In this paper, we develop algorithmic datalog-based foundations for the incremental processing of tree-pattern queries over active documents, i.e. document with incoming streams of data. We define query satisfiability for such documents based on a logic with 3-values: ``true'', ``false forever'', and ``false for now''. Also, given an active document and a query, part of the document (and in particular, some incoming streams) may become irrelevant for the query of interest. We introduce an incremental algorithm for detecting such useless data and streams, essential for implementing garbage collection. We also provide complexity analysis for the problems we study.