Analog computers and recursive functions over the reals

@article{Graa2003AnalogCA,
  title={Analog computers and recursive functions over the reals},
  author={Daniel Silva Graça and Jos{\'e} F{\'e}lix Costa},
  journal={J. Complex.},
  year={2003},
  volume={19},
  pages={644-664}
}

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This work starts by analyzing the first known model of this kind, the General Purpose Analog Computer, and proposes an alternative approach that originates a more robust model than the GPAC, maintaining its principal properties such as the equivalence with differentially algebraic functions.
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This work presents a mathematical definition of an analog generable function of a real variable in terms of a simultaneous set of nonlinear differential equations possessing a "domain of generation," which includes functions generated by existing general-purpose analog computers.
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It is shown that G, the set G of GPAC-computable functions is not closed under iteration, but a simple extension of it is, which includes all primitive recursive functions and has the additional closure property that if T(x) is in G+?k, then any function ofx computable by a Turing machine in T( x) time is also.
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Shannon's General Purpose Analog Computer is considered, which is a model of computation by differential equations in continuous time, and it is shown that several classical computation classes have natural analog counterparts.
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