1,602 research outputs found
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
About the globular homology of higher dimensional automata
We introduce a new simplicial nerve of higher dimensional automata whose
homology groups yield a new definition of the globular homology. With this new
definition, the drawbacks noticed with the construction of math.CT/9902151
disappear. Moreover the important morphisms which associate to every globe its
corresponding branching area and merging area of execution paths become
morphisms of simplicial sets.Comment: 44 pages ; LaTeX2e, 1 figure ; final version to appear in CTGD
Labeled homology of higher-dimensional automata
We construct labeling homomorphisms on the cubical homology of higher-dimensional
automata and show that they are natural with respect to cubical dimaps and compatible
with the tensor product of HDAs. We also indicate two possible applications of labeled
homology in concurrency theory.info:eu-repo/semantics/publishedVersio
Classification of Quantum Cellular Automata
There exists an index theory to classify strictly local quantum cellular
automata in one dimension. We consider two classification questions. First, we
study to what extent this index theory can be applied in higher dimensions via
dimensional reduction, finding a classification by the first homology group of
the manifold modulo torsion. Second, in two dimensions, we show that an
extension of this index theory (including torsion) fully classifies quantum
cellular automata, at least in the absence of fermionic degrees of freedom.
This complete classification in one and two dimensions by index theory is not
expected to extend to higher dimensions due to recent evidence of a nontrivial
automaton in three dimensions. Finally, we discuss some group theoretical
aspects of the classification of quantum cellular automata and consider these
automata on higher dimensional real projective spaces.Comment: 53 pages, 15 figures; v2: minor corrections, final version in pres
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
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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