Polynomial differential equations compute all real computable functions on computable compact intervals |
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Authors: | Olivier Bournez Manuel L Campagnolo Daniel S Graça Emmanuel Hainry |
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Institution: | 1. Inria Lorraine, France;2. LORIA (UMR 7503 CNRS-INPL-INRIA-Nancy2-UHP), Campus Scientifique, BP 239, 54506 Vandœuvre-Lès-Nancy, France;3. DM/ISA, Technical University of Lisbon, 1349-017 Lisboa, Portugal;4. SQIG/IT, Technical University of Lisbon, 1049-001 Lisboa, Portugal;5. DM/FCT, Universidade do Algarve, C. Gambelas, 8005-139 Faro, Portugal;6. Institut National Polytechnique de Lorraine, France |
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Abstract: | In the last decade, there have been several attempts to understand the relations between the many models of analog computation. Unfortunately, most models are not equivalent. Euler's Gamma function, which is computable according to computable analysis, but that cannot be generated by Shannon's General Purpose Analog Computer (GPAC), has often been used to argue that the GPAC is less powerful than digital computation. However, when computability with GPACs is not restricted to real-time generation of functions, it has been shown recently that Gamma becomes computable by a GPAC. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions over compact intervals can be defined by such models. |
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Keywords: | Analog computation Computable analysis General Purpose Analog Computer Church&ndash Turing thesis Differential equations |
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