227 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
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
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
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
Weak equivalence of higher-dimensional automata
This paper introduces a notion of equivalence for higher-dimensional
automata, called weak equivalence. Weak equivalence focuses mainly on a
traditional trace language and a new homology language, which captures the
overall independence structure of an HDA. It is shown that weak equivalence is
compatible with both the tensor product and the coproduct of HDAs and that,
under certain conditions, HDAs may be reduced to weakly equivalent smaller ones
by merging and collapsing cubes.This research was partially supported by FCT (Fundacao para a Ciencia e a Tecnologia, Portugal) through project UID/MAT/00013/2013
Weak equivalence of higher-dimensional automata
This paper introduces a notion of equivalence for higher-dimensional
automata, called weak equivalence. Weak equivalence focuses mainly on a
traditional trace language and a new homology language, which captures the
overall independence structure of an HDA. It is shown that weak equivalence is
compatible with both the tensor product and the coproduct of HDAs and that,
under certain conditions, HDAs may be reduced to weakly equivalent smaller ones
by merging and collapsing cubes
A Kleene Theorem for Higher-Dimensional Automata
We prove a Kleene theorem for higher-dimensional automata (HDAs). It states that the languages they recognise are precisely the rational subsumption-closed sets of interval pomsets. The rational operations include a gluing composition, for which we equip pomsets with interfaces. For our proof, we introduce HDAs with interfaces as presheaves over labelled precube categories and use tools inspired by algebraic topology, such as cylinders and (co)fibrations. HDAs are a general model of non-interleaving concurrency, which subsumes many other models in this field. Interval orders are used as models for concurrent or distributed systems where events extend in time. Our tools and techniques may therefore yield templates for Kleene theorems in various models and applications
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