Termination Detection of Local Computations

Contrary to the sequential world, the processes involved in a distributed system do not necessarily know when a computation is globally finished. This paper investigates the problem of the detection of the termination of local computations. We define four types of termination detection: no detection, detection of the local termination, detection by a distributed observer, detection of the global termination. We give a complete characterisation (except in the local termination detection case where a partial one is given) for each of this termination detection and show that they define a strict hierarchy. These results emphasise the difference between computability of a distributed task and termination detection. Furthermore, these characterisations encompass all standard criteria that are usually formulated : topological restriction (tree, rings, or triangu- lated networks ...), topological knowledge (size, diameter ...), and local knowledge to distinguish nodes (identities, sense of direction). These results are now presented as corollaries of generalising theorems. As a very special and important case, the techniques are also applied to the election problem. Though given in the model of local computations, these results can give qualitative insight for similar results in other standard models. The necessary conditions involve graphs covering and quasi-covering; the sufficient conditions (constructive local computations) are based upon an enumeration algorithm of Mazurkiewicz and a stable properties detection algorithm of Szymanski, Shi and Prywes

The Mann-Su theorem

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Ignasi Mundet i Riera[en] In this text, we give the necessary tools to prove and understand the Mann-Su theorem. In the context of transformation groups theory, the Mann-Su theorem gives a restriction on which finite groups can act effectively on a manifold. Particularly, we will find an upper bound $N$ that only depends on the manifold $M$ such that groups of the form $(\mathbb{Z}_p )^{r}$ can not act effectively on $M$ if $r > N$. Restricting ourselves to the case of smooth manifolds and actions, we will take a slightly different approach compared to the original paper where L.N Mann and J.C. Su proved the theorem