On subrecursive representation of irrational numbers: Contractors and Baire sequences

Abstract

We study the computational complexity of three representations of irrational numbers: standard Baire sequences, dual Baire sequences and contractors. Our main results: Irrationals whose standard Baire sequences are of low computational complexity might have dual Baire sequences of arbitrarily high computational complexity, and vice versa, irrationals whose dual Baire sequences are of low complexity might have standard Baire sequences of arbitrarily high complexity. Further- more, for any subrecursive class S closed under primitive recursive oper- ations, the class of irrationals that have a contractor in S is exactly the class of irrationals that have both a standard and a dual Baire sequence in S. Our results implies that a subrecursive class closed under primitive recursive operations contains the continued fraction of an irrational number α if and only if there is a contractor for α in the class.