Reducibilities on real numbers

Abstract

The concept of reducibility in recursive function theory and computational complexity theory is applied to real numbers to investigate the notion of relative computability and relative complexity of real numbers. Several common types of reducibility such as Turing, truth-table and many-one reducibilities are considered. We also consider reducibilities defined by various sub-classes of recursive real functions. Some equivalence results among these reducibilities are obtained: The reducibility defined by recursive real functions is equivalent to the generalized truth-table redicibility; and the reducibility defined by recursive increasing real functions is equivalent to the generalized many-one reducibility. Similar equivalence results on polynomial time reducibilities are also proved. Different reducibilities are distinguished.