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We consider the computation of all respectively a polynomial subset of the explanations of an abductive query from a Horn theory, and pay particular attention to whether the query is a positive or negative letter, the explanation is based on literals from an assumption set, and the Horn theory is represented in terms of formulas or characteristic models. We derive tractability results, one of which refutes a conjecture by Selman and Levesque, as well as intractability results, and furthermore also semi-tractability results in terms of solvability in quasi-polynomial time. Our results complement previous results in the literature, and elucidate the computational complexity of generating the set of explanations.