  We study the problem of finite ontology mediated query answering (FOMQA), the
variant of OMQA where the represented world is assumed to be finite, and thus
only finite models of the ontology are considered. We adopt the most typical
setting with unions of conjunctive queries and ontologies expressed in
description logics (DLs). The study of FOMQA is relevant in settings that are
not finitely controllable. This is the case not only for DLs without the finite
model property, but also for those allowing transitive role declarations. When
transitive roles are allowed, evaluating queries is challenging: FOMQA is
undecidable for SHOIF and only known to be decidable for the Horn fragment of
ALCIF. We show decidability of FOMQA for three proper fragments of SOIF: SOI,
SOF, and SIF. Our approach is to characterise models relevant for deciding
finite query entailment. Relying on a certain regularity of these models, we
develop automata-based decision procedures with optimal complexity bounds.
