13,761 research outputs found
Homotopy Bisimilarity for Higher-Dimensional Automata
We introduce a new category of higher-dimensional automata in which the
morphisms are functional homotopy simulations, i.e. functional simulations up
to concurrency of independent events. For this, we use unfoldings of
higher-dimensional automata into higher-dimensional trees. Using a notion of
open maps in this category, we define homotopy bisimilarity. We show that
homotopy bisimilarity is equivalent to a straight-forward generalization of
standard bisimilarity to higher dimensions, and that it is finer than split
bisimilarity and incomparable with history-preserving bisimilarity.Comment: Heavily revised version of arXiv:1209.492
History-Preserving Bisimilarity for Higher-Dimensional Automata via Open Maps
We show that history-preserving bisimilarity for higher-dimensional automata
has a simple characterization directly in terms of higher-dimensional
transitions. This implies that it is decidable for finite higher-dimensional
automata. To arrive at our characterization, we apply the open-maps framework
of Joyal, Nielsen and Winskel in the category of unfoldings of precubical sets.Comment: Minor updates in accordance with reviewer comments. Submitted to MFPS
201
Higher Dimensional Transition Systems
We introduce the notion of higher dimensional transition systems as a model of concurrency providing an elementary, set-theoretic formalisation of the idea of higher dimensional transition. We show an embedding of the category of higher dimensional transition systems into that of higher dimensional automata which cuts down to an equivalence when we restrict to non-degenerate automata. Moreover, we prove that the natural notion of bisimulation for such structures is a generalisation of the strong history preserving bisimulation, and provide an abstract categorical account of it via open maps. Finally, we define a notion of unfolding for higher dimensional transition systems and characterise the structures so obtained as a generalisation of event structures
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
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
Homotopy Bisimilarity for Higher-Dimensional Automata
We introduce a new category of higher-dimensional automata in which the morphisms are functional homotopy simulations, i.e. functional simulations up to concurrency of independent events. For this, we use unfoldings of higher-dimensional automata into higher-dimensional trees. Using a notion of open maps in this category, we define homotopy bisimilarity. We show that homotopy bisimilarity is equivalent to a straight-forward generalization of standard bisimilarity to higher dimensions, and that it is finer than split bisimilarity and incomparable with history-preserving bisimilarity
Formal Relationships Between Geometrical and Classical Models for Concurrency
A wide variety of models for concurrent programs has been proposed during the
past decades, each one focusing on various aspects of computations: trace
equivalence, causality between events, conflicts and schedules due to resource
accesses, etc. More recently, models with a geometrical flavor have been
introduced, based on the notion of cubical set. These models are very rich and
expressive since they can represent commutation between any bunch of events,
thus generalizing the principle of true concurrency. While they seem to be very
promising - because they make possible the use of techniques from algebraic
topology in order to study concurrent computations - they have not yet been
precisely related to the previous models, and the purpose of this paper is to
fill this gap. In particular, we describe an adjunction between Petri nets and
cubical sets which extends the previously known adjunction between Petri nets
and asynchronous transition systems by Nielsen and Winskel
Combinatorics of labelling in higher dimensional automata
The main idea for interpreting concurrent processes as labelled precubical
sets is that a given set of n actions running concurrently must be assembled to
a labelled n-cube, in exactly one way. The main ingredient is the
non-functorial construction called labelled directed coskeleton. It is defined
as a subobject of the labelled coskeleton, the latter coinciding in the
unlabelled case with the right adjoint to the truncation functor. This
non-functorial construction is necessary since the labelled coskeleton functor
of the category of labelled precubical sets does not fulfil the above
requirement. We prove in this paper that it is possible to force the labelled
coskeleton functor to be well-behaved by working with labelled transverse
symmetric precubical sets. Moreover, we prove that this solution is the only
one. A transverse symmetric precubical set is a precubical set equipped with
symmetry maps and with a new kind of degeneracy map called transverse
degeneracy. Finally, we also prove that the two settings are equivalent from a
directed algebraic topological viewpoint. To illustrate, a new semantics of
CCS, equivalent to the old one, is given.Comment: 47 pages, LaTeX2e, no figure
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
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