4,405 research outputs found
A model category for the homotopy theory of concurrency
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 S-homotopy
equivalent. This result provides an interpretation of the notion of S-homotopy
equivalence in the framework of model categories.Comment: 45 pages ; 4 figure ; First paper corresponding to the content of
math.AT/0201252 ; final versio
Towards a homotopy theory of process algebra
This paper proves that labelled flows are expressive enough to contain all
process algebras which are a standard model for concurrency. More precisely, we
construct the space of execution paths and of higher dimensional homotopies
between them for every process name of every process algebra with any
synchronization algebra using a notion of labelled flow. This interpretation of
process algebra satisfies the paradigm of higher dimensional automata (HDA):
one non-degenerate full -dimensional cube (no more no less) in the
underlying space of the time flow corresponding to the concurrent execution of
actions. This result will enable us in future papers to develop a
homotopical approach of process algebras. Indeed, several homological
constructions related to the causal structure of time flow are possible only in
the framework of flows.Comment: 33 pages ; LaTeX2e ; 1 eps figure ; package semantics included ; v2
HDA paradigm clearly stated and simplification in a homotopical argument ; v3
"bug" fixed in notion of non-twisted shell + several redactional improvements
; v4 minor correction : the set of labels must not be ordered ; published at
http://intlpress.com/HHA/v10/n1/a16
The homotopy branching space of a flow
In this talk, I will explain the importance of the homotopy branching space
functor (and of the homotopy merging space functor) in dihomotopy theory. The
paper is a detailed abstract of math.AT/0304112 and math.AT/0305169.Comment: Expository paper ; 11 pages ; to appear in GETCO'03 proceedin
Combinatorics of branchings in higher dimensional automata
We explore the combinatorial properties of the branching areas of execution
paths in higher dimensional automata. Mathematically, this means that we
investigate the combinatorics of the negative corner (or branching) homology of
a globular -category and the combinatorics of a new homology theory
called the reduced branching homology. The latter is the homology of the
quotient of the branching complex by the sub-complex generated by its thin
elements. Conjecturally it coincides with the non reduced theory for higher
dimensional automata, that is -categories freely generated by
precubical sets. As application, we calculate the branching homology of some
-categories and we give some invariance results for the reduced
branching homology. We only treat the branching side. The merging side, that is
the case of merging areas of execution paths is similar and can be easily
deduced from the branching side.Comment: Final version, see
http://www.tac.mta.ca/tac/volumes/8/n12/abstract.htm
T-homotopy and refinement of observation (III) : Invariance of the branching and merging homologies
This series explores a new notion of T-homotopy equivalence of flows. The new
definition involves embeddings of finite bounded posets preserving the bottom
and the top elements and the associated cofibrations of flows. In this third
part, it is proved that the generalized T-homotopy equivalences preserve the
branching and merging homology theories of a flow. These homology theories are
of interest in computer science since they detect the non-deterministic
branching and merging areas of execution paths in the time flow of a higher
dimensional automaton. The proof is based on Reedy model category techniques.Comment: 30 pages ; final preprint version before publication ; see
http://nyjm.albany.edu:8000/j/2006/Vol12.ht
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