1,134 research outputs found
Investigating The Algebraic Structure of Dihomotopy Types
This presentation is the sequel of a paper published in GETCO'00 proceedings
where a research program to construct an appropriate algebraic setting for the
study of deformations of higher dimensional automata was sketched. This paper
focuses precisely on detailing some of its aspects. The main idea is that the
category of homotopy types can be embedded in a new category of dihomotopy
types, the embedding being realized by the Globe functor. In this latter
category, isomorphism classes of objects are exactly higher dimensional
automata up to deformations leaving invariant their computer scientific
properties as presence or not of deadlocks (or everything similar or related).
Some hints to study the algebraic structure of dihomotopy types are given, in
particular a rule to decide whether a statement/notion concerning dihomotopy
types is or not the lifting of another statement/notion concerning homotopy
types. This rule does not enable to guess what is the lifting of a given
notion/statement, it only enables to make the verification, once the lifting
has been found.Comment: 28 pages ; LaTeX2e + 4 figures ; Expository paper ; Minor typos
corrections ; To appear in GETCO'01 proceeding
Trace spaces in a pre-cubical complex
AbstractIn directed algebraic topology, directed irreversible (d)-paths and spaces consisting of d-paths are studied from a topological and from a categorical point of view. Motivated by models for concurrent computation, we study in this paper spaces of d-paths in a pre-cubical complex. Such paths are equipped with a natural arc length which moreover is shown to be invariant under directed homotopies. D-paths up to reparametrization (called traces) can thus be represented by arc length parametrized d-paths. Under weak additional conditions, it is shown that trace spaces in a pre-cubical complex are separable metric spaces which are locally contractible and locally compact. Moreover, they have the homotopy type of a CW-complex
A convenient category for directed homotopy
We propose a convenient category for directed homotopy consisting of
preordered topological spaces generated by cubes. Its main advantage is that,
like the category of topological spaces generated by simplices suggested by J.
H. Smith, it is locally presentable
Strictifying and taming directed paths in Higher Dimensional Automata
Directed 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, eg involving synchronizations. This note shows that this is not the
case, ie that one may operate with nice schedulings without inflicting any
harm.
Several of the results in this note had previously been obtained by
Ziemianski. 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.Comment: 21 pages, 9 figure
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