Author: Léa Meyer.
Source: Theoretical Computer Science Vol. 185, No. 1, 1997, 81-128.
Abstract. The present paper deals with probabilistic identification of indexed families of uniformly recursive languages from positive data under monotonicity constraints. Thereby, we consider conservative, 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.
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. ). 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. ), 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 proper conservative probabilistic learning and proper strong-monotonic probabilistic learning, respectively, we are able to show the probabilistic hierarchies to be dense in [0, 1]. For proper monotonic probabilistic learning, we show that the probabilistic hierarchy is dense in (2/3,1]. Considering class preserving conservative or monotonic probabilistic learning, we show the corresponding probabilistic hierarchies to be strictly decreasing with increasing probability. These results extend the previous work considerably (cf. [50, 51]). For class preserving strong-monotonic probabilistic learning, we show that every indexed family of uniformly recursive languages strong-monotonically identifiable with a probability p > 2/3 with respect to a class preserving hypothesis space is deterministically strong-monotonically identifiable with respect to a class preserving hypothesis space, i.e., the corresponding probabilistic hierarchy has a "gap" beginning with p = 2/3. In the class comprising case, each of the investigated probabilistic hierarchies has a "gap". 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.
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