108 research outputs found

### 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

### 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

### Sliced Max-Flow on Circular Mapper Graphs

In this work we develop a novel global invariant of circle-parametrized
mapper graphs in order to analyse periodic sets, which often arise in materials
science applications. This invariant describes a flow in a graph, slicing it
with the fibers of the associated map onto the circle

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