311 research outputs found
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
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
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
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
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