158 research outputs found
A convenient category for directed homotopy
We propose a convenient category for directed homotopy consisting of
preordered topological spaces generated by cubes. Its main advantage is that,
like the category of topological spaces generated by simplices suggested by J.
H. Smith, it is locally presentable
Trace Spaces: an Efficient New Technique for State-Space Reduction
State-space reduction techniques, used primarily in model-checkers, all rely
on the idea that some actions are independent, hence could be taken in any
(respective) order while put in parallel, without changing the semantics. It is
thus not necessary to consider all execution paths in the interleaving
semantics of a concurrent program, but rather some equivalence classes. The
purpose of this paper is to describe a new algorithm to compute such
equivalence classes, and a representative per class, which is based on ideas
originating in algebraic topology. We introduce a geometric semantics of
concurrent languages, where programs are interpreted as directed topological
spaces, and study its properties in order to devise an algorithm for computing
dihomotopy classes of execution paths. In particular, our algorithm is able to
compute a control-flow graph for concurrent programs, possibly containing
loops, which is "as reduced as possible" in the sense that it generates traces
modulo equivalence. A preliminary implementation was achieved, showing
promising results towards efficient methods to analyze concurrent programs,
with very promising results compared to partial-order reduction techniques
Simplicial models for concurrency
We model both concurrent programs and the possible executions from one state
to another in a concurrent program using simplices. The latter are calculated
using necklaces of simplices in the former.Comment: 12 pages, Section 4 from v1 omitted since quasi-category equivalences
are too strong: they induce equivalences of path categorie
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
Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again
The geometric models of concurrency - Dijkstra\u27s PV-models and V. Pratt\u27s Higher Dimensional Automata -
rely on a translation of discrete or algebraic information to geometry.
In both these cases, the translation is the geometric realisation of a semi cubical complex,
which is then a locally partially ordered space, an lpo space.
The aim is to use the algebraic topology machinery, suitably adapted to the fact
that there is a preferred time direction.
Then the results - for instance dihomotopy classes of dipaths, which model
the number of inequivalent computations should be used on the discrete model and give the corresponding discrete objects.
We prove that this is in fact the case for the models considered:
Each dipath is dihomottopic to a combinatorial dipath
and if two combinatorial dipaths are dihomotopic, then they are combinatorially equivalent.
Moreover, the notions of dihomotopy (LF., E. Goubault, M. Raussen)
and d-homotopy (M. Grandis) are proven to be equivalent for these models
- hence the Van Kampen theorem is available for dihomotopy.
Finally we give an idea of how many spaces have a local po-structure given by cubes.
The answer is, that any cubicalized space has such a structure
after at most one subdivision.
In particular, all triangulable spaces have a cubical local po-structure
Classification of dicoverings
AbstractThe dicoverings of a “well pointed” d-space are classified as quotients of the universal dicovering space under congruence relations. We prove that the subcategory of d-spaces generated by the subcategory of directed cubes is equal to the category generated by the interval and the directed interval. Similarly, the category of topological spaces generated by simplices may be generated by the interval
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