Many Web applications are based on dynamic interactions between Web components exchanging flows of information. Such a situation arises for instance in mashup systems or when monitoring distributed autonomous systems. Our work is in this challenging context that has generated recently a lot of attention; see Web 2.0. We introduce the axlog formal model for capturing such interactions and show how this model can be supported efficiently. The central component is the axlog widget defined by one tree-pattern query or more, over an active document (in the Active XML style) that includes some input streams of updates. A widget generates a stream of updates for each query, the updates that are needed to maintain the view corresponding to the query. We exploit an array of known technologies: datalog optimization techniques such as Differential or MagicSet, constraint query languages, and efficient XML filtering (YFilter). The novel optimization technique we propose is based on fundamental new notions: a relevance (different than that of MagicSet), satisfiability and provenance for active documents. We briefly discuss an implementation of an axlog engine, an application that we used to test the approach, and results of experiments