Abduction is a fundamental mode of reasoning, which has taken on increasing importance in Artificial Intelligence (AI) and related disciplines. Computing abductive explanations is an important problem, and there is a growing literature on this subject. We contribute to this endeavor by presenting new results on computing multiple resp. all of the possibly exponentially many explanations of an abductive query from a propositional Horn theory represented by a Horn CNF. Here the issues are whether a few explanations can be generated efficiently and, in case of all explanations, whether the computation is possible in polynomial total time (or output-polynomial time), i.e., in time polynomial in the combined size of the input and the output. We explore these issues for queries in CNF and important restrictions thereof Among the results, we show that computing all explanations for a negative query literal from a Horn CNF is not feasible in polynomial total time unless P = NP, which settles an open issue. However, we show how to compute under restriction to acyclic Horn theories polynomially many explanations in input polynomial time and all explanations in polynomial total time, respectively. Complementing and extending previous results, this draws a detailed picture of the computational complexity of computing multiple explanations for queries on Horn theories.
