Polynomial Time Computation in the Context of Recursive Analysis

Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955] as an approach for investigating computation over the real numbers. It is based on enhancing the Turing machine model by introducing oracles that allow the machine to access finitary portions of the real infinite objects. Classes of computable real functions have been extensively studied as well as complexity-theoretic classes of functions restricted to compact domains. However, much less have been done regarding complexity of arbitrary real functions. In this article we give a definition of polynomial time computability of arbitrary real functions. Then we present two main applications based on that definition. The first one, which has already been published, concerns the relationships between polynomial time real computability and the corresponding notion over continuous rational functions. The second application, which is a new contribution to this article, concerns the construction of a function algebra that captures polynomial time real computability.