Algebraic Characterization of Complexity Classes of Computable Real Functions

Several algebraic machine-independant characterizations of computable functions over the reals have been obtained recently. In particular nice connections between the class of computable functions (and some of its sub and sup-classes) over the reals and algebraically defined (suband sup-) classes of R-recursive functions à la Moore 96 have been obtained. We provide in this paper a framework that allows to relate these classes to classical computability and complexity classes over the integers. While our setting provides a new reading of some of the existing characterizations, this also provides new results: in particular, we provide an algebraic characterization of polynomial time computable real functions.