The elementary computable functions over the real numbers: applying two new techniques

The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). We provide general tools for investigating these issues, using two techniques we call approximation and lifting. We use these methods to obtain two main theorems. First we provide an alternative proof of the result from Campagnolo, Moore and Costa [3], which precisely relates the Kalmar elementary computable functions to a function algebra over the reals. Secondly, we build on that result to extend a result of Bournez and Hainry [1], which provided a function algebra for the C2 real elementary computable functions; our result does not require the restriction to C2 functions. In addition to the extension, we provide an alternative approach to the proof. Their proof involves simulating the o...

• Type: Journal
• Year: 2008
• Where: AML
• Authors: Manuel Lameiras Campagnolo, Kerry Ojakian
• Keywords: AML 2008, Elementary Computable Functions, Function Algebra, Turing Machine