Probabilistic limit identification up to “small” sets
In this paper we study limit identification of total recursive functions in the case when “small” sets of errors are allowed. Here the notion of “small” sets we formalize in a very general way, i.e. we define a notion of measure for subsets of natural numbers, and we consider as being small those sets, which are subsets of sets with zero measure.
We study relations between classes of functions identifiable up to “small” sets for different choices of measure. In particular, we focus our attention on properties of probabilistic limit identification. We show that regardless of particular measure we always can identify a strictly larger class of functions with probability 1/(n+1) than with probability 1/n. Besides that, for computable measures we show that, if there do not exist sets with an arbitrary small non zero measure, then identifiability of a set of functions with probability larger than 1/(n+1) implies also identifiability of the same set with probability 1/n. Otherwise (in the case when there exist sets with an arbitrary small non zero measure), we always can identify a strictly larger class of functions with probability (n+1)/(2n+1) than with probability n/(2n-1), and identifiability with probability larger than (n+1)/(2n+1) implies also identifiability with probability n/(2n-1).
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