A Hybrid Abductive Inductive Proof Procedure OLIVER RAY, KRYSIA BRODA, ALESSANDRA RUSSO, Department of Computing, Imperial College London, 180 Queen’s Gate, South Kensington Campus, SW7 2AZ,UK. E-mail: {or,kb,ar3}@doc.ic.ac.uk Abstract This paper introduces a proof procedure that integrates Abductive Logic Programming (ALP) and Inductive Logic Programming (ILP) to automate the learning of first order Horn clause theories from examples and background knowledge. The work builds upon a recent approach called Hybrid Abductive Inductive Learning (HAIL) by showing how language bias can be practically and usefully incorporated into the learning process. A proof procedure for HAIL is proposed that utilises a set of user specified mode declarations to learn hypotheses that satisfy a given language bias. A semantics is presented that accurately characterises the intended hypothesis space and includes the hypotheses derivable by the proof procedure. An implementation is described that combines an extension of the Kakas-Mancarella ALP procedure within an ILP procedure that generalises the Progol system of Muggleton. The explicit integration of abduction and induction is shown to allow the derivation of multiple clause hypotheses in response to a single seed example and to enable the inference of missing type information in a way not previously possible. Keywords : Abductive Logic Programming, Inductive Logic Programming, Machine Learning. 1 Introduction Inductive Logic Programming (ILP) [17] is the sub-field of Machine Learning (ML) [14] that formalises and automates the task of learning first order theories from examples and prior background knowledge. The benefits of relational learning methods that use expressive logical representations are well illustrated by the ILP system Progol [15], which has been successfully deployed in several applications, most notably in the area of computational bioinformatics [1, 11, 27]. Underlying Progol is an inference method known as Bottom Generalisation (BG) [15, 29] that is used to construct hypotheses incrementally one clause at a time. Given a background theory, B, and a single seed example, e, this inference method generalises a clause, called a Bottom Clause [15], to return a hypothesis, h, that logically entails e relative to B. The role played by the Bottom Clause is of great theoretical and practical importance as it serves to bound a hypothesis space that would otherwise be intractable to search. The success of Progol is to a large extent due to the efficient use that is made of both language bias and search bias during the construction and generalisation of the Bottom Clause. In ML, language bias is the name given to any syntactic constraints on hypothesised clauses, while search bias refers to any procedural preferences that result in certain hypotheses being preferred over others [14, 18]. Progol’s language bias is determined by a set of so-called mode declarations [15] that allow the user to focus the search on interesting and potentially useful subsets of the hypothesis 1 L. J. of the IGPL, Vol. 0 No. 0, pp. 1–27 0000 c  Oxford University Press