  We investigate the entailment problem in ELHr ontologies annotated with
provenance information. In more detail, we show that subsumption entailment is
in NP if provenance is represented with polynomials from the Trio semiring and
in PTime if the semiring is not commutative. The proof is based on the
construction of a weighted tree automaton which recognises a language that
matches with the corresponding provenance polynomial.
