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We compute the homology of the spaces of directed paths on a certain class of cubical subcomplexes of the directed Euclidean space $$\mathbb{R }^n$$ by a recursive process. We apply this result to calculate the homology and cohomology of the space of directed loops on the $$(n-1)$$ -skeleton of the directed torus $$\vec {T}^n$$ . [ABSTRACT FROM AUTHOR]

Higher dimensional automata (HDA) are highly expressive models for concurrency in Computer Science, cf van Glabbeek (Theor Comput Sci 368(1-2):168-194, ). 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, ), 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. [ABSTRACT FROM AUTHOR]

In [1], we performed a systematic investigation of reparametrizations of continuous paths in a Hausdorff space that relies crucially on a proper understanding of stop data of a (weakly increasing) reparametrization of the unit interval. I am grateful to Marco Grandis (Genova) for pointing out to me that the proof of Proposition 3.7 in [1] is wrong. Fortunately, the statement of that Proposition and the results depending on it stay correct. It is the purpose of this note to provide correct proofs. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]