Learning Abductive Theories Yannis Dimopoulos and Antonis Kakas 1 Abstract. In this paper we study the problem of learning abductive theories with particular interest in learning theories for the problem of attribute-based classification as studied in the area of machine learning. The paper proposes a new alternative formulation of this class of learning problems where abduction takes an integral part in the formulation of the appropriate theories and more importantly in the definition of the learning problem itself. We present a general algorithm that learns abductivetheories for classificationand examine its main features. We show how within our abductive approach it is possible to formulate and handle in a natural way cases of the problem with incomplete information. We also study the relation of our approach to other existing approachesfor these learning problems, notably that of decision trees, and argue that our approach could provide a useful link between abduction and machine learning. 1 INTRODUCTION Over the last decade many authors have pointed out the importance of abductive reasoning in problems of Artificial Intelligence and other areas of Computer Science (see the surveys [5], [9] and references therein). Abduction has been shown to be a useful form of reason- ing for many applications ranging from diagnosis and planning to database updates andusermodeling. Althoughrecently there has been some work trying to relate abduction and learning (or induction) there has been very little work in studying the problem of learning abduc- tive theories i.e. theories whose main form of reasoning is abduction suited for the kind of applications mentioned above. In this paper we will address this problem of learning abductive theories by studying the special, but characteristic case, of learning abductive theories for classification (or diagnosis). In particular, we will investigate a simplified case of this problem that is essentially the same problem as that of attribute-based learning (or discrimination) studied in Machine Learning. From the point of view of learning, this paper aims to propose a new approach for learning classification theories based on an alternative abductive perspective in which the learning problem itself is formalized in terms of abduction rather than deduction as it is usually done in conventionallearning theories. We will also provide a basic computational model for such learn- ing problems based on computing suitable abductive theories and compare this to the well-known method of decision trees. Intuitively, within this abductive perspective the classification of an objectis understood as a task of best explaining the (given)features of the object by assuming that the object belongs to a certain class (or concept). Therefore in the learning phase we need to construct (abductive) theories which can provide abductive explanations of the features in terms of concepts. Instead of learning rules of the form that can be used deductively to infer (classify) 1 Departmentof Computer Science, University of Cyprus, CY-1678, Nicosia, Cyprus. Email: yannis,antonis @turing.cs.ucy.ac.cy concepts from their features (attributes ) we will be learning rules of the form that can be used abductivelyto explain the features. These abductive theories will be triples , where is a set of rules of the form , -the set of abducible predicates- will contain the concept predicates that we are learning to discriminate among and is a set of integrity constraints. Hence the abducible predicates, which usually capture the incompleteness of the problem, are the concepts that we are trying to learn and an abductive theory learns these concepts when it is able to explain correctly (typical) sets of features in terms of their corresponding concepts. A formula is abductively explained or entailed by an abductive theory iff there exists such that with , provided that does not violate any of the constraints in . Coverage of a training example for the concept is defined accordingly by saying that a theory covers " " if the hypothesis is an abductive explanation of the given features of " ". Let us illustrate these ideas by means of an example. Consider the simple situation where we want to learn (to discriminate) the concepts " " and " " from their features. Our intention is to construct a theory like 1 , where : and the set of abducibles is . Existing learning approaches could construct a theory 2 composed of the rules A new object "Tweety"described by the set of observations(features) will be classified as a bird by 1 since abductively explains . On the other hand the object "Eagle" with the features will be classified either as a bird or as a plane. A main characteristic of abductive theories like 1 is that we can reason under incomplete information. In the previous example even though we have no information about attribute we can still classify Tweety as a bird. Similarly, for the example of Eagle but here this results in a disjunctive classification. Note also that classification within the abductive theories framework is a goal-directed reasoning task. In the above example the classifi- cation of as a is obtained by the goal-directed process of explaining its given features. Deductive theories like 2 that de- fine the concepts by rules, in general require forward reasoning with all of these rules to see which one is satisfied in order to arrive at a classification. The rest of the paper is organized as follows. In section 2 we present the languageand basic notions of abductive theories neededto formulate classification problems. Then in section 3 we formalize the c 1996 Yannis Dimopoulos and Antonis Kakas ECAI 96. 12th European Conferenceon Artificial Intelligence Edited by W. Wahlster Published in 1996 by John Wiley & Sons, Ltd.