Randomization in Learning Theory

We consider inductive inference learning of recursive computable functions Here the power of probabilistic algorithms increases as the probability of success decrease The pattern of this increase depends on the exact requirements about learning algorithm Typically the interval of success probabilities can be split into subintervals such that the power of algorithms with the success probabilities in the same subinterval is the same The structure of these intervals called probability hierarchy is simple in some cases and very complicated in some other cases Finite identi cation is one of simplest criteria of success for inductive inference It was the rst criteria for which probabilistic identi cation was studied However it also generates one of most complicated probability hierarchy and a large part of this structure remains unknown after years of study since The structure is simpler when larger probabilities of success are required However when the probabilities of success decrease it becomes more complex and even small advances require large e orts We take an approach di erent from the previous work Instead of trying to describe the probability hierarchy by explicit formulas we study the properties of the whole hierarchy We show several interesting results for Popperian nite identi cation a special case of nite identi cation Our main result is a decision algorithm for the probability hierarchy This algorithm takes two probabilities as the input and answers whether the learning with these two probabilities has equal power We also determine the ordering type of the probability hierarchy It is the ordering type of all expressions possible in the rst order arithmetic This shows how complex the probability hierarchy is and explains why it is not feasible to describe all cutpoints in the hierarchy explicitly