  Modal logics are widely used in multi-agent systems to reason about actions,
abilities, norms, or epistemic states. Combined with description logic
languages, they are also a powerful tool to formalise modal aspects of
ontology-based reasoning over an object domain. However, the standard
relational semantics for modalities is known to validate principles deemed
problematic in agency, deontic, or epistemic applications. To overcome these
difficulties, weaker systems of so-called non-normal modal logics, equipped
with neighbourhood semantics that generalise the relational one, have been
investigated both at the propositional and at the description logic level. We
present here a family of non-normal modal description logics, obtained by
extending ALC-based languages with non-normal modal operators. For formulas
interpreted on neighbourhood models over varying domains, we provide a modular
framework of terminating, correct, and complete tableau-based satisfiability
checking algorithms in NExpTime. For a subset of these systems, we also
consider a reduction to satisfiability on constant domain relational models.
Moreover, we investigate the satisfiability problem in fragments obtained by
disallowing the application of modal operators to description logic concepts,
providing tight ExpTime complexity results.
