Subrecursive incomparability of the graphs of standard and dual Baire sequences

Abstract

Our main question of interest is the existence or the non-existence of a subrecursive reduction between different representations of the irrational numbers. For any representation, considered as a total function, we consider the characteristic function of its graph. The graph is computably equivalent to the function itself, but not subrecursively equivalent. In some cases, the graph of a representation is subrecursively equivalent to an already known representation, but in other cases it is a new representation. In the present paper we undertake a systematic study of the graphs of standard and dual Baire sequences. By combining our new results with the previously known results on the graph of the continued fraction, we obtain a total of eight new subrecursive degrees, which lie strictly between the Dedekind cut and the continued fraction.