158 research outputs found
The Directed Homotopy Hypothesis
The homotopy hypothesis was originally stated by Grothendieck: topological spaces should be "equivalent" to (weak) infinite-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g., for rewriting, distributed systems, (homotopy) type theory etc.
But an essential feature in the work set up in concurrency theory, is that time should be considered irreversible, giving rise to the field of directed algebraic topology. Following the path proposed by Porter, we state here a directed homotopy hypothesis: Grandis\u27 directed topological spaces should be "equivalent" to a weak form of topologically enriched categories, still very close to (infinite,1)-categories. We develop, as in ordinary algebraic topology, a directed homotopy equivalence and a weak equivalence, and show invariance of a form of directed homology
Directed Homology and Persistence Modules
In this note, we give a self-contained account on a construction for a
directed homology theory based on modules over algebras, linking it to both
persistence homology and natural homology. We study its first properties, among
which some exact sequences
Bisimulations and Unfolding in P-Accessible Categorical Models
In this paper, we propose a categorical framework for bisimulations and unfoldings that unifies the classical approach from Joyal and al. via open maps and unfoldings. This is based on a notion of categories accessible with respect to a subcategory of path shapes, i.e., for which one can define a nice notion of trees as glueing of paths. We prove that transitions systems and pre sheaf models are a particular case of our framework. We also prove that in our framework, several characterizations of bisimulation coincide, in particular an "operational one" akin to the standard definition in transition systems. Also, accessibility is preserved by coreflexions. We then design a notion of unfolding, which has good properties in the accessible case: its is a right adjoint and is a universal covering, i.e., initial among the morphisms that have the unique lifting property with respect to path shapes. As an application, we prove that the universal covering of a groupoid, a standard construction in algebraic topology, coincides with an unfolding, when the category of path shapes is well chosen
The tropical double description method
We develop a tropical analogue of the classical double description method
allowing one to compute an internal representation (in terms of vertices) of a
polyhedron defined externally (by inequalities). The heart of the tropical
algorithm is a characterization of the extreme points of a polyhedron in terms
of a system of constraints which define it. We show that checking the
extremality of a point reduces to checking whether there is only one minimal
strongly connected component in an hypergraph. The latter problem can be solved
in almost linear time, which allows us to eliminate quickly redundant
generators. We report extensive tests (including benchmarks from an application
to static analysis) showing that the method outperforms experimentally the
previous ones by orders of magnitude. The present tools also lead to worst case
bounds which improve the ones provided by previous methods.Comment: 12 pages, prepared for the Proceedings of the Symposium on
Theoretical Aspects of Computer Science, 2010, Nancy, Franc
Directed Homotopy in Non-Positively Curved Spaces
A semantics of concurrent programs can be given using precubical sets, in
order to study (higher) commutations between the actions, thus encoding the
"geometry" of the space of possible executions of the program. Here, we study
the particular case of programs using only mutexes, which are the most widely
used synchronization primitive. We show that in this case, the resulting
programs have non-positive curvature, a notion that we introduce and study here
for precubical sets, and can be thought of as an algebraic analogue of the
well-known one for metric spaces. Using this it, as well as categorical
rewriting techniques, we are then able to show that directed and non-directed
homotopy coincide for directed paths in these precubical sets. Finally, we
study the geometric realization of precubical sets in metric spaces, to show
that our conditions on precubical sets actually coincide with those for metric
spaces. Since the category of metric spaces is not cocomplete, we are lead to
work with generalized metric spaces and study some of their properties
Static Analysis of Programs with Imprecise Probabilistic Inputs
International audienceHaving a precise yet sound abstraction of the inputs of numerical programs is important to analyze their behavior. For many programs, these inputs are probabilistic, but the actual distribution used is only partially known. We present a static analysis framework for reasoning about programs with inputs given as imprecise probabilities: we define a collecting semantics based on the notion of previsions and an abstract semantics based on an extension of Dempster-Shafer structures. We prove the correctness of our approach and show on some realistic examples the kind of invariants we are able to infer.Il est important de disposer d'une abstraction précise mais correcte des entrées d'un programme numérique pour analyser ses comportements. Pour de nombreux programmes, ces entrées sont probabilistes, mais la distribution réellement utilisée n'est connue que partiellement. Nous présentons un cadre d'analyse statique permettant le raisonnement sur des programmes dont les entrées sont données sous forme de probabilités imprécises: nous définissons une sémantique collectrice fondée sur la notion de prévisions et une sémantique abstraite fondée sur une extension des structures de Dempster-Shafer. Nous démontrons la correction de notre approche et montrons sur des exemples réalistes le genre d'invariants que nous sommes capables d'inférer
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