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TOPOLOGICAL DEFORMATION OF HIGHER DIMENSIONAL AUTOMATA
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL.5(2), 2003, PP.39–82
, 2003
"... A local pospace is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are ..."
Abstract

Cited by 40 (18 self)
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A local pospace is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are two distinct notions of deformation of higher dimensional automata, “spatial” and “temporal”, leaving invariant computer scientific properties like presence or absence of deadlocks. Unfortunately, the formalization of these notions is still unknown in the general case of local pospaces. We introduce here a particular kind of local pospace, the “globular CWcomplexes”, for which we formalize these notions of deformations and which are sufficient to formalize
A model category for the homotopy theory of concurrency
 Homology, Homotopy and Applications
"... Abstract. We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are Shomotopy equivalent. This result provides an interpretation of the notion ..."
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Cited by 37 (13 self)
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Abstract. We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are Shomotopy equivalent. This result provides an interpretation of the notion of Shomotopy equivalence in the framework of model
Investigating The Algebraic Structure of Dihomotopy Types
, 2001
"... 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 ..."
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Cited by 8 (1 self)
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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.
Homotopy Branching Space and Weak Dihomotopy
"... The branching space of a flow is the topological space of germs of its nonconstant execution paths beginning in the same way. However, there exist weakly Shomotopy equivalent flows having non weakly homotopy equivalent branching spaces. This topological space is then badly behaved from a computersc ..."
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Cited by 5 (5 self)
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The branching space of a flow is the topological space of germs of its nonconstant execution paths beginning in the same way. However, there exist weakly Shomotopy equivalent flows having non weakly homotopy equivalent branching spaces. This topological space is then badly behaved from a computerscientific viewpoint since weakly Shomotopy equivalent flows must correspond to higher dimensional automata having the same computerscientific properties. To overcome this problem, the homotopy branching space of a flow is introduced as the left derived functor of the branching space functor from the model category of flows to the model category of topological spaces. As an application, we use this new functor to correct the notion of weak dihomotopy equivalence, which did not identify enough flows in its previous version.