J. Cussens and A. Frisch (Eds.): ILP 2000, LNAI 1866, pp. 93-111, 2000.  Springer-Verlag Berlin Heidelberg 2000 Induction of Recursive Theories in the Normal ILP Setting: Issues and Solutions Floriana Esposito, Donato Malerba, and Francesca A. Lisi Dipartimento di Informatica, Università degli Studi di Bari, Via Orabona 4, I-70126 Bari, Italy {esposito | malerba | lisi}@di.uniba.it Abstract. Induction of recursive theories in the normal ILP setting is a complex task because of the non-monotonicity of the consistency property. In this paper we propose computational solutions to some relevant issues raised by the multiple predicate learning problem. A separate-and-parallel-conquer search strategy is adopted to interleave the learning of clauses supplying predicates with mutually recursive definitions. A novel generality order to be imposed to the search space of clauses is investigated in order to cope with recursion in a more suitable way. The consistency recovery is performed by reformulating the current theory and by applying a layering technique based on the collapsed dependency graph. The proposed approach has been implemented in the ILP system ATRE and tested in the specific context of the document understanding problem within the WISDOM project. Experimental results are discussed and future directions are drawn. 1 Introduction Inductive learning of recursive logical theories is equivalent to learning multiple predicate definitions from a set of examples. De Raedt et al. [9] have showed that learning multiple predicates is more difficult than learning a single predicate. In fact, the former task is not limited to the generation of several independent predicate definitions, but involves the discovery of concept dependencies. A wrong hypothesis on concept dependencies may significantly affect the learning results. Moreover, the ordering typically used in inductive logic programming (ILP), namely θ-subsumption [26], is not sufficient to guarantee the completeness and consistency of learned definitions with respect to logical entailment. The main problems raised by multiple/recursive predicate learning can be explained in terms of an important property of the normal ILP problem setting: Whenever two individual clauses are consistent on the data, their conjunction need not to be consistent on the same data [11]. As a consequence, clauses supplying predicates with multiple/recursive definitions should not be learned individually but, in principle, they should be generated all together. In order to overcome these problems, it has been proposed to work on a weak setting of ILP [14], in which the monotonicity property is satisfied: Whenever two individual clauses are valid on the data, their conjunction will also be valid on the