analysis of concurrent systems

This article intends to provide some new insights into concurrency using ideas from the theory of dynamical systems. Inherently discrete concurrency corresponds to a parallel continuous concept: a discrete state space corresponds to a differential manifold, an execution path corresponds to a flow line of a dynamical system. To model non-determinacy within dynamical systems, we introduce a new geometrical object, a section cone. A section cone is a convex set in the space of vector fields, all elements having the same singular points. We show that it is enough to consider flow lines of a single vector field in order to capture the behavior of all flow lines in the section cone up to homotopy (corresponding to equivalence of executions).

Homology of Spaces of Directed Paths in Euclidean Pattern Spaces

Let \(\mathcal{F}\) be a family of subsets of {1, …, n} and let

Directed Topological Models of Concurrency

\displaystyle{ Y _{\mathcal{F}} =\bigcup _{F\in \mathcal{F}}\{(x_{1},\ldots,x_{n}) \in \mathbb{R}^{n}: x_{ i} \in \mathbb{Z}\text{for all}i \in F\}. }