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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.arXiv:2108.12774v1 fatcat:2xykbvp56rgxpmetzjtwu6titi