An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Abstract We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers.

• Type: Conference
• Year: 2004
• Where: ICALP
• Authors: Olivier Bournez, Emmanuel Hainry
• Keywords: Computable Functions, ICALP 2004, Real Numbers, Recursive Analysis, Theoretical Computer Science