Abstract
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 et al. (J Complex 18:977–1000, 2002), which precisely relates the Kalmar elementary computable functions to a function algebra over the reals. Second, we build on that result to extend a result of Bournez and Hainry (Theor Comput Sci 348(2–3):130–147, 2005), which provided a function algebra for the \({\mathcal{C}}^2\) real elementary computable functions; our result does not require the restriction to \({\mathcal{C}}^2\) functions. In addition to the extension, we provide an alternative approach to the proof. Their proof involves simulating the operation of a Turing Machine using a function algebra. We avoid this simulation, using a technique we call lifting, which allows us to lift the classic result regarding the elementary computable functions to a result on the reals. The two new techniques bring a different perspective to these problems, and furthermore appear more easily applicable to other problems of this sort.
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Bournez O., Campagnolo M.L., Graça D.S. and Hainry E. (2007). Polynomial differential equations compute all real computable functions on computable compact intervals. J. Complex. 23(3): 317–335
Bournez O. and Hainry E. (2005). Elementarily computable functions over the real numbers and \({\mathbb{R}}\) -sub-recursive functionsTheor. Comput. Sci. 348(2–3): 130–147
Bournez O. and Hainry E. (2006). Recursive analysis characterized as a class of real recursive functions. Fundamenta Informaticae 74(4): 409–433
Campagnolo M.L., Moore C. and Costa J.F. (2002). An analog characterization of the Grzegorczyk hierarchy. J. Complex. 18: 977–1000
Campagnolo, M.L., Ojakian, K.: The methods of approximation and lifting in real computation. In: Cenzer, D., Dillhage, R., Grubba, T., Weihrauch, K. (eds.) Third International Conference on Computability and Complexity in Analysis. Electronic Notes in Theoretical Computer Science, vol. 167, pp 387–423 (2007)
Graça D.S. (2004). Some recent developments on Shannon’s general purpose analog computer. Math. Logic Q. 50(4–5): 473–485
Grzegorczyk A. (1955). Computable functionals. Fund. Math. 42: 168–202
Ko K.-I. (1991). Complexity Theory of Real Functions. Birkhaüser, Basel
Moore C. (1996). Recursion theory on the reals and continuous-time computation. Theor. Comput. Sci. 162: 23–44
Mycka J. (2003). μ-Recursion and infinite limits. Theor. Comput. Sci. 302: 123–133
Mycka J. and Costa J.F. (2004). Real recursive functions and their hierarchy. J. Complex. 20(6): 835–857
Weihrauch K. (2000). Computable Analysis: An Introduction. Springer, Heidelberg
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Campagnolo, M.L., Ojakian, K. The elementary computable functions over the real numbers: applying two new techniques. Arch. Math. Logic 46, 593–627 (2008). https://doi.org/10.1007/s00153-007-0059-x
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DOI: https://doi.org/10.1007/s00153-007-0059-x