The power of probabilism in Popperian FINite learning

Abstract

Abstract We consider the capabilities of probabilistic and team learners who (a) are allowed to conjecture only one hypothesis when given a function to learn; and (b) must always produce a program (i.e. hypotheses) that halts on every input. Learners satisfying only (a) are termed FIN-type learners, and learners satisfying both (a) and (b) are termed PFIN-type learners. We show that the structure of the learning capability of probabilistic and team learning with success ratio above 1/2 in PFIN-type learning is analogous to the structure observed in FIN-type learning. On the contrary, the structure of probabilistic and team learning with success ratio below 1/2 is more sparse for PFIN-type learning than for FIN-type learning. For n ≥ 2, we show that the probabilistic hierarchy below 1/2 for PFIN-type learning is denned by the sequence 4n/(9n − 2), which has an accumulation point at 4/9. We also show that the redundancy type at the accumulation point 4/9 is different from the one observed at 1/2. More inter...