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**1 - 4**of**4**### Learning Recursive Concepts with Anomalies

, 2000

"... This paper provides a systematic study of inductive inference of indexable concept classes in learning scenarios in which the learner is successful if its final hypothesis describes a finite variant of the target concept – henceforth called learning with anomalies. As usual, we distinguish between l ..."

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This paper provides a systematic study of inductive inference of indexable concept classes in learning scenarios in which the learner is successful if its final hypothesis describes a finite variant of the target concept – henceforth called learning with anomalies. As usual, we distinguish between learning from only positive data and learning from positive and negative data. We investigate the following learning models: finite identification, conservative inference, set-driven learning, and behaviorally correct learning. In general, we focus our attention on the case that the number of allowed anomalies is finite but not a priori bounded. However, we also present a few sample results that affect the special case of learning with an a priori bounded number of anomalies. We provide characterizations of the corresponding models of learning with anomalies in terms of finite tell-tale sets. The varieties in the degree of recursiveness of the relevant tell-tale sets observed are already sufficient to quantify the differences in the corresponding models of learning with anomalies. In addition, we study variants of incremental learning and derive a complete picture concerning the relation of all models of learning with and without anomalies mentioned above.

### Inductive Inference of Approximations for Recursive Concepts

, 2005

"... This paper provides a systematic study of inductive inference of indexable concept classes in learning scenarios where the learner is successful if its final hypothesis describes a finite variant of the target concept, i.e., learning with anomalies. Learning from positive data only and from both pos ..."

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This paper provides a systematic study of inductive inference of indexable concept classes in learning scenarios where the learner is successful if its final hypothesis describes a finite variant of the target concept, i.e., learning with anomalies. Learning from positive data only and from both positive and negative data is distinguished. The following learning models are studied: learning in the limit, finite identification, set-driven learning, conservative inference, and behaviorally correct learning. The attention is focused on the case that the number of allowed anomalies is finite but not a priori bounded. However, results for the special case of learning with an a priori bounded number of anomalies are presented, too. Characterizations of the learning models with anomalies in terms of finite tell-tale sets are provided. The observed varieties in the degree of recursiveness of the relevant tell-tale sets are already sufficient to quantify the differences in the corresponding learning models with anomalies. Finally, a complete picture concerning the relations of all models of learning with and without anomalies mentioned above is derived.

### Learning Approximations of Recursive Concepts

, 2001

"... This paper provides a systematic study of inductive inference of indexable concept classes in learning scenarios where the learner is successful if its final hypothesis describes a finite variant of the target concept, i.e., learning with anomalies. Learning from positive data only and from both ..."

Abstract
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This paper provides a systematic study of inductive inference of indexable concept classes in learning scenarios where the learner is successful if its final hypothesis describes a finite variant of the target concept, i.e., learning with anomalies. Learning from positive data only and from both positive and negative data is distinguished. The following learning models are studied: learning in the limit, finite identification, set-driven learning, conservative inference, and behaviorally correct learning. The attention is focused on the case that the number of allowed anomalies is finite but not a priori bounded. However, results for the special case of learning with an a priori bounded number of anomalies are presented, too. Characterizations of the learning models with anomalies in terms of finite telltale sets are provided. The observed varieties in the degree of recursiveness of the relevant tell-tale sets are already sufficient to quantify the differences in the corr...

### Probabilistic Learning of Indexed Families under Monotonicity Constraints -- Hierarchy Results and Complexity Aspects

"... We are concerned with probabilistic identification of indexed families of uniformly recursive languages from positive data under monotonicity constraints. Thereby, we consider con-servative, strong-monotonic and monotonic probabilistic learning of indexed families with respect to class comprising, c ..."

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We are concerned with probabilistic identification of indexed families of uniformly recursive languages from positive data under monotonicity constraints. Thereby, we consider con-servative, strong-monotonic and monotonic probabilistic learning of indexed families with respect to class comprising, class preserving and proper hypothesis spaces, and investigate the probabilistic hierarchies in these learning models. In the setting of learning indexed families, probabilistic learning under monotonicity constraints is more powerful than deterministic learning under monotonicity constraints, even if the probability is close to 1, provided the learning machines are restricted to proper or class preserving hypothesis spaces. In the class comprising case, each of the investigated probabilistic hierarchies has a threshold. In particular, we can show for class comprising conservative learning as well as for learning without additional constraints that probabilistic identification and team identification are equivalent. This yields discrete probabilistic hierarchies in these cases. In the second part of our work, we investigate the relation between probabilistic learn-