The learning system Progol5 and the underlying inference method of Bottom Generalisation are firmly established within Inductive Logic Programming (ILP). But despite their success, it is known that Bottom Generalisation, and therefore Progol5, are restricted to finding hypotheses that lie within the semantics of Plotkin's relative sub-sumption. This paper exposes a previously unknown incompleteness of Progol5 with respect to Bottom Generalisation, and proposes a new approach, called Hybrid Abductive Inductive Learning, that integrates the ILP principles of Progol5 with Abductive Logic Programming (ALP). A proof procedure is proposed, called HAIL, that not only overcomes this newly discovered incompleteness, but further generalises Progol5 by computing multiple clauses in response to a single seed example and deriving hypotheses outside Plotkin's relative subsumption. A semantics is presented, called Kernel Generalisation, which extends that of Bottom Generalisation and includes the hypotheses constructed by HAIL.
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