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

Cut-off Theorems for the PV-model

We prove cut-off results for deadlocks and serializability of a $PV$-thread $T$ run in parallel with itself: For a $PV$ thread $T$ which accesses a set $\mathcal{R}$ of resources, each with a maximal capacity $\kappa:\mathcal{R}\to\mathbb{N}$, the PV-program $T^n$, where $n$ copies of $T$ are run in parallel, is deadlock free for all $n$ if and only if $T^M$ is deadlock free where $M=\Sigma_{r\in\mathcal{R}}\kappa(r)$. This is a sharp bound: For all $\kappa:\mathcal{R}\to\mathbb{N}$ and finite $\mathcal{R}$ there is a thread $T$ using these resources such that $T^M$ has a deadlock, but $T^n$ does not for $n<M$. Moreover, we prove a more general theorem: There are no deadlocks in $p=T1|T2|\cdots |Tn$ if and only if there are no deadlocks in $T_{i_1}|T_{i_2}|\cdots |T_{i_M}$ for any subset $\{i_1,\ldots,i_M\}\subset [1:n]$. For $\kappa(r)\equiv 1$, $T^n$ is serializable for all $n$ if and only if $T^2$ is serializable. For general capacities, we define a local obstruction to serializability. There is no local obstruction to serializability in $T^n$ for all $n$ if and only if there is no local obstruction to serializability in $T^M$ for $M=\Sigma_{r\in\mathcal{R}}\kappa(r)+1$. The obstructions may be found using a deadlock algorithm in $T^{M+1}$. These serializability results also have a generalization: If there are no local obstructions to serializability in any of the $M$-dimensional sub programs, $T_{i_1}|T_{i_2}|\cdots |T_{i_M}$, then $p$ is serializable

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

Towards Directed Collapsibility

In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have non-trivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs

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

Directed topological complexity

International audienceIt has been observed that the very important motion planning problem of robotics mathematically speaking boils down to the problem of finding a section to the path-space fibration, raising the notion of topological complexity, as introduced by M. Farber. The above notion fits the motion planning problem of robotics when there are no constraints on the actual control that can be applied to the physical apparatus. In many applications, however, a physical apparatus may have constrained controls, leading to constraints on its potential future dynamics. In this paper we adapt the notion of topological complexity to the case of directed topological spaces, which encompass such controlled systems, and also systems which appear in concurrency theory. We study its first properties, make calculations for some interesting classes of spaces, and show applications to a form of directed homotopy equivalence