Probabilistic language learning under monotonicity constraints

Abstract

The present paper deals with probabilistic identification of indexed families of uniformly recursive languages from positive data under various monotonicity constraints. Thereby, we consider strong-monotonic, monotonic and weak-monotonic probabilistic learning of indexed families with respect to class comprising, class preserving and exact hypothesis spaces and investigate the probabilistic hierarchies of these learning models. Earlier results in the field of probabilistic identification established that — considering function identification — each collection of recursive functions identifiable with probability p}>1/2 is deterministically identifiable (cf. [16]). In the case of language learning from text, each collection of recursive languages identifiable from text with probability p}>2/3 is deterministically identifiable (cf. [14]), but when dealing with the learning models mentioned above, we obtain probabilistic hierarchies highly structured without a “gap” between the probabilistic and deterministic learning classes. In the case of exact probabilistic learning, we are able to show the probabilistic hierarchy to be dense for every mentioned monotonicity condition. Considering class preserving weak-monotonic and monotonic probabilistic learning, we show the respective probabilistic hierarchies to be strictly decreasing for probability p→1, p}<1. These results extend previous work considerably (cf. [16], [17]). For class comprising weak-monotonic learning as well as for learning without additional constraints, we can prove that probabilistic identification and team identification are equivalent. This yields discrete probabilistic hierarchies in these cases.

In the previous work we gave a complete picture of exact probabilistic learning under monotonicity constraints and some results on class preserving and class comprising probabilistic learning. Further work will establish the probabilistic hierarchies for the cases not considered in this paper.

The author wishes to thank Thomas Zeugmann for suggesting the topic and helpful discussion. Furthermore, I wish to thank Britta Schinzel and Arun Sharma for helpful comments.