Trace spaces in a pre-cubical complex

Raussen, Martin

doi:10.1016/j.topol.2009.02.003

Cited by 27 publications

(56 citation statements)

References 18 publications

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“…Instead, it is probably necessary to use the method described in Raussen [28] with a higher combinatorial complexity still to be sorted out.…”

Section: Structure and Overview Of Resultsmentioning

confidence: 99%

“…Reparametrization equivalent d-paths [4] in X have the same directed image (= trace) in X . Dividing out the action of the monoid of (weakly-increasing) reparametrizations of the parameter interval E I , we arrive at trace space E T .X /.c; d/ (cf Fahrenberg and Raussen [4; 27]) which is shown in Raussen [28] to be homotopy equivalent to path space E P .X /.c; d/ for a far wider class of directed spaces X . In the latter paper, it is also shown that trace spaces enjoy nice properties: They are metrizable, locally compact, locally contractible, and they have the homotopy type of a CW-complex.…”

Section: A Simple Higher Dimensional Automatonmentioning

confidence: 99%

“…As to the functor D , we need to verify the conditions of the projection lemma: It was shown in Raussen [28], that E T .X /.0; 1/ is paracompact -even under much weaker assumptions to X . Furthermore, Proposition 2.12 allows us to replace the cover given by the subspaces E T .X j 1 ;:::;j l /.0; 1/ to that given by the homotopy equivalent open subspaces E T .Y j 1 ;:::;j l /.0; 1/ from Definition 2.11, with the same colimit and a homotopy equivalent homotopy colimit.…”

Section: Trace Spaces and Prodsimplicial Complexes As Colimitsmentioning

confidence: 99%

“…General topological properties of spaces of d-paths and of traces (Dd-paths up to monotone reparametrizations; cf Fahrenberg and Raussen [4; 27]) in semicubical complexes were investigated in Raussen [28]. But so far, apart from low-dimensional examples with convincing drawings, there have been very few explicit examples of actual computations of spaces of such traces (for an attempt in dimension two, see Raussen [24]); let alone a general method to perform such computations.…”

Section: Introductionmentioning

confidence: 99%

“…The overall philosophy reminds of the analysis of the topology of path spaces in CW-complexes in Milnor's article [22]: Also the spaces of d-paths in a precubical complex with given end points are equi locally convex (ELCX) (cf Raussen [28]) and thus locally contractible; for the PV-models analysed here, suitably chosen contractible subsets can be described explicitly, by a blend of order and combinatorics. They and their intersections form a poset category with a geometric realization that is homotopy equivalent to the path space under consideration.…”

Section: Introductionmentioning

confidence: 99%

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Simplicial models of trace spaces

Raussen

2010

Algebr. Geom. Topol.

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Directed algebraic topology studies topological spaces in which certain directed paths (d-paths) are singled out; in most cases of interest, the reverse path of a d-path is no longer a d-path. We are mainly concerned with spaces of directed paths between given end points, and how those vary under variation of the end points. The original motivation stems from certain models for concurrent computation. So far, homotopy types of spaces of d-paths and their topological invariants have only been determined in cases that were elementary to overlook.In this paper, we develop a systematic approach describing spaces of directed pathsup 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.55P10, 55P15, 55U10; 68Q55, 68Q85

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“…Instead, it is probably necessary to use the method described in Raussen [28] with a higher combinatorial complexity still to be sorted out.…”

Section: Structure and Overview Of Resultsmentioning

confidence: 99%

Section: A Simple Higher Dimensional Automatonmentioning

confidence: 99%

Section: Trace Spaces and Prodsimplicial Complexes As Colimitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Simplicial models of trace spaces

Raussen

2010

Algebr. Geom. Topol.

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Execution spaces for simple higher dimensional automata

Raussen

2012

AAECC

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ABSTRACT. Higher Dimensional Automata (HDA) are highly expressive models for concurrency in Computer Science, cf van Glabbeek [26]. 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 [25], 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 this 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.

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Spaces of directed paths on pre-cubical sets

Ziemiański

2017

AAECC

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The spaces of directed paths on the geometric realizations of pre-cubical sets, called also -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 ) w v 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 ) w v are provided.

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Trace spaces in a pre-cubical complex

Cited by 27 publications

References 18 publications

Simplicial models of trace spaces

Simplicial models of trace spaces

Execution spaces for simple higher dimensional automata

Spaces of directed paths on pre-cubical sets

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