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 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. 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 learning and oracle identification under monotonicity constraints. We deal with the question how much additional information provided by oracles is sufficient and necessary for compensating the additional power of probabilistic learning machines. In this context, we introduce a complexity measure in order to measure the additional power of the probabilistic machines in qualitative terms. One main result is that for each oracle A ≤T K, there exists an indexed family LA which is properly conservatively identifiable with p = 1/2, and which exactly reflects the Turing degree of A, i.e., LA is properly conservatively identifiable by an oracle machine M [B] iff A ≤T B. However, not every indexed family which is conservatively identifiable with probability p = 1/2 reflects the Turing degree of an oracle. Hence, the conservative probabilistic learning classes are higher structured than the Turing degrees below K. Finally, we prove that there exist learning problems which are conservatively (monotonically) identifiable with probability p = 1/2 (p = 2/3), but conservatively (monotonically) identifiable only by oracle machines having access to T OT . Dekan: Prof. Dr. Gerald Urban Vorsitzender der Prüfungskommission: Prof. Dr. Thomas Ottmann Erstreferent: Prof. Dr. Britta Schinzel Zweitreferent: Prof. Dr. Thomas Zeugmann (Uni Lübeck) Datum der Disputation: 31.Mai 2001