We identify three basic modalities for expressing boolean queries using the expressions of a query language: nonemptiness, emptiness, and containment. For the class of first-order queries, these three modalities have exactly the same expressive power. For other classes of queries, e.g., expressed in weaker query languages, the modalities may differ in expressiveness. We propose a framework for studying the expressive power of boolean query modalities. Along one dimension, one may work within a fixed query language and compare the three modalities, e.g., we can compare a fixed query language F under emptiness to F under nonemptiness. Here, we identify crucial query features that enable us to go from one modality to another. Furthermore, we identify semantical properties that reflect the lack of these query features to establish separations. Along a second dimension, one may fix a modality and compare different query languages. This second dimension is the one that has already received quite some attention in the literature, whereas in this paper we emphasize the first dimension. Combining both dimensions, it is interesting to compare the expressive power of a weak query language using a strong modality, against that of a seemingly stronger query language but perhaps using a weaker modality. We present some initial results within this theme. As an important auxiliary result, we establish a preservation theorem for monotone containments of conjunctive queries.
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