Decreasing Diagrams for Confluence and Commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

L'Europe en quête d'une politique de migration : les contraintes de la mondialisation et de la restructuration des marchés du travail

This paper argues that the development towards a common migration policy in the European Union reflects the emergence of a new form of regionalism resulting from the recent structural transformations in the global political economy. The European governments are caught in a web of contradictory interests and tendencies. On one side, the logic of global economic restructuring dictates continued deregulation and flexibilisation of the labour market, implying increased high levels of immigration. On the other hand, the political backlash against globalisation pushes towards a closure of the external borders. The result is the construction of a Fortress Europe, with a set of specific cooperation agreements with the regions surrounding the European Union in order to regulate the inflow of migrants